Download dialogues between abelard and eloise

DIALOGUES BETWEEN ABELARD AND ELOISE
First-Order Modal Logic
and Independence Friendly Logic:
A Dialogical and Game Theoretical Introduction
Shahid Rahman and Tero Tuleneheimo
Draft (12 January 2007)
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Table of contents
Preface
Introduction
Preface
I
Introduction
I.1 Philosophical remarks:
I.1.1 The case against modal logic
I.1.2 What is a possible world
I.2 A brief historical outline
II Games and Dialogues for Classical and Intuitionisitic Logic
An Overview
II.1 GTS-truth and the dialogical proof
II.1.0 Preliminaries
II.1.1 The languages considered: Classical and Intuitionistic First-order logic
II.1.2 GTS and truth in a model
II.1.3 Dialogues and validity
II.1.4 From games to dialogues and back
II.2 Dialogues, tableaux and sequent-calculus
II.2.1 Dialogues and Tableaux
II.2.2 Dialogues and sequent-calculus
II.3 History
III IF Logic
IV Modal Propositional logic: Basic Semantic Notions
V Validity on Frames
V.1 The model theoretical approach
V.2 The game theoretical approach
V.3 The dialogical approach
V.4 Propositional modal dialogic and sequent-calculus
V.6 Soundness and completeness of propositional modal dialogic
VI
The Name of the World: Dialogues and Hybrid Languages
VII
Non-Normal Dialogics for a Wonderful World
VII.1 Dialogics and non-normal logics
VII.2 Hybrid languages and non-normal dialogics
VII.3 Subnormal logics:
VII.3.1 The logic N
VII.3.2 Conditional logics
VIII Bisimulation and IF-Propositional Modal Logic
VIII.1 Bisimulation and modal logic
VIII.2 IF-propositional modal logic
IX First Order Modal Logic
IX.1 Varying and constant domains
IX.2 Dialogues and games for varying and constant domains
IX.3 Equality, existence and Barcan again
IX.3.1 Equality and Frege-Kanger’s challenge
IX.3.2 Fitting and Mendelsohn on necessary equalities
IX.3.3 Equality and dialogues
IX.3.4 Existence, Equality and dialogues
IX.4 From arbitrary objects to arbitrary constants
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IX.5 The syntax and semantics of predicate abstraction
IX.5.1 Syntax
IX.5.2 Semantics
IX.5.3 Fitting and Mendelsohn on equality, abstraction and rigidity
IX.5.4 To know and not to know and predicate abstracts
IX.6 Reasoning about nothing
IX.7 Modal scope and the subjunctive marker
IX.8 Dialogues and sequent calculi for first order modal logic
X. IF and First-Order Modal logic
X.1 IF and first-order modal logic
X.2 IF and generalizing the notion of scope in modal logic
XI Higher Order Modal Logic
APPENDIX:
A Functions
B Axiomatics for Modal Logic
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I Introduction
The propositional modal language is an extension of the pure propositional classical
language by adding new 1-ary connectives (known as the necessity and the possibility
connectives or more abstract as box ( ) and diamond (◊) connectives).
In basic modal logic box and diamond are interdefinable:
ϕ iff ¬◊¬ϕ
◊ϕ iff ¬ ¬ϕ
Unlike the propositional connectives of classical logic box and diamond do not have a
uniform and fixed interpretation. In fact different readings of these connectives suggest
different semantics and proof systems. E.g.:
:
ϕ is known
ϕ is necessary
ϕ will be always true
ϕ was always true
ϕ is obligatory
ϕ is provable
◊:
¬ϕ: ϕ is not known
ϕ is possible
ϕ will be someday true
ϕ was someday true
ϕ is permissible
ϕ is consistent (with some formal system
of arithmetic)
In these readings, it should be clear, at least from the first four pairs of the list, that they are
intended as kinds of universal quantifiers over information states (or scenarios), possible
worlds and over time contexts. Modal languages have been also used to analyse behaviour of
computer programs and the state transitions of (finite) automata. Furthermore many modal
languages such as temporal languages combine the different readings of the modal
connectives.
The first developments and applications of modal logic where philosophical and connected to
various philosophical notions of necessity (sometimes identified with the temporal reading of
the box). The first attempts to formalise modal logic by the end of the 19th century by the
French logician of Scottish origin Hugh MacColl (1837-1909) were overtaken and
axiomatised by Clarence Irving Lewis by 1938. As early as 1946 Carnap explored the idea of
analysing modality as quantification over possible worlds, but he did not have the relation of
accessibility which defines possible world semantics. The nowadays possible world semantics
of modal logic was born by the confluence of the model theoretical approach to formal
semantics of the polish tradition with the axiomatics of Lewis and followers. Actually there
was another link less visible but also very important, namely link between algebraic logic and
the model-theoretical approach to the semantics of modal logic. This link is a result of
Stephen Kanger achieved in his seminars at the university of Stockholm by 1955 and
published 1957 under by that university the title Provability in Logic; and of Richard
Montague. Indeed, in a lecture of 1955 at the UCLA Montague gave a full model theoretical
interpretation of propositional modal logic.. Kanger referred in a footnote (1957c, 39) to the
work of Jónnson and Tarski (1951), from whose his use of the relational apparatus seems to
have derived. As Copeland writes (2006, 392) “with hindsight, these theorems can be viewed
as in effect a treatment of all the basic modal axioms and corresponding properties of the
accessibility relation”. Now in the work of Kanger and in the work of Montague the notion of
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relation used was that of a relation between models and not between possible worlds.The
standard approach to the semantics of basic modal logic of nowadays was developed
independently by various logicians between 1955 and 1959, notably by the work of Carew
Meredith, Arthur Prior, Jaakko Hintikka and Saul Kripke. Jack Copeland (2006) gives the
priority to the joint work of Meredith and Copeland in 1956. In fact the work of Jaakko
Hintikka and Saul Kripke was the better known version of the possible-world semantics.
While the former delved in the epistemic reading of the box (to know) the second studied an
ontological interpretation of the (Leibnizian) notion of necessity. Furthermore, in his early
work Hintikka called the relation an alternativeness relation between possible states of affairs.
In the context of deontic logic Hintikka (1957) calls the relation a copermissibility relation.
Richard Montague initiated around 1975 a systematic application of modal languages for the
formalization of natural language. Prior (1962) after a suggestion of Peter Geach (1960)
called the relation “accessibility”, which is now the standard name for the binary relation
between possible worlds.1 Hans Kamp extended the so-called Montague grammars to
Discourse Representation Theory (DRT) which is nowadays the most influential paradigm
to formalize natural language with application in various fields as, computer linguistics,
artificial intelligence and philosophy. Through the work of Johan van Benthem; modal logic
is understood as a formal language for the study of structures. This is being been combined on
one hand with the approach of the DRT and on the other hand with the game theoretical
semantics, which is the main theoretical framework of the present book.
(Wittgenstein: semantics: GTS, proof theory: dialogues)
I.1 Philosophical remarks:
I.1.1 The case against modal logic
I.1.2 What is a possible world
I.2 A brief historical outline and further reading
Carnap, R. “Modalities and Quantification”. Journal of Symbolic Logic, vol. 11, 1946, 33-64.
Copeland, J. B. “Meredith, Prior and the History of Possible World Semantics”. Synthese, vol.
150/3, June 2006, 373-397.
Kanger, S. “The Morning Star Paradox”. Theoria, vol. 23, 1957a, 1-11.
Kanger, S. Provability in Logic. Almqvist and Wiksell: Stockholm, 1957b.
1
cf. Copeland 2006, 384.
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II Dialogues and Games for Classical and Intuitionisitic Logic
An Overview
The fact that game-theoretical semantics (GTS) and dialogic are sisters has been widely
acknowledged. The differences between the original approaches have been discussed too:
while GTS relates to the study of truth in a model, dialogic has explored the possibilities of a
certain type of proof-theoretical approach to validity. Despite the close relationship between
the two approaches, no detailed, thorough analysis of their interaction has yet been
undertaken. The insightful article of Saarinen (1978) is, however, a notable early attempt at a
comparison of the two approaches. The aim of this chapter is to introduce to GTS and
dialogues. We will also discuss some results obtained recently by the authors of this book
(2005) in relation to a systematic comparison of dialogical logic and GTS, from the viewpoint
of analyzing the notion of validity.
II.1 GTS-truth and the dialogical proof
II.1.0 Preliminaries
Truth and validity - or material truth and logical truth, respectively - are the two most
important semantic properties that logics deal with. Semantically, logics are used for making
assertions about models, and a part of the specification of a semantics for a logic is telling
under which conditions a formula of such-and-such logic is true, relative to a given model.
Logical truth then means truth with respect to all models relative to which the semantics is
defined. Truth is sometimes qualified as material truth, to convey the idea that this notion of
truth is relative to a contingent context. For some logics the notions of truth and validity admit
of generalization. Hence in First-order logic, truth (in a model) is a special case of satisfaction
(in a model and under a variable assignment); and logical truth is an instance of satisfaction in
every model and under all variable assignments.
A logic for which a semantics is specified in some way, typically admits of conceptually
different ways of capturing the notions of truth and validity appropriate to that logic. For
instance, the most common way of defining the semantics of First-order logic is by defining
satisfaction conditions of its formulae relative to a model and a variable assignment, by
recursion on the structure of a formula. This was Tarski's original approach in defining the
semantics of First-order logic (Tarski 1933, Tarski & Vaught 1956). An alternative to the
Tarskian way of specifying the semantics would be game-theoretical semantics (Hintikka
1973, Hintikka & Sandu 1997), which captures the very same satisfaction conditions in terms
of the existence of a winning strategy for a certain player in a semantical game, associated
with a formula, a model and a variable assignment.
Alternative ways of specifying a semantics are said to characterize the notions defined by the
specification of the logic. In this sense, GTS serves to characterize the semantics of Firstorder logic that is defined by the Tarskian semantics. Similarly, dialogues associated with
first-order sentences serve to characterize validity in First-order logic, i.e. the same property
that under the Tarskian approach is captured by the condition `true in every model', or, prooftheoretical ly, as derivability in a complete and sound proof system from the empty set of
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premises.2 This means that the same logic would have been obtained, had any of the
characterizations been used in place of the original definition of truth or validity for the logic
in question.
Yet the different ways of capturing the same notions may make it possible to pose questions
that would otherwise not have appeared. For example, various questions whose original
motivation derives from game theory, arise in connection with logics whose semantics is
defined game-theoretical ly. Cases in point are issues of determinacy (whether always one of
the players has a winning strategy), imperfect information (whether a player is always fully
informed of a past course of a play), and strategic action (what are the different ways in which
a true sentence may be verified), which all have turned out to function as grounds for
interesting generalizations of first-order and modal logics - as witnessed by Hintikka's socalled IF, or `independence-friendly', logic (see e.g. Hintikka & Sandu 1989, Hintikka 1996,
2002), and the research pursued within the framework of the Games and Logic paradigm of
van Benthem and other Dutch logicians (see e.g. van Benthem 2001a,b, 2006). The usefulness
of the dialogical approach in connection with linear logic is another example of the
fruitfulness of the game-theoretical approach for novel perspectives in logic (Blass 1992).
Thus alternative formulations of semantics may enable asking new questions; what is more,
the requisite new conceptual tools may actually make it possible to study logics that could not
even be formulated in terms of the traditional tools - or whose formulation using the received
tools would in any event be clumsier. Examples are logics with Henkin quantifiers (Henkin
1961, Krynicki \& Mostowski 1995), infinitely deep languages (Hintikka & Rantala 1976,
Karttunen 1984, Hyttinen 1990) and Vaught formulae (Vaught 1973, Makkai 1977), which all
extend First-order logic. They all are very naturally defined using games.
In this chapter we will be concerned with one family of alternative ways of defining
semantically important notions in logics: game-theoretical methods. More specifically, we
aim at building a bridge between two research traditions - those that go under the names
game-theoretical semantics and dialogical logic. The former is concerned with characterizing
truth in a model for various logics, while in the latter case the notion of validity is being
characterized, likewise for various logics. Our concern is systematic. Both traditions make use
of game theory to characterize semantic notions involving the notion of truth. Yet it is far
from evident how to turn the results of one tradition to serve the other.
While GTS has been from the beginning clearly model-theoretical ly oriented, the dialogical
approach has a strong connection, both historically and philosophically, to proof theory. For
logics admitting a complete proof system (such as Propositional logic and First-order logic),
the gap between model theory and proof theory is of course bridgeable, but even so the two
backgrounds lead easily to different types of development, so much so that even the
corresponding ultimate understanding of what semantics is may be affected: there are
constructivistically oriented philosophers who would consider proof theoretical inference
rules as meaning constitutive and who would speak of proof-conditional semantics for logical
operators (see Ranta 1988, Sundholm 2002), whereas from the viewpoint of classical model
theory there is no proof theoretical component at all to the semantics of logical operators. A
sense of an existing common ground between GTS and the dialogical approach is still
2
A system of inference rules and/or axioms is complete, if any logically true sentence is derivable in that
system; and sound, if any derivable sentence is logically true. Such a system is said to be strongly complete, if
from the fact that a sentence A is a logical consequence of the sentences in Σ it follows that A is derivable from
the set Σ of premises. (As is well known, such a proof system exists for First-order logic.). We will come to this
notions with details later on.
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unmistakable. So what to make of it? We attempt to move towards a systematic comparison
of these traditions, which both were originally motivated by philosophical considerations, and
which both use formal tools from the same branch of mathematics: game theory.3
Specifically, we establish for classical Propositional logic and classical First-order logic an
exact connection between `intuitionistic dialogues with hypotheses' and semantical games.
Basically, we show how the existence of a winning strategy for one of the players (called
Proponent) in a dialogue D(A;H1,…,Hn) corresponding to a sentence A with a finite number
of hypotheses Hi of a certain type, gives rise to a family of Eloise's winning strategies in
semantical games G(A,M), one strategy for each model M; and, conversely, how to construct
a winning strategy for Proponent in the dialogue D(A;H1,…,Hn) out of Eloise's winning
strategies in games G(A,M). The proofs are constructive in the sense that we explicitly show,
by providing a suitable algorithm, how a strategy for one type of game is built using a strategy
for the other type of game. In fact, these algorithms are the real content of our results - it is
well known that abstractly, validity in one sense (dialogic) coincides with validity in the other
sense (GTS), simply because they both characterize the notion `true in all models'.
II.1.1 The languages considered: Classical and Intuitionistic firstorder logic
Propositional Logic
Given a countable set prop of propositional atoms (denoted p,q,…), we consider
Propositional logic (PL) with the connectives conjunction (∧), disjunction (∨) and negation
(¬). (By ‘countable’ we mean ‘finite or of size ℵ0’.) Semantics of PL is defined as usual
relative to models
M : prop x {true, false}
Such a model M partitions the propositional atoms into two classes: those that are true in the
model, and those that are false. We assume that the reader is familiar with the semantics of
classical propositional logic.
In our official syntax, the conditional sign (→) does not appear4 For classical Propositional
logic this is no restriction, as there conditional can be defined from disjunction and negation:
A→B := ¬A∨B
For intuitionistic logic this is a genuine restriction, however, since intuitionistically
conditional is not definable from the other connectives. In particular, as we will follow from
the paragraph on intuitionistic dialogic from
¬A∨B
it follows intuitionistically that
A→B,
3
As a mathematical discipline, game theory can be seen to have been created by J. von Neumann, in his 1928
paper “Zur Theorie der Gesellschaftsspiele“. The single most inluential book on game theory is von Neumann
and O. Morgenstern's Theory of Games and Economic Behavior (1944). E. Zermelo (1913), E. Borel (1921,
1924) and H. Steinhaus (1925) can be considered as precursors of game theory.
4
The conditional sign, “→“ is not to be confused with the sign “x“ used to indicate the domain and range of a
function.
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but the other sense does not hold.
In the chapter on the transformation between GTS and dialogues, we consider languages
without conditional. Sometimes, however, we phrase definitions for the extended language
involving conditional, to give a fuller picture of the logical situation and we will assume
conditional for the dialogical formulations of classical and intuitionistic logic.
The notion of (proper) subformula of a formula is defined in the usual way:
Sub(p) = π;
Sub(A∨B) = Sub(A→B) = Sub(A∧B) = {A,B}4Sub(A)4Sub(B); and
Sub(¬A) = {A}4Sub(A).
We will say that a propositional formulaA is in negation normal form, if the negation sign (¬)
appears in A only prefixed to atomic subformulae. It is not difficult to verify that every
propositional formula has an equivalent in negation normal form:
FACT 1:
For every A of the propositional language there is formula B of the same language
such that B is in negation normal form, and A is logically equivalent to B.
First-order logic
Let τ be a finite vocabulary, i.e. a finite set consisting of constants k0, k1, … and relation
symbols R0, R1, …Each relation symbol is associated with a positive natural number, called its
arity.
Let a set of individual variables, Var = {k0, k1, …}, be fixed. Constants and variables are
jointly referred to as terms (ti). Atomic first-order formulae are strings of the form
Rit1 … tn
where Ri is n-ary, and each tj is a term.
The class of formulae of First-order logic of vocabulary τ, or FO[τ], is obtained by closing the
set of atomic formulae under conjunction, disjunction, conditional and negation, as well as
under universal , and existential quantification.
We use capital letters A, B, C, … from the beginning of the alphabet for arbitrary (atomic or
complex) formulae.
The notion of (proper) subformula is obtained by extending the definition of subformula of a
PL-formula by the clauses:
Sub(∀x1B) = Sub(∃x1B) = {B}4Sub(B).
The set Free[B] of free variables of a formula is defined recursively as usual:
•
Free[Rit1 … tn] = {t1 … tn} ∩ Var.
•
Free[¬B] = Free[B].
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•
Free[B∧C] = Free[B∨C] = Free[B→C] = Free[B]4Free[C].
•
Free[∀x1B] = Free[∃x1B] = Free[B] \ {xi}.
Formulae whose set of free variables is empty, are sentences. Sometimes we will write A(x1
… xn) to indicate that x1 … xn are among the free variables of A.
Semantics of FO[τ] is defined relative to τ-structures, i.e. structures M consisting of a nonempty domain M together with interpretations of the symbols appearing in the vocabulary τ:
interpretation of a constant in the structure is simply an element of the domain,
while the interpretation of relation-symbol is a relation on the domain, i.e. a subset of the
product Mn. The free variables will be related to the domain via assignments functions.
We assume that the reader is familiar with the semantics of First-order logic, though we will
come to it thoroughly while introducing first-order modal logic.
The negation normal form result of Fact 1 extends straighforwardly to FO[τ].
FACT 2:
For every A of FO[τ] there is formula B of the same language such that B is in
negation normal form, and A is logically equivalent to B (that is, satisfied in exactly
the same τ-structures by precisely the same variable assignements).
II.1.2 GTS and truth in a model
II.1.3 Dialogues and validity
Formal dialogues
Let us see what is at stake in dialogical logic by reconstructing in dialogical terms the
notion of validity in First-order logic.5 We first define a language L[τ]; this language will
basically be obtained from First-order logic (of vocabulary τ) by adding certain metalogical
symbols.
We introduce special force symbols ? and !.
An expression of L[τ] is either a formula of FO[τ], or one of the following strings:
L, R, ∨, ∀xi/kj or ∃xi/kj
where xi is any variable and kj any constant. We refer to the latter type of expressions as
attack markers.
5
This version is esentially from Rahman/Tulenheimo [2006]. For somewhat different accounts, see Rahman
\Keiff [2005], and Keiff [2006].)
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In addition to expressions and force symbols, for L[τ] we have available labels O and P,
standing for the players (Proponent, Opponent) of dialogues.
Every expression e of L[τ] can be augmented with labels P or O on the one hand, and by the
force symbols ? and ! on the other, so as to yield the strings
P-!-e, O-!-e, P-?-e, O-?-e
These strings are said to be (dialogically) signed expressions. Their role is to signify that in
the course of a dialogue, the move corresponding to the expression e is to be made by P or O,
respectively, and that the move is made as a defence (!) or an attack (?). We will use X and Y
as variables for P and O, always assuming X≠Y.
Particle rules
An argumentation form or particle rule is an abstract description of the way a formula,
according to its outmost form, can be criticized, and how to answer the critique. It is abstract
in the sense that this description can be carried out without reference to a specified context. In
dialogical logic, these rules are said to state the local semantics, for they show how the game
runs locally: what is at stake is only the critique and the answer corresponding to a given
logical constant, rather than the whole context where the logical constant is embedded.6 The
particle rules fix the dialogical semantics of the logical constants of L[τ] in the following way:
Assertion
X-!-A∨B
Attack
Y-?-∨
X-!-A∧B
Y-?-L
or
Y-?-R
(the challenger chooses)
Y-!-A
Y-!-A
X-!-A→B
X-!-¬A
Defence
X-!-A
or
X-!-B
(the defender chooses)
X-!-A
respectively
X-!-B
X-!-B
⊗
No defence possible. Only
a counterattack is available
X-!-∀xA
Y-?-∀x/k
For any k available to Y
X-!-A[x/k]
For any k chosen earlier by
Y
X-!-∃xA
Y-!-∃
X-!-A[x/k]
For any k available to Y
In the diagram, A[x/k] stands for the result of substituting the constant k for every occurrence
of the variable x in the formula A.
6
There can be no particle rule corresponding to atomic formulae. But it is possible to add a set of Opponent's
initial concessions to the particle rules. This is done in `material dialogues' (see Rahman/Tulenheimo [2006].
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A more thorough way to stress the sense in which the particle rules determine local semantics
is to see these rules as defining the notion of state of a (structurally not yet determined) game.
DEFINITION [State of a dialogue]:
Let A be a formula of FO[τ], and let a countable set {k0, k1, …} of individual constants be
fixed. A state of the dialogue D(A) corresponding to the formula A is a quintuple
<B, X, ϒ, e, σ> such that:
•
B is a (proper or improper) subformula of A.
•
X,-ϒ-e is a dialogically signed expression. Thus, X is either O or P and , ϒ ∈ {?, !},
and e ∈ L[τ].
•
σ : Free[B] x {k0, k1, …} is a function mapping the free variables of B to individual
constants.
The component e is either a formula of FO[τ], or an attack marker. We stipulate that in
the former case, always e:=B
Given a force ϒ,, let us write ϒ ’, for the opposite force, i.e. let
ϒ ’∈{?, !} \ ϒ.
Each state <B, X, ϒ, e, σ> has an associated role assignment, indicating which player
occupies the role of challenger and which the role of defender. In fact, the role assignment is
a function ρ : {P, O} x {?, !} such that ρ(X) = ϒ and ρ(Y) = ϒ’’.
We say that the state <B2, X2, ϒ2, e2, σ2> is reachable from state <B1, X1, ϒ1, e1, σ1> if it is a
result of X1 making a move in accordance with the appropriate particle rule in the role ϒ1. If
the role is that of challenger (ϒ1=?), the player states an attack, whereas if the role is that of
defender (ϒ1=!), the player poses a defence.
Let us take a closer look at the transitions from one state to another. Particle rules determine
which state S2 of a dialogue is reachable from a given other state S1. Notice that the player
who defends need not be the same at both states. In order for state S2 to be reachable from
state S1=<B, X, ϒ, e, σ>, it must satisfy the following.
•
Particle rule for negation: If B=e, ϒ= ! and B is of the form ¬C, then
S2=<C, Y, !, C, σ>
So if P is defender of ¬C at S1, then O is defender of C at S2, and P will challenge
(counterattack) C; and dually, if P is challenger of ¬C at S1.
Notice that here state S2 involves the claim that C can be defended; however, this claim has
been asserted in the course of an attack, and the whole move from S1 to S2 counts as an attack
on the initial negated formula, i.e. an attack on C. Actually this follows from the fact that at
S2, the roles of the players are inverted as compared with S1. Counterattack may yield from S2
a further state, S3= <C, X, ?, *, σ>, where C is the formula considered, and the attack pertains
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to the relevant logical constant of C, for which * is a suitable attack marker determined by the
logical form of C.
•
Particle rule for conjunction:} If B=e, ϒ= ! and B is of the form C∧D, then
S2=<C, X, !, C, σ> or S2=<D, X, !, D, σ>
according to the choice of the challenger between the attacks ?-L and ?-R. (Here the
challenger is Y: Y's role is ? here.)
•
Particle rule for disjunction: If B=e, ϒ= ! and B is of the form C∨D, then
S2=<C, X, !, C, σ> or S2=<D, X, !, D, σ>
according to the choice of the defender, reacting to the attack ?- ∨ of the challenger. (Here the
defender is X: X's role is ! here.)
•
Particle rule for conditional: If B=e, ϒ= ! and B is of the form C→D, then
S2=<C, Y, !, C, σ>
and, further, state
S3=<D, X, !, D, σ>
is reachable from S2. So if P is the defender of C→D at S1, and hence O is the defender of C
at S2, it is P who will be the defender of D at S3.
To attack a conditional amounts to being prepared to defend its antecedent, and so it should be
noticed that the defence of C at state S2 counts as an attack. If P is the defender of C→D at S1,
then at state S3 reachable from S2, either P may defend D, or else P may counterattack C, thus
yielding a further state, S4=<C, X, ?, *, σ>, where C is the formula considered, and the attack
pertains to the relevant logical constant of C, for which * is a suitable attack marker
determined by the logical form of C.
•
Particle rule for universal quantifier: If B=e, ϒ= ! and B is of the form ∀xDx, then
S2=<Dx, X, !, Dx, σ[x/ki]>
where ki is the constant chosen by the challenger (who here is Y) as a response to the attack ?∀x/ki.
As usual, the notation ‘σ[x/ki]’ stands for the function that is otherwise like σ, but maps the
variable x to ki.Hence if σ is already defined on x, σ[x/ki] is the result of reinterpreting x by ki
otherwise it is the result of extending σ by the pair (x, ki)
•
Particle rule for existential quantifier: If B=e, ϒ= ! and B is of the form ∃xDx, then
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S2=<Dx, X, !, Dx, σ[x/ki]>
where ki is the constant chosen by the defender (who here is X) reacting to the attack ?-∃x of
the challenger (Y).
Structural rules
As we analyze dialogues, we will make use of the following notions: dialogue, dialogical
game, and play of a dialogue. It is very important to keep them conceptually distinct.
Dialogical games are sequences of dialogically signed expressions, i.e. expressions of the
language L[τ] equipped with a pair of labels, P-!, O-!, P-?, or O-?. The labels carry
information about how the dialogue proceeds. Dialogical games are a special case of plays: all
dialogical games are plays, but not all plays are dialogical games. However, all plays are
sequences of dialogical games. Finally, dialogues are simply sets of plays.
A complete dialogue is determined by game rules. They specify how dialogical games in
particular, and plays of dialogues in general, are generated from the thesis of the dialogue.
Particle rules are among the game rules, but in addition to them there are so-called structural
rules, which serve to specify the general organization of the dialogue.
Different types of dialogues have different kinds of structural rules. When the issue is to test
validity - as it is for the dialogues considered in the present paper - a dialogue can be thought
of as a tree, whose (maximal) branches are (finished) plays relevant for establishing the
validity of the thesis. The structural rules will be chosen so that Proponent succeeds in
defending the thesis against all allowed critique of Opponent if, and only if, the thesis is valid
in the standard sense of the term (`true in every model'). In dialogical logic the existence of
such a winning strategy for Proponent is typically taken as the definition of validity; however,
this dialogical definition indeed captures the standard notion (see the discussion in connection
with the definition of validity below).
Each split into two branches - into two plays - in a dialogue tree should be considered as the
outcome of a propositional choice made by Opponent. Any choice by O in defending a
disjunction, attacking a conjunction, and reacting to an attack against a conditional, gives rise
to a new branch: a new play. By contrast, Proponent's choices do not generate new plays; and
neither do Opponent's choices for quantifiers (defending an existential quantifier, attacking a
universal quantifier).
The participants P and O of the dialogues that we are here interested in - the dialogues used
for characterizing validity - are of course idealized agents. If real-life agents took their place,
it might happen that one of the players was cognitively restricted to the point of following a
strategy which would make him lose against some, or even every sequence of moves by the
opponent - even if a winning strategy would be available to him. The idealized agents of the
dialogues are not hence restricted: their `having a strategy' means simply that there exists, by
combinatorial criteria, a certain kind of function; it does not mean that the agent possesses a
strategy in any cognitive sense.
Plays of a dialogue are sequences of dialogically signed expressions, and they share their first
member, the thesis of the dialogue. In particular, plays can always be analyzed into dialogical
games: any play is of the form ∆1 … ∆n, where the ∆1 are dialogical games (i := 1, … ,n). The
14
members of plays other than the thesis are termed moves. A move is either an attack or a
defence.
The particle rules stipulate exactly which moves are to be counted as attacks. Exactly those
moves X-ϒ-e whose expression component e is a first-order formula, are said to have
propositional content. Recall that in the case of conditional and negation some moves with
propositional content count as attacks. (In the actual design of a dialogue there usually is a
notational device to differentiate between those moves with propositional content that are
attacks and those that are not.)
We move on to introduce a number of structural rules for dialogues designed for the language
L[τ]. We will write D(A) for the dialogue about A, i.e. the dialogue whose thesis is A.
Further, we will write ∆[n] for the member of the sequence ∆ with the position n. Let A be a
first-order sentence of vocabulary τ. We have the following structural rules (SR-0) to (SR-6)
regulating plays τ in ∆∈D(A), i.e. members of the dialogue D(A)
(SR-0) (Starting rule)
a) The dialogically signed expression <P-!-A> belongs to the dialogue D(A): the thesis
A stated by Proponent is itself a play in the dialogue about A.
b) If ∆ is any play in the dialogue D(A), then the thesis A has position 0 in ∆. If ∆∈D(A),
then
∆[0]= <P-!-A>.
c) At even positions P makes a move, and at odd positions it is O who moves. That is,
each ∆[2n] is of the form <P-ϒ-B> for some ϒ∈{?, !} and B∈Sub(A); and each
∆[2n+1] is similarly of the form <O-ϒ-B>. Every move after ∆[0] is a reaction to an
earlier move made by the other player, and is subject to the particle rules and the other
structural rules.
(SR-1.I) (Intuitionistic round closing rule).
Whenever player X has a turn to move, he may attack any (complex) formula asserted by his
opponent, Y, or he may defend himself against the last not already defended attack (i.e. the
attack by Y with the greatest associated natural number such that X has not yet responded to
that attack).
A player may postpone defending himself as long as he can perform attacks. Only the latest
attack that has not yet received a response may be answered: If it is X's turn to move at
position n, and positions l and m both involve an unanswered attack (l<m<n), then player X
may not at position n defend himself against the attack of position l.
(SR-1.C) (Classical round closing rule)
Whenever player X has turn to move, he may attack any (complex) formula asserted by his
opponent, Y, or he may defend himself against any attack, including those which have already
been defended. That is, here even redoing earlier defences is allowed.
(SR-2) (Branching rule for plays)
15
If in a play ∆∈D(A) it is O's turn to make a propositional choice, that is, to defend a
disjunction, attack a conjunction, or react to an attack against a conditional, then ∆ extends
into two plays ∆1, ∆2∈D(A),7
∆1=∆ fα and ∆2=∆ fβ
differing in the chosen disjunct, conjunct resp. reaction, α vs. β. More precisely: Let
{n≤max{m : ∆[m]}.
•
If ∆[n]= <O-!-B∨C> and ∆[max] = <P-?-∨>, then
α := <O-!-B> and β := <O-!-C>.
•
If ∆[n]= ∆[max]=<P-!-B∧C>, then
α := <O-!-L> and β := <O-!-R>.
•
If ∆[n]= <O-!-B→C> and ∆[max] = <P-?-B>, then
α := <O-?-*> and β := <O-!-C>.
where * is an attack marker corresponding to the logical form of the formula B.
No moves other than propositional moves made by O will trigger branching.
(SR-3) (Shifting rule)
When playing a dialogue D(A), O is allowed to switch between ‘alternative’ plays ∆,
∆’∈
∈D(A). More exactly, if O loses a play ∆, and ∆ involves a propositional choice made by O,
then O is allowed to continue by switching to another play - existing by the Branching rule
(SR-2). Concretely this means that the sequence ∆∩∆’ will, then, be a play, i.e. an element of
D(A).
It is precisely the Shifting rule that introduces plays which are not plain dialogical games.
(Dialogical games are a special case of plays: they are identified with unit sequences of
dialogical games.)
As an example of applying the Shifting rule, consider a dialogue D(A) proceeding from the
hypotheses (or initial concessions of O) B, ¬C, with the thesis A := B∧C. If O decides to
attack the left conjunct, the result will be the play
(<P-!-B∧C>, <O-?-L>, <P-!-B>)
and O will lose (because he has already conceded B). But then, by the Shifting rule, O may
decide to do have another try. This time he wishes to choose the right conjunct. The result is
the play
(<P-!-B∧C>, <O-?-L>, <P-!-B>, <P-!-B∧C>, <O-?-R>, <P-!-C>)
7
) If ā=(a0, … ,an) is a finite sequence and an+1 is an object, ā∩an+1 is by definition the sequence (a0, … ,an, an+1).
If ā=ā1∩ā2, then, ā1 is said to be an initial segment of ā, and, if the sequence ā2 is not empty, then we say that ā1 is
the initial proper segment of ā.
16
Observe that this play consists of two dialogical games, namely
(<P-!-B∧C>, <O-?-L>, <P-!-B>) and <P-!-B∧C>, <O-?-R>, <P-!-C>
By contrast, this play is not itself a dialogical game.
(SR-4) (Winning rule for plays)
A play ∆∈D(A) is closed, if ∆=( ∆1, …, ∆n), where the ∆i are dialogical games, and in the most
recent dialogical game ∆n there appears the same positive literal in two positions, one stated
by X and the other one by Y. That is, ∆ is closed if for some k, m<ω and some positive literal
ℓ∈Sub(A)∪{A}, we have:
∆n[k]=ℓ= ∆n[m]
where k<m and furthermore, k is odd if, and only if m is even or equal to zero. If this
condition is not satisfied, ∆ is open.
If a play is closed, the player who stated the thesis (that is, P) wins the play; otherwise he
loses it. A play is finished, if it is either closed, or else such that no further move is allowed by
the particle rules or (other) structural rules. If a play is finished and open, O wins the play.
Observe that whenever a play ∆∈D(A) is finished, there is no further play ∆’∈D(A) such that
∆ is an initial segment of ∆’.
(SR-5) (Formal use of atomic formulae)
P cannot introduce positive literals: any positive literal must be stated by O first. Positive
literals cannot be attacked.
In the following, when introducing material dialogues we will consider too, when speaking of
First-order logic, intuitionistic dialogues with additional hypotheses introduced as initial
concessions by O, such as:
∀x1 …∀xn(Rx1 … xn ∨ ¬Rx1 … xn)
where R is a relation symbol of a fixed vocabulary τ. That is, the relevant hypotheses are
instances of (a universal closure of) tertium non datur. In the presence of such hypotheses, we
may use a more general formulation of the rule (SR-5):
(SR-5*)
P cannot introduce literals: any literal (positive or not) must be stated by O first. Positive
literals cannot be attacked.
Before we can state the structural rule (SR-6), or the No delaying tactics rule}, we need some
definitions.
DEFINITION [Strict repetition of an attack / a defence]
a) We speak of a strict repetition of an attack, if a move is being attacked although the
same move has already been challenged with the same attack before. (Notice that even
17
though choosing the same constant is a strict repetition, the choice of ?-L and ?-R are
in this context different attacks.)
In the case of moves where a universal quantifier has been attacked with a new
constant, the following type of move must be added to the list of strict repetitions:
A universal quantifier move is being attacked using a new constant, although the
same move has already been attacked before with a constant which was new at the
time of that attack.
b) We speak of a strict repetition of a defence, if a challenging move (attack) m1, which
has already been defended with the defensive move (defence) m2 before, is being
defended against the challenge m1 once more with the same defensive move. (Notice
that the left part and the right part of a disjunction are in this context two different
defences.)
c) In the case of moves where an existential quantifier has been defended with a new
constant, the following type of move must be added to the list of strict repetitions:
An attack on an existential quantifier is being defended using a new constant,
although the same quantifier has already been defended before with a constant
which was new at the time.
Notice that according to these definitions, neither a new defence of an existential
quantifier, nor a new attack on a universal quantifier, represents a strict repetition,
emph it uses a constant that is not new but is however different from the one used in
the first defence (or in the first attack).
(SR-6) (`No delaying tactics' rule) This rule has two variants, classical and intuitionistic,
depending on whether the dialogue is played with the classical structural rule (SR-1.C), or
with the intuitionistic structural rule (SR-1.I).
Classical: No strict repetitions are allowed.
Intuitionistic: If O has introduced a new atomic formula which can now be used by P, then P
may perform a repetition of an attack. No other strict repetitions are allowed.
DEFINITION [Validity] A first-order sentence A is said to be dialogically valid in the
classical (intuitionistic) sense, if all plays belonging to the classical (resp. intuitionistic)
dialogue D(A) are closed.
It is possible to prove that the dialogical definition of validity coincides with the standard
definition, both in the classical and in the intuitionistic case. First formulations of the proof
were developed in the PhD-Thesis by Kuno Lorenz (reprinted in Lorenzen/Lorenz 1978),
Haas (1980) and Felscher (1985) proved the equivalence for intuitionistic First-order logic (by
proving the correspondence between intuitionistic dialogues and intuitionistic sequent
calculi); while Stegmüller (1964) established the equivalence in the case of classical Firstorder logic. Rahman (1994: 88-107), who stressed the idea that dialogues for validity could be
seen as a proof-theoretical frame to build tableaux systems, proved directly the equivalence
18
between the two types of dialogues and the corresponding semantic tableaux, from which the
result extends to the corresponding sequent calculi.
Philosophical remarks: propositions as games.
Particle rules determine dynamically how to extend a set of expressions from an initial
assertion. In the game perspective, one of the more important features of these rules is that
they determine, whenever there is a choice to be made, who will choose. This is what can be
called the pragmatic dimension of the dialogical semantics for the logical constants. Indeed,
the particle rules can be seen as a proto-semantics, i.e. a game scheme for a not yet
determined game which when completed with the appropriate structural rules will render the
game semantics, which in turn will build the notion of validity.
Actually by means of the particle rules games have been assigned to sentences (that is, to
formulæ). But sentences are not games, so what is the nature of that assignment? The games
associated to sentences are meant to be propositions (i.e. the constructions grasped by the
(logical) language speakers). What is connected by logical connectives are not sentences but
propositions. Moreover, in the dialogic, logical operators do not form sentences from simpler
sentences, but games from simpler games. To explain a complex game, given the explanation
of the simpler games (out) of which it is formed, is to add a rule which tells how to form new
games from games already known: if we have the games A and B, the conjunction rule shows
how we can form the game A∧B in order to assert this conjunction.
Now, particle rules have another important function: they not only set the basis of the
semantics, and signalise how it could be related to the world of games – which is an outdoor
world if the games are assigned to prime formulæ, but they also show how to perform the
relation between sentences and propositions. Sentences are related to propositions by means
of assertions, the content of which are propositions. Assertions are propositions endowed with
a theory of force, which places logic in the realm of linguistic actions. The forces performing
this connection between sentences and propositions are precisely the attack (?) and the
defence (!). An attack is a demand for an assertion to be uttered. A defence is a response (to
an attack) by acting so that you may utter the assertion (e.g. that A). Actually the assertion
force is also assumed: utter the assertion that A only if you know how to win the game A.
Certainly the "know" introduces an epistemic moment, typical of assertions made by means of
judgements. But it does not presuppose in principle the quality of knowledge required. The
constructivist moment is only required if the epistemic notion is connected to a tight
conception of what means that the player X knows that there exists a winning game or
strategy for A.
Let us take examples of dialogues, classical and intuitionistic.
EXAMPLE: Consider the classical dialogue D(p∨¬p). Its thesis is p∨¬p, where p is an atomic sentence. In
Figure 1, a dialogical game from dialogue D(p∨¬p) is described. This dialogical game is won by P:
O
1
3
[1]
?-∨
P
[?-∨]
P
p∨¬p
¬p
—
p
0
2
[0]
II.1.3.f1. Classical rules, P wins.
19
0
2
4
The outer columns indicate the position of the move inside the dialogical game, while the inner columns state the
position of the earlier move which is being attacked. The defence is written on the same line with the
corresponding attack: an attack together with the corresponding defence constitutes a so-called closed round.
The sign ‘— ‘ indicates that there is no possible defence against an attack on a negation.
In the dialogical game of the example, P wins because after O's last attack in move 3, P is allowed - according to
the classical rule SR-1.C - to defend (once more) himself against O's attack made in move 1, which was certainly
not the last attack of O, and so the game in question is closed. P states his new defence in move 4. (Actually O
does not repeat his attack of move 1: what we have written between square brackets simply serves to remind of
the attack against which P is re-acting.)
• In fact the described dialogical game is the only finished play of the dialogue D(p∨¬p): O could not
prolong the play any further by making different moves. Hence not only does P win the described
particular dialogical game - in fact he has a winning strategy in the dialogue, i.e. he is able to win no
matter what O does. In other words, the sentence p∨¬p is dialogically valid in the classical sense (cf.
Definition Def:validity).
Here is an example concerning Peirce's Law and which requires to consider two plays:
EXAMPLE:
In the version of strategy dialogues what actually happens is that O generates two dialogical games one
defending and the other counterattacking. Both dwill be closed and thus won by P.
O
1
(p→q)→p
0
1
2
p
P
((p→q)→p)→p
p
p→q
0
4
2
2
II.1.3.f2. Classical rules, P wins.
O
1
3
(p→q)→p
p
0
1
P
((p→q)→p)→p
p
p→q
0
4
2
II.1.3.f3 Classical rules, P wins.
Actually this produces a play with two dialogical games. Let us label each dialogical game with a roman letter and put all in
only one graphic. In the graphic below we splitted the play in two showing the dialogical games produced - simpler would be
to eliminate the outer columns and add the label directly to the formulae, but this notation will make it easier to show the
relation to (the branches produced by a correspondent) sequent calculi.
The expression between the signs ‘<’ and ‘>’ signalise that the Opponent has decided in the choice I.2 not to counterattack
the expession inside those signs but defend himself. This expression is then not at stake in the play I.2 and can be considered
as an attack marker rather than a formula
Because of the tree-like structure of the proof we will assume that the thesis (move 0),t he first challenge of O (move 1),
which occur before the splitting takes place, and the answer (move 4) are shared by both dialgical games and will neither
repeat them:
O
1
I.3 
 II.3
(p→q)→p
p
p
0
1
P
((p→q)→p)→p
p
<p→q>  p→q
0
I.4  II.4
I.2  II.2
II.2
II.1.3.f4 Classical rules, P wins.
Let us consider now the intuitionistic variant of the dialogue of the first example.
20
EXAMPLE: In figure below, a dialogical game from the intuitionistic dialogue D(p∨¬p) is described. This game
is won by O:
O
1
3
P
p∨¬p
¬p
—
0
2
?-∨
p
0
2
II.1.3.f5. Intuitionistic rules, Owins.
It is O who wins the dialogical game of the example: the game is open, and no further move is possible
following the intuitionistic structural rules. In particular remaking an earlier move (i.e., answering to an attack
which was not the last one - as in the above example of a classical dialogue - is not possible.
In fact O has trivially a winning strategy in the intuitionistic dialogue D(p∨¬p): P cannot prevent, by making
different moves, O from generating precisely the described play won by O.
• Observe, in particular, that the sentence p∨¬p is not dialogically valid in the intuitionistic sense. (This
does not mean, of course, that thereby the sentence ¬(p∨¬p) would be intuitionistically valid!)
The following example shows the fail of double negation in intuitionistic logic
EXAMPLE
D¬¬p→p
O
1
¬¬p
0
¬p
2
0
—
3
P
¬¬p→p
1
p
2
—
II.1.3.f6. Intuitionistic rules, O wins.
O wins because P is not allowed to use the atomic formula stated by O at move 3 to defend the
challenge of move 1. Indeed, move 3 is the last attack of O and P must answer now to this attack.
Unfortunately, by the particle rule of negation, there is no defence to challenged negation. Only
counterattacks are possible. But p is an atomic formula which cannot be counterattacked!
To come back to a success story for P let us see a more trickier case. Namely, an intuitionistic
dialogue for D(¬¬(p∨¬p)) where P should have a winning strategy. Indeed, the double
negation of any valid classical formula is valid intuitionically too!
EXAMPLE
D(¬¬(p∨¬p))
O
1
3
5
7
¬(p∨¬p)
—
?-∨
p
—
?-∨
0
1
2
4
1
6
21
P
¬¬(p∨¬p)
—
p∨¬p
¬p
—
p∨¬p
p
0
2
4
6
8
II.1.3.f7. Intuitionistic rules, Pwins.
The tricky point is move 6 where P is allowed to repeat the attack on the first move of O because since move 1,
O introduced a new atomic formula (see SR-6)S. Indeed at move 5 O introduced the positive literal p and this
can be now used to defend the new occurrence of the disjunction.
The way to build a winning strategy for dialogues for first-order logic is not really different
from the propositional case: Here the Proponent will try to wait so long as he can before
choosing a value for the variables. More precisely, he will wait until the Opponent has chosen
first the value for the variables at stake and later on he will simply copy-cat them.Let us show
examples of dialogues for first order logic:
EXAMPLE
D((∀x((Ax∨Bx)∧¬Ax)))→∀x (¬¬Bx∨Cx))
O
1
∀x((Ax∨Bx)∧¬Ax))
0
3
?-∀x/k
2
5
7
9
11
13
?-∨
(Ak∨Bk)∧¬Ak
Ak∨B
Ak
¬Ak
—
4
6
P
(∀x((Ax∨Bx)∧¬Ax)))→∀x (¬¬Bx∨Cx)
?-∀x(¬¬Bx∨Cx)
¬¬Bk∨Ck
1
5
9
7
13
?-∀x/k
?-L
?-∨
?-R
Ak
0
2
4
6
8
10
12
14
II.1.3.f7. Classical rules, P wins.
EXERCISE II.1.3.e1:
1. Justify the moves in the play above.
• Notice that in this play P does not need to defend the challenge on his disjunction at 4.
2. Complete the proof with the play where O chooses the right side of the disjunction
3. Run an intuitionistic dialogue
Let us consider now a very tricky dialogue for classical logic:
EXAMPLE
D(∃x(Ax→∀x Ax))
O
1
?-∃
0
3
Ak1
2
P
∃x(Ax→∀xAx)
Ak1→∀xAx
∀xAx
5
?-∀x/k2
4
Ak2
8
[1]
7
[?-∃]
Ak2
0
6
Ak2→∀xAx
6
II.1.3.f7. Classical rules, P wins.
EXERCISE II.1.3.e2:
1. Justify the moves in the play above.
• Notice that in this play P does not need to defend the challenge on his conditional of 6.
22
0
2
4
2.
In fact it would be a mistake to defend himself by playing once more ∀x Ax. Do you see why?
Run an intuitionistic dialogue for the same formula.
EXERCISE II.1.3.e3:
Notation : we will use expressions such as
∀xy.. as abbreviation for
∀x ∀y ∀…
Run classical dialogues for
1. ∃x(∃yAy→Ax)
2. ∃x∀yRxy→∀y∃xRxy
• Recall that Rkikj is an atomic formula
3. ∀xy(Rxy→¬Ryx) → ∀x ¬Rxx
4. ∀xyz(Rxy∧Ryz) →¬Rxz) → ∀x ¬Rxx
5. ∀xyz((Rxy∧Ryz) →Rxz))∧∀xy(Rxy→Ryx) → ∀x Rxx
6. ∀xyz((Rxy∧Ryz) →Rxz))∧∀xy(Rxy→Ryx) ∧∀x∃y(Rxy)→ ∀x Rxx
Run intuitionistic dialogues for
7. ∀x(a∨Ax)→ a∨∀x Ax
8. ∃x(∃yAy→Ax)
9. ¬¬∃x(∃yAy→Ax)
10. (∃x(∃yAy→Ax) ∨ ¬∃x(∃yAy→Ax)) → ∃x(∃yAy→Ax)
11. ∀xyz((Rxy∧Ryz) →Rxz))∧∀xy(Rxy→Ryx) ∧∀x∃y(Rxy)→ ¬¬∀x Rxx
II.2 Dialogues, tableaux and sequent-calculus
II.2.1 Dialogues and tableaux
As already mentioned, the strategy dialogical games introduced above, furnish the elements of
building a tableau notion of validity. Following the seminal idea at the foundation of dialogic,
this notion is attained via the game-theoretical notion of winning strategy. X is said to have a
winning strategy if there is a function which, for any possible Y-move, gives the correct Xmove ensuring the wining of the game.28
Indeed, it is a well known fact that the usual semantic tableaux for intuitionistic and classical logic,
as reformulated 1968 in a tree-shaped structure by Raymond Smullyan and 1969 by Melvin Fitting,
are directly connected with the tableaux (and the correspondent sequent calculus) for strategies
generated by dialogue games, played to test validity in the sense defined by these logics (cf. Rahman
1993).
A systematic description of the winning strategies available can be obtained from the following
considerations:
If P is to win against any choice of O, we will have to consider two main different situations,
namely the dialogical situations in which O has stated a (complex) formula and those in which P
has stated a (complex) formula. We call these main situations the O-cases and the P-cases
respectively.
In both of these situations another distinction has to be examined:
1. P wins by choosing an attack in the O-cases or a defence in the P-cases, iff he can win at least
one of the dialogues he can choose.
2. When O can choose a defence in the O-cases or an attack in the P-cases, P can win iff he can
win all of the dialogues O can choose.
23
The closing rules for dialogical tableaux are the usual ones: a branch is closed iff it contains two
copies of the same atomic formula, one stated by O and the other one by P. A tableau for (P)A (i.e.
starting with (P)A) is closed iff each branch is closed. This shows that strategy systems for classical
and intuitionistic logic are nothing other than the very well known tableau systems for these logics.
For the intuitionistic tableau system, the structural rule about the restriction on defences has to be
considered. The idea is quite simple: the tableau system allows all the possible defences (even the
atomic ones) to be written down, but as soon as determinate formulae (negations, conditionals,
universal quantifiers) of P are attacked all other P-formulae will be deleted  this is an
implementation of the structural rule RI4 for intuitionistic logic. Clearly, if an attack on a P-statement
causes the deletion of the others, then P can only answer the last attack. Those formulae which compel
the rest of P’s formulae to be deleted will be indicated with the expression ‘∑[O]’ which reads: in the
set ∑ save O’s formulae and delete all of P’s formulae stated before.i
However the resulting tableaux are not quite the same as the standard ones. A special feature
of dialogue games is the notorious formal rule, which is responsible for many of the
difficulties of the proof of the equivalence between the dialogical notion and the truthfunctional notion of validity. The role of the formal rule, in this context, is to induce dialogue
games which will generate a tree displaying the (possibly) winning strategy of P, the branches
of which do not contain redundancies. Thus the formal rule actually works as a filter for
redundancies, producing a tableau system with some flavour of natural deduction (cf.
Rahman/Keiff 2005).
Classical Tableaux
•
(O)-Cases
(P)-Cases
∑, (O)A∨B
------------------------------∑, <(P)?-∨>(O)A | ∑, <(P)?-∨>(O)B
∑, (P)A∨B
-------------------∑, <(O)?-∨>(P)A
∑, <(O)?-∨>(P)B
∑, (O)A∧B
--------------------∑, <(P)?-L>(O)A
∑, <(P)-?R>(O)B
∑, (P)A∧B
------------------------------∑, <(O)?-L>(P)A | ∑, <(O)?-R>(P)B
∑, (O)A→B
------------------------------∑, (P)A ... | <(P)A>(O)B
∑, (P)A→B
------------------∑, (O)A; ∑,(P)B
∑, (O)¬A
-----------------∑, (P)A; —
∑, (P)¬A
--------------∑, (O)A; —
∑, (O)∀xA
-------------------∑, <(P)?-∀x/ki>(O)A[x/ki]
∑, (P)∀xA
-------------------∑, <(O)?-∀x/ki >(P)A[x/ki]
ki is new
∑, (O)∃xA
-------------------∑, <(P)?-∃>(O)A[x/ki]
ki is new
∑, (P)∃xA
-------------------∑, <(O)?-∃>(P)A[x/ki]
If ∑ is a set of dialogically signed formulae and X is a single dialogically signed
formula, we will write ∑, X for ∑4{X}.
24
•
•
•
•
Observe that the formulae below the line always represent pairs of attack and defence
moves. In other words, they represent rounds.
The vertical bar ‘|’ indicates alternative choices for O, P's strategy must have a
defence for both possibilities (dialogical games defining two possible plays).
The rules containing yielding two lines indicate that it is P who has the choice – an he
thus might need only one of both possible choices.
Note that the expressions between the symbols "<" and ">", such as <(P)?> or <(O)?>
are moves – more precisely they are attacks – but not formule (assertions) which could
be attacked.
Intuitionistic Tableaux
(O)-Cases
(P)-Cases
∑, (O)A∨B
------------------------------∑, <(P)?-∨>(O)A | ∑, <(P)?-∨>(O)B
∑, (P)A∨B
-------------------∑[O], <(O)?-∨>(P)A
∑[O], <(O)?-∨>(P)B
∑, (O)A∧B
--------------------∑, <(P)?-L>(O)A
∑, <(P)?-R>(O)B
∑, (P)A∧B
------------------------------∑[O], <(O)?-L>(P)A | ∑[O], <(O)?-R>(P)B
∑,(O)A→B
------------------------------∑[O], (P)A ... | <(P)A>(O)B
∑, (P)A→B
------------------∑[O], (O)A; (P)B
∑, (O)¬A
-----------------∑[O], (P)A; —
∑, (P)¬A
--------------∑[O], (O)A; —
∑, (O)∀xA
-------------------∑, <(P) ?-∀x/ki>(O)A[x/ki]
∑, (P)∀xA
-------------------∑[O], <(O) ?-∀x/ki >(P)A[x/ki]
ki is new
∑, (O)∃xA
-------------------∑, <(P)?-∃>(O)A[x/ki]
ki is new
∑, (P)∃xA
-------------------∑[O], <(O)?-∃>(P)A[x/ki]
Intuitionistic tableaux are generated with an extra notational device. Some of the expressions
are labelled with (O), for instance (P)(O)A. the intuitionistic deduction rule includes this: the
totality of the previous P-formulæ on the same branch of the tree are eliminated. The (O)
label marks every assertion of O.
Let us look at two examples, namely one for classical logic and one for intuitionistic logic. We use the
tree shape of the tableau made popular by Smullyan ([18]):
EXAMPLE
(P) ∀x(¬¬Ax→Ax)
25
<(O)?-∀x/k > (P) ¬¬Ak→Ak
(O)¬¬Ak
(P) Ak
(P)¬Ak
(O)Ak
The tableau closes: P wins
EXERCISE II.2.1.e1:
Reconstruct the dialogue which originated the tableau of the example above writing down the exact
correspondence move by move
• Notice that the order of the tableau is the order of rounds not of moves!
The following intuitionistic tableau makes use of the deletion rule:
EXAMPLE:
(P) ∀x(¬¬Ax→Ax)
<(O)?-∀x/k > (P)[O] ¬¬Ak→Ak
(O)[O]¬¬Ak
(P) Ak
(P)¬Ak
(O)[O]Ak
The tableau remains open: O wins
EXERCISE II.2.1.e2:
Reconstruct the dialogue which originated the tableau of the example above writing down the exact
correspondence move by move. Indicate the correspondence between the deleting rule of the tableau and
the use of the intuitionistic structural rule in the dialogue
II.2.2 Dialogues and sequent-calculus
The standard way to produce sequent-calculus from tableaux systems is to writte the rules
for the logical constants of latter upside down, replace the rule of closing a branch with an
axiom and adding structural rules which make explicit the properties of the consequence
relation at stake. The last point relates to the advantages and disadvantages of sequentcalculus in relation to tableaux systems and dialogues. While the tree-like structure of
tableaux systems and dialogues allows to avoid some re-writing, the sequent-calculus - where
indeed in every branch every formula in use has to be rewritten - has the advantage that it
makes explicit the use of the precise structural rule required by the definition of the
consequence relation assumed in the proof.
Rules of the sequent-calculus consist of a premiss-sequent and a conclusion-sequent.
Let us start by the following rewriting of the tableaux for classical logic.
Classical sequent-calculus
1. We add the sequent sign ‘⇒’ and write the O-signed formula unsigned at the left of
the sequent sign and the P-signed formulae at the right of the sequent.
Thus,
∑, (O)A∨B
---------------------------------------------
26
∑, <(P)?-∨>(O)A | ∑, <(P)?-∨>(O)B
will be written as
∑, A∨B⇒Θ
------------------------------------------∑, A ⇒Θ, <?-∨> | ∑, B ⇒Θ, <?-∨>
2. We write the version of the rules achieved by 1 upside down
Thus,
∑, A∨B⇒Θ
-------------------------------------------∑, A ⇒Θ, <?-∨> | ∑, B ⇒Θ, <?-∨>
will be written as
∑, A ⇒Θ, <?-∨> | ∑, B ⇒Θ, <?-∨>
--------------------------------------------∑, A∨B⇒Θ
3. The tableu-rules containing two lines will rewritten in one line separated by a colon.
Thus,
∑, (P)A∨B
-------------------∑, <(O)?-∨>(P)A
∑, <(O)?-∨>(P)B
will yield:
∑, <(O)?-∨> ⇒ Θ, A, B
------------------------------∑ ⇒ Θ, A∨B
•
Notice that this device corresponds in the dialogues to the structrual rule which allows
the Proponent to re-answer a challenge by the choice of another disjunct (see the
example of the dialogue for third excluded in the preceding chapter)
4. Replace the closing rules with the following axiom-schema:
---------------p⇒p
•
Notice that we formulated the axiom-schema with atomic formulae. The axiom-schema
could be generalized for complex formulae but we prefer this version because of the
analogy with the formal rule for dialogues.
Rules for logical constants
LEFT-CASES
RIGHT-CASES
∑, A ⇒Θ <?-∨> | ∆, B ⇒Π, <?-∨>
--------------------------------------------∑, ∆, A∨B⇒Θ, Π
∑, <?-∨> ⇒ Θ, A, B
------------------------------∑ ⇒ Θ, A∨B
27
∑, A, B ⇒ Θ, <?-L>, <?-R>
---------------------------------------------∑, A∧B ⇒ Θ
∑, <?-L> ⇒Θ, A | ∆, <?-R> ⇒Π, B
--------------------------------------------∑, ∆ ⇒Θ, Π, A∧B
∑ ⇒Θ, A | ∆, B ⇒Π, <A>
--------------------------------∑, ∆, A→B ⇒ Θ, Π
∑, A ⇒ Θ, B
----------------∑ ⇒ Θ, A→B
∑ ⇒ Θ, A
---------------∑, ¬A ⇒ Θ
∑, A ⇒ Θ
-----------------∑ ⇒ Θ, ¬A
∑, A[ki] ⇒ Θ, <?-∀x/ki>
-----------------------------∑, ∀xA ⇒ Θ
∑, <?-∀x/ki>⇒ Θ, A[ki]
---------------------------------∑ ⇒ Θ, ∀xA
]
ki is new: it does not occur in the conclusion
∑, A[ki] ⇒ Θ, <?-∃>
-------------------∑, ∃xA ⇒ Θ
ki is new: it does not occur in the conclusion
•
∑, <?-∃> ⇒ Θ, A[ki]
-------------------∑ ⇒ Θ, ∃xA
As we will discuss below the rules have been designed to match the tableau-rules (and
the dialogues) as close as possible. Actually the sequent-calculus describe above has
been conceived to be developed bottom up, this applies particularly to the formulation
to the quantifier rules. If we prefer to take the sequent-calculus top-down then
alternatives to the rules for ∃A ⇒ and ⇒ Θ, ∀, where the conclusion records the
substitution of the given constant by an adequate variable (and not the other way
round) are more appropriate:
∑, A ⇒ Θ
-------------------∑, ∃xA[ki/x] ⇒ Θ
∑, >⇒ Θ, A
-------------------∑ ⇒ Θ, ∀xA[ki/x]
ki is new: it does not occur in the conclusion
ki is new: it does not occur in the conclusion
Let us now come to the rules defining the classical consequence relation:
Structural Rules for classical logic
LEFT-CASES
RIGHT-CASES
Weakening
Weakening
∑⇒Θ
--------∑ A ⇒Θ
∑⇒Θ
-----------∑ ⇒ Θ, A
Contraction
Contraction
∑, A, A ⇒ Θ
-----------------
∑ ⇒ Θ, A, A
-----------------
28
∑ A ⇒Θ
∑ ⇒Θ, A
Interchange
Interchange
∑, A, B ⇒ Θ
----------------∑ B, A ⇒Θ
∑ ⇒ Θ, A, B
----------------∑ ⇒Θ, B, A
Cut
∑ ⇒ Θ, A
∆, A ⇒ Π
-------------------------------------∑, ∆ ⇒ Θ, Π
The interchange rules permit the rearrangement of formulae in the left and the right side of the
sequent.
Cut differs from the other structural rules, of course, in that it has two premises. It also differs
from that from all the other rules in that a formula appearing in a premise (A) fails to occur
itself or as a subformula of another formula in the conclusion of the rule. It is a rule about
which we shall be hearing a considerable amount of times and details. We leave it like that for
the moment.
The other rules deserve some general comments.
As already mentioned the present sequent calculus has been formulated in a way, which
matches as closely as possible the tableau-rules. Indeed, we will assume that the proofs will
be developed bottom-up, i.e., from the sequent to be proven up to the axiom-schema(ta)
which delivers the foundations of such a proof. Indeed, instead of viewing the rules as
descriptions for legal derivations in predicate logic, one may also consider them as
instructions for the construction of a proof. In this case the rules can be read bottom-up. For
example, the rule conjunction to the right says that, in order to prove that A ∧B follows from
the assumptions Σ and ∆, it suffices to prove that A can be concluded from Σ and B can be
concluded from ∆, respectively. Note that, given some antecedent, it is not clear how this is to
be split into ∆ and Σ. However, there are only finitely many possibilities to be checked since
the antecedent by assumption is finite. This also illustrates how proof theory can be viewed as
operating on proofs in a combinatorial fashion: given proofs for both A and B, one can
construct a proof for A ∧B.
•
Cut and the bottom up reading: As already mentioned, when looking for some proof,
most of the rules offer more or less direct recipes of how to do this. The rule of cut is
different: It states that, when a formula A can be concluded and this formula may also
serve as a premise for concluding other statements, then the formula A can be "cut out"
and the respective derivations are joined. When constructing a proof bottom-up, this
creates the problem of guessing A (since it does not appear at all below). This issue is
addressed in the theorem of cut elimination. 8
8 Cf. http://en.wikipedia.org/wiki/Sequent_calculus. See too Girard, Jean-Yves; Paul Taylor, Yves Lafont [1989]
(1990). Proofs and Types. Cambridge University Press (Cambridge Tracts in Theoretical Computer Science, 7). ISBN 0521-37181-3.
29
Let us stress the following points elucidating both perspectives in relation to the reading of the
rules: top down and bottom up:
•
The top down perspective (from the axiom-schemata to the thesis):
1.
Weakening rules permit the addition of an arbitrary wff to either the left or the
right side of the sequent.
2.
Contraction rules permit the dropping of duplicated formulae.
3.
The rules for the logical constants are introduction rules: their conclusions
contain formulae with fresh logical constants.
•
The bottom up perspective (from the thesis to the axiom-schemata):
1.
Weakening rules permit to eliminate arbitrary wff to either the left or the right
side of the sequent in order to obtain the appropriate axiom schemata.
In fact, if we build our proof bottom-up, the weakening will allow us to regard
some formulae as redundant.
The dialogical (and tableau) counterpart of the use of this rule is implicit in
the definition of closing a dialogical game (branch of a tableau) which allows
to close despite other subformulae have not been used in the game (branch) at
stake. Take as an example the dialogical (tableau) proof of (p∧Q)→p, where
the complex formula Q will not be used to close the dialogical game (branch)
which yields a winning strategy (tableau-proof) for P.
2.
Contraction rules permit to duplicate formulae.
The dialogical (and tableau) counterpart of the use of this rule (if we do not
assume that cut has been used) is implicit in the tree-like structure of a play
(tableau) which allows to close two different dialogical games (branches)
using twice the same O-signed formula. We will come to a detailed comment
of some examples below but for the moment think in a dialogue (tableau) for
(p→(p→q))→(p→q). In the dialogical (tableau) proof the signed atomic
formula (O)p will be used twice. Namely in each of the dialogical games
(branches) generated by the splitting of (O) (p→(p→q).
3.
The rules for the logical constants are elimination rules: their conclusions
contain subformulae from the formula of the premise where the logical
constant at stake occurs.
From dialogues to sequents
We have allready described how to convert dialogical tableau rules in sequent rules. We
would like now to show how to produce a proof in a sequent calculus from a dialogical proof.
Let us do it bottom up (from the thesis to the axiom schema):
First a definition
DEFINITION
• A pair of literals are said to have been used in dialogical (tableau) proof iff these
pairs motivate the closing of a dialogical game (branch).
30
•
A complex formula A is said to have been used in a dialogical proof iff all the
possible aggressive and defensive moves related to A allowed by the structural and
particle rules have been stated in the course of that proof. (A complex formula A is
said to have been used in a tableau iff a tableau rule has been applied to the formula
in the course of that tableau).
0. Rewrite the dialogical (tableau) proof using the following device:
• Tick those formula which have been used (see definition above).
• If a given formula has been used twice tick it twice. Tick twice any quantifer which
has been instantiated twice.
• Expressions between between the signs ‘<’ ‘>’ do not need to be ticked because
they can’t be used.
1. Start the calculus by writing the thesis at the right of the sequent sign.
2. Follow the order of the rules of the dialogical proof in the orther of the rounds not of
the moves but apply the sequent versions of the rules.
3. Add to each formula the label it carries at the dialogue.
4. If the result of the process of applying the sequent versions of the rules yields formulae
without a label (because they do not occur in the dialogue), apply the adequate
weakening rule.
• The use of weakening can be recognized already in the ticked notation of the
dialogical proof of a valid formula. Indeed if a (sub)formula has not been ticked
then weakening will be needed. Moreover this notation will indicate if the weaking
required is a left (O-side) or rather a right (P-side) one.
Expressions between the signs ‘<’ ‘>’ can be eliminated without the use of
weakening. Furthermore
those expressions don’t need to be eliminated from an
axiom schema.9
5. If a formula carries in the dialogical proof more than one tick, apply an appropriate
instantiation of the contraction rule.
EXAMPLES
Let us take once more the dialogical proof of Peirce’s law:
O
1
I.3 
 II.3
(p→q)→p√
p√ 
 p√
0
1
P
((p→q)→p)→p√
p√
<p→q>  p→q
0
4
I.2  II.2
II.2
II.2.2.f1 Classical rules, P wins.
The ticking device already shows that at least one weakening is needed. Indeed the P-formula p→q stated at move II.2,
has not been ticked because the challenge of move II.3 has never been responded.
If we apply the bottom-up procedure described above we obtain in the sequent calculus the following proof of Peirce’s law,
where we added small roman numbers to keep track of the order of the stages in our proof:
v
iv
iii
p (II.3) ⇒ p (1)
-----------------------⇒ weakening
p (II.3) ⇒ q, p (1)
-----------------------⇒ conditional
⇒ p→q (II.2), p (1)
|
p (I.3) ⇒ p (1), <p→q> (I.2)
9
. Actually these expressions are not part of the object language of the calculus. If, as we will
discuss in ???, those kinds of expressions associated to quantifiers are imported into the object
language: free logic obtains. That is, the result of developing a device to convert these kinds
of expressions in well formed formulae for quantifiers is a logic where terms might be related
also to fictional objects.
31
ii
i
------------------------------------------------------------------------------conditional ⇒
(p→q)→p (1) ⇒ p
(4)
-------------------------------------- ⇒ conditional
⇒ ((p→q)→p)→p (0)
Let us see now two cases of contraction:
O
1
3
I. 5

I.i.7
(p→(p→q) √
p √√
p→q 
q√ 
P
(p→(p→q))→(p→q)√
p→q√
q√
<p>  p√
<p>  p√
0
2
1
0
2
8
I.4  II.4
I.i.6  I.ii.6
II.2.2.f2 Classical rules, P wins.
The double ticking indicates that it is move 3 which needs to be used twice by P:
vi
v
iv
iii
ii
i
I.i.7 q ⇒ q (8), <p> (I.4), <p> (I.i.6)
|
|
|
p (3) ⇒ <p> (I.4), p(I.ii.6)
-----------------------------------------------------------------------------conditional ⇒
p→q (I.5), p (3) ⇒ q (8), <p> (I.4)
|
p (3) ⇒ p (II.4)
----------------------------------------------------------------------------------------------------------------------------conditional ⇒
(p→(p→q) (1), p (3), p (3) ⇒ q (8)
------------------------------------------contraction ⇒
(p→(p→q) (1), p (3) ⇒ q (8)
-------------------------------------- ⇒ conditional
(p→(p→q) (1) ⇒ p→q (2)
-------------------------------------- ⇒ conditional
⇒ (p→(p→q))→(p→q) (0)
EXAMPLE
D(∃x(Ax→∀x Ax))
O
1
?-∃
0
3
Ak1
2
P
∃x(Ax→∀xAx) √√
Ak1→∀xAx √
∀xAx √
5
?-∀x/k2
4
Ak2 √
8
[1]
7
[?-∃]
Ak2 √
0
6
Ak2→∀xAx
6
0
2
4
II.2.2.f3. Classical rules, P wins.
The ticking device already shows that the we will need two applications of weakening, namely, one for move 3
(not ticked), and for the subformula ∀xAx of the (unticked) conditional stated at move 6.
Furthermore the double ticks on the formula stated at the thesis shows that this formula required the application
of a contraction rule.
viii
vii
vi
v
Ak2 (7) ⇒ Ak2 (8)
------------------------- weakening ⇒
Ak2 (7) , Ak1 (3) ⇒ Ak2 (8)
------------------------- ⇒ weakening
Ak2 (7), Ak1 (3) ⇒ ∀xAx, Ak2 (8)
----------------------------------------- ⇒ conditional
<?-∃>, Ak1 (3)⇒ Ak2→∀xAx (6) Ak2 (8)
Notice that the duplicate of the attack to the thesis has no label
32
------------------------------------------ ⇒ existential
iv
iii
ii
ii
i
<?-∀x/k2> (5), Ak1 (3) ⇒ ∃x(Ax→∀xAx), Ak2 (8)
------------------------------------------ ⇒ universal
Ak1 (3) ⇒ ∃x(Ax→∀xAx), ∀xAx (4)
-------------------------------------- ⇒ conditional
<?-∃> (1) ⇒ ∃x(Ax→∀xAx), Ak1→∀xAx (2)
----------------------------------------------⇒ existential
⇒ ∃x(Ax→∀xAx), ∃x(Ax→∀xAx) (0)
Notice that the duplicate of the thesis has no label
----------------------------------------------⇒ contraction
⇒ ∃x(Ax→∀xAx) (0)
EXERCISE II.2.2.e1:
Prove with sequent calculus the following :
1. ⇒∃x(∃yAy→Ax)
2. ⇒∃x∀yRxy→∀y∃xRxy
3. ∀xy(Rxy→¬Ryx) ⇒ ∀x ¬Rxx
4. ∀xyz(Rxy∧Ryz) →¬Rxz) ⇒ ∀x ¬Rxx
5. ∀xyz((Rxy∧Ryz) →Rxz))∧∀xy(Rxy→Ryx) ∧∀x∃y(Rxy)⇒∀x Rxx
EXERCISE II.2.2.e2:
Formulate a procedure alternative to the one described above giving up the formal rule.
Discuss the advantages of using such alternative.
EXERCISE II.2.2.e3:
Formulate the other sense of the procedure. That, is, formulate a procedure to reconstruct the dialogues from the proof in the
sequent-calculus.
HINTS: In the sense from dialogues sequents one has to make explcit what it is implicit. In this case exactly the contrary
applies. The point is to regulate how which is explicit in the sequent does not appear in the dialogue, because it migth be
redundant from the point of view of the dialogue or even impossible to make explicit (in view of the formal rule).
There are may ways to achieve this: one would be to buid up explicit dialogues, giving up the formal rule.
33
IV Modal propositional logic: Basic semantic notions
Once upon a time modal logics were characterized by giving different sets of axioms for
them (see Appendix C: Axioms for modal logic). The success of the semantic approach to
modal logics as developed by Hintikka’s and Kripke’s is partly due to the fact that within this
model theoretical approach most standard modal logics could be characterized by placing
simple mathematical conditions on frames. It is important to notice this: conditions are placed
on frames. Although models are what we deal with most often, frames play a central role.
DEFINITION : Model, Frame, Truth
•
A model <W,R,v> for modal propositional logic consists of
1. a non empty set W of possible worlds (contexts or scenarios: like temporal states,
states of information etc.)
2. a binary relation R on W called accessibility relation
3. a valuation function v which assigns a truth value v(a) to each propositional letter of
the propositional language in each world w∈W
•
A set of possible worlds W with a suitable accessibility relation is called a frame or
structure. Thus given a frame <W,R> we can turn it into a model by the addition of the
valuation function v. Moreover any given frame can be turned into a variety of different
models, depending on the valuation function which is added. For a frame only establishes
the worlds we are dealing with and fixes are accessible from which. A valuation is needed
to establish what is the case in each of the possible worlds and in general there will be
many ways to do that. Each of this ways is a model establishing the factual conditions
under which our logical explorations will take place. The frame will provide the basis of
anyone of a variety of such factual conditions.
•
The truth definition of modal logic tell us what formulae are true in what wi of any given
model. The valuation function gives us the values of the propositional letters and the truth
definition extends this to the complex formulae. The difference of this truth definition to
the classical case is that the truth is here made relative to the value of the possible worlds
of the model at stake. Furthermore the evaluation is dependent too on the interrelations
between the given possible worlds. More precisely:
DEFINITION : If M is model <W,R,v>, then vMw(ϕ) – i.e. the truth value of ϕ in w given
M, is defined in the following way:
vMw(a)= vMw(a), for all propositional letters a
vMw(¬ϕ)= 1 iff vMw(ϕ)= 0
vMw(ϕ→ψ)= 1 iff vMw(ϕ)= 0 or vMw(ψ)= 1
vMw(ϕ∨ψ)= 1 iff vMw(ϕ)= 1 or vMw(ψ)= 1
vMw(ϕ∧ψ)= 1 iff vMw(ϕ)= 1 and vMw(ψ)= 1
vMw(◊ϕ)=1 iff for at least one w’∈W such that wRw’: vMw’(ϕ)= 1
vMw( ϕ)=1 iff for at all w’∈W such that wRw’: vMw’(ϕ)= 1
V Validity on Frames
V.1 The model theoretical approach
34
DEFINITION : L-Valid
We say that the model <W, R, v> is based on the frame <W,R>.
• A formula ϕ is valid in a model <W, R, v> if it is true at every world of W.
• A formula ϕ is valid in a frame if it is true at every world of every model based on that
frame. For short: a formula ϕ is valid in a frame if it is valid in every model based on
that frame.
• If L is a collection of frames, ϕ is L -valid if ϕ is valid in every frame in L
Different modal logics are characterized semantically as the L-valid formulae, for particular
classes L of frames. To give a first simple example, the logic called T is characterized by the
class of frames having the property that each world is accessible from itself (this amounts to
assume that the relation R is reflexive).
DEFINITION : Properties of a frame. We say that a frame <W,R> is :
• reflexive if wiRwi ; for every wi of W
• symmetric if wiRwj implies wjRwi, for all wi, wj of W
• transitive if wiRwj and wjRwk together imply wiRwk , for all wi, wj, wk of W
• serial if for each wi of W there is at least one wj of W such that wiRwj
• linear if for all wi, wj of W either wiRwj or wjRwi.
These definitions yield the frame-characterization of the modal logics which have the best
reputation, namely: K, D, T, B, K4, S4, S4.3, S5.
LOGIC
K
D
T
B
K4
S4
S4.3
S5.
FRAME CONDITIONS
no conditions are imposed on the frame
serial
reflexive
reflexive, symmetric
transitive
reflexive, transitive
reflexive, transitive, linear
reflexive, symmetric, transitive.
V.1.t1
This brings us to one of the main occupations of modal logicians, which is laying bare the
relations between the validity of the formulae and the properties of frames. Indeed, the
validity of the following formulae characterize the frames mentioned above, in the sense that
they are valid iff the conditions at stake hold :
FORMULA
FRAME CONDITIONS
1. (A→B)→( A→ B)
no conditions
2. A→◊A
serial
3. A→A
reflexive
4. A→ ◊A
symmetric
5. ◊A→ ◊A
symmetric and transitive
6. ( A→ B)∨ ( B→ A)
linear
35
7.
8.
A→
A
A→ ◊ ◊A
transitive
reflexive and transitive
V.1.t2
Remark: It is important to notice that equivalence showed in table V.1.t2 hold for frames.
That is for any model based on the correspondent frame. Not for every model. That is, it is not
the case that e.g. every model for the formula A→A is reflexive. Indeed, consider a model
with only one world not reflexive and where A is true. In such a model; the formula is true but
the world, by assumption, not reflexive.
PROOFS
Let do now what modal logicians like even more than simply lying bare the relations between
the validity of the formulae and the properties of the frames, namely proving this :
Transitivity:
Let us take the model <W, R, v> and show that R is transitive iff for each wi of W the formula
valid at wi.
We start from left to right : If R is transitive in <W, R, v> then
A→
A→
A is
A is valid
Assume that A→
A is true at w1.
If that is the case then we must show that
if A is true at w1, then
A is also true at w1.
1. Assume thus that A is true at w1.
2. If
A is true at w1, then we must show that
A is true at w2 where w1Rw2.
3. If A is true at w2, then we must show that
A is true at w3 where w2Rw3.
4. Because of the assumption at 1 we know that A is true at w1.
5. Because of the assumption of transitivity if A is true at w1, then A it is true at w3, which is what we
needed to prove.
Let us prove the other sense : If
A→
A is valid in <W, R, v> then R must be transitive.
But this is amounts to show that at least one counterexample to
where R is not transitive.
A→
A can be constructed on any frame
So let assume a frame which is not transitive and let us construct a model <W, R, v> based on such a frame:
W: {w1, w2, w3}
R: {<w1, w1>, <w1, w2>, <w2, w3>} – NOTICE that w1 is not in relation to w3
vw1(A) =1 – i.e. A is true at w1
vw2(A) =1 – i.e. A is true at w2
vw3(A) =0 – i.e. A is false at w3
In a diagram:
• •W •W
W1
v(A)=1
2
3
v(A)=1 v(A)=0
V.1.f1
In this model on one hand we have that
true there.
A is true at w1, since w1 and w2 are the accessible worlds and A is
36
On the other hand
false there.
A is false at w1, because
A is false at w2, since w3 is the only accessible world and A is
We thus constructed a counterexample to the formula at stake in a non transitive frame and this amounts to the
proof of our thesis.
K -Frame
For the proof in relation to K, consider that there is no specific frame condition for K. Hence, in normal modal
logic (see chapter VII) amounts to show that the corresponding formula is valid for any of the possible frame
conditions. More precisely
If <W, R> is any frame then (A→B)→( A→ B) is valid
If it were not valid on every frame, there would have to be a model <W, R, v> such that for some wi of W,
1 v( (A→B)=1 at wi
2 v( A→ B)=0 at wi
That is,
3 v( A)=1 and v( B)=0 at wi
But there is no such model. Indeed
If v( B)=0 at wi, then for a wj, such that wiRwj
4 v(B)=0 at wj
Since wiRwj
5 v(A)=1 at wj
Also from wiRwj
6 v((A→B)=1 at wj
But from 4 and 5 we get that v((A→B)=0 at wj, which contradicts 6.
For the other direction of the proof it sufficient to observer that if that the corresponding formula is valid then the
frame could have any possible frame conditions. Indeed, the only way to refute this claim would be to have the
case where the K-formula is valid, but no one of the possible frame conditions (for normal modal logic) hold.
EXERCISE
Prove that the validity of the formulae 2, 3, 4, 5, 6, 8 of table X.2 characterize the indicated
frames.
V.2 Validity on frames: the game theoretical approach
LABELS: In the game-theoretical and in the dialogical approaches to modal logic we will
make use of a device for labelling formulae. The intuitive idea, if we express it in the modeltheoretical language, is that a label i, names a world in some model, and iA tells us that A is
true at the world i names. Moreover, our labels will be finite sequences of positive integers
such as 1.1.1 and 1.1.2 which should indicate that the worlds named by 1.1.1 and 1.1.2 are
accessible from 1.1.
DEFINITION
• A label is finite sequence of positive integers. A labelled formula is an expression of the
form iϕ, where i is the label of the formula ϕ.
37
•
If the label i is a sequence of length >1 the positive integers of the sequence will be
separated by periods. Thus, if i is a label and an n is positive integer, then i.n is a new
label, called an extension of i. The label is then an initial segment of i.n.
GTS
K4:
1) Let us assume that Eloise states ¬( A) v ( A) at world i
2) If there is not accessible world from 1, then Eloise choses right and wins.
3) If Abelard wants to refute the disjunction he must produce a counter model where both sides of the disjunction
are false.
4) If the left side has to be false, then A has to be true in all worlds accessible from i. That is, Abelard has a
winning strategy on the left side iff A is true in all j such that iRj.
5) Let us assume an arbitrary model where the world 1.1.1 is accessible from 1.1, and both are accessible from 1
6) Either A or not A holds at 1.1.1.
7) Because of 4 we know that A has to be true there. (If A would not hold there, Abelard could not have a
winning strategy for the left side of the conjunction and Eloise would win choosing left)
8) Then Eloise chooses right at 1. But then this engage her to show A at 1.1. The latter amounts to defend A at
1.1.1. But A is true and thus she wins there.
To proof of the other sense, that is: If A→
A is valid in <W, R, v> then R must be transitive, amounts to
show that at least one counterexample to A→
A can be constructed on any frame where R is not transitive.
The proof follows from the same countermodel we used in the non-GTS proof. Let us thus describe once more
the following countermodel:
W: {w1, w2, w3}
R: {<w1, w2>, <w2, w3>} – NOTICE that w1 is not in relation to w3
vw1(A) =1 – i.e. A is true at w1
vw2(A) =1 – i.e. A is true at w2
vw3(A) =0 – i.e. A is false at w3
EXERCISE: We leave to the reader to show that Eloise does not have a winning strategy
Example2: Prove that A→ ◊A characterizes B-frames (symmetric frames).
Let us prove that if the frame is symmetric Eloise has a winning strategy for at ¬A∨ ◊A 1
Indeed, if the A is false at 1, then Eloise wins by choosing the left disjunct.
If A is true Eloise wins by choosing right. Since for any world chosen by Abelard, say 1.1,
Eloise can defend ◊A there by choosing 1 (by the assumption of symmetry 1 is accessible
from 1.1) as witness of the possibility operator.
EXERCISE: We leave it to the reader to prove the other direction of the proof.
V.3 Validity on frames: the dialogical approach
Introduction to modal dialogic
Modal dialogic is a systematic account of an explicit notion of context, in the sense that the
latter is introduced by an explicit label. Modal moves are hence dialogical expressions with a
supplementary label, indicating the context in which the move has been made. The usual
modal operators are then defined in the following way:
38
,◊
Attack
Defence
X Ai
( A has been stated by
player X at context i)
Y ? j (iRDj) i
(at the context i the
challenger Y attacks by
choosing a dialogically
accessible context j)
XAj
(the defender claims that
A holds at the label j)
X ◊A i
(◊A has been stated by
player X at context i)
Y?◊ i
(at the context i the
challenger Y attacks
asking X to choose a j
where A holds)
XA j(iRDj)
(the defender chooses the
context j such that j is
dialogically accessible
from i)
V.3.t1
[add states of the game for modal dialogic]
In modal dialogic the frame conditions implemented as special structural rules which allow
the Proponent to increase his choice possibilities while challenging a necessity operator or
defending a possibility operator.
DEFINITION
If at i the Opponent while challenging a necessity operator of defending a possibility operator
chooses a new label j such that i is a proper initial segment of j we say that the Opponent has
introduced j and conceded that the label j is dialogically accessible from the label i (for short
iRDj):
,◊
Attack
D
Defence
P Ai
( A has been stated by P
at context i)
O ? j (iR j) i
(at the context i O
introduces j and
concedes iRDj))
PAj
O ◊A i
(◊A has been stated by O
at context i)
P?◊ i
(at the context i P asks O
to choose a j where A
holds)
OA j (iRDj)
(at the context i O
introduces j and
concedes iRDj)
V.3.t2
MODAL FORMAL RULE
At label i the Proponent may choose a label j such that iRDj iff j has been introduced by the
Opponent before or this choice has been allowed by the appropriate modal structural rule.
39
,◊
Attack
D
j (iR j)
O Ai
P?
i
( A has been stated by O at the context i P chooses
at context i)
a j iRDj such that j has
been introduced by O or
has been allowed by the
appropriate structural
rule)
P ◊A i
(◊A has been stated by P
at context i)
Defence
OAj
O?◊ i
PA j (iRDj)
(at the context i O asks P (at the context i P
to choose a j where A
chooses a j iRDj such
holds)
that j has been
introduced by O or has
been allowed by the
appropriate structural
rule)
V.3.t3
LOGIC
MODAL STRUCTURAL RULES
K
No conditions
D
The Proponent may choose a label i though it has NOT been chosen
by the Opponent before.
Assume that P is at i. P may then choose i.
Assume that P is at i.n. P may then choose i.n and
P may then choose also i (i.n is the immediate extension of i).
Assume that P is at i. P may then choose a j such that i is an initial
segment of j.
Assume that P is at i. P may then choose i and he may choose j such
that i is an initial segment of j .
For short an formulating both conditions at once: P may choose a j such
that i is an initial segment (proper or otherwise) of j.
For the reflexive and transitive cases take the structural rules for S4frames. For the linear condition: Assume that P is at i and also
assume that O conceded that iRDj and iRDk. P may then ask O to choose between
conceding either jRDk or kRDj.
Take the structural rules for the reflexive-, symmetric- and transitiveframes.
For short: Assume that P is at i. Then P can choose any (already introduced) label j
(even j=i).
T
B
K4
S4
S4.3
S5.
V.3.t4
Notice that if we add the classical and the intuitionistic structural rules we obtain
classical and intuitionist versions for each modal logic.
Remarks:
The dialogues for the logic D might be infinite. Now, one way to think proofs with dialogues is as attempting to
find a countermodel. If the the dialog is infinite the countermodel will be infinite too. This does not mean that
40
the only countermodels in D to, say, A are infinite. For example take a model where there is only one world
with reflexive accessibility and where A is false. The standard dialogue counter proof will not find this
countermodel. The point is that in the standard description of the modal dialogue the Opponent has to construct
the countermodel with the choices allowed by the correspondent structural rules. Similar can happen in K4 and
S4. There is way to avoid such dialogues with help of an appropriate repetition rule and a finishing rule. The
following will do:
DEFINITION:
Given a dialogue; say a modal formula has been strictly repeated if:
A possibility (necessity) operator is defended (attacked) with the introduction of a new label i.k though the same
operator (with the label i) has been already defended (attacked) with the introduction of the then new prefix i.j.
Modal Repetition Rule:
No strict repetitions of modal formulae are allowed
Modal Finishing Rule:
Add to the standard definition of finished game the following:
We say that a dialogical game is finished if the set of formulae associated with the prefix i...n duplicates the set
associated with some proper initial segment of i…n.
This device is closely related to the notion of “termination of a branch” implemented by M. Fitting (1983) who
proves that every branch of a special systematic tableau terminates (Fitting 1983, 414-416) in the sense that we
can tell that a branch will become periodic without constructing all infinitely all much of it. We will prove this
too for dialogues while proving completeness.
There are two particles which have a modal flavour and which will useful for our proof. We
will come to them later more thoroughly while discussing connexive logic
THE OPERATORS V AND F
We introduce here the defensibility operator V and the attackability operator F. The operator
F is related to the well-known failure operator of Prolog.ii
We will first introduce the corresponding particle rules:
1. The operator V
In stating the formula VA the argumentation partner X asserts that A can be defended under
certain conditions. The other argumentation partner Y challenges VA by asserting that there is
no condition under which A can be defended, that is, the challenger asserts that attacks on A
can be played successfully independent of what the conditions are. Thus, the challenge of Y
compels X (who stated VA in the so-called upper section) to open a subdialogue where he (X)
states A and Y attacks A. Now, because of the scope of challenge which extends to any
condition, the challenger must play formally. Graphically:
41
Attack
VA
Defence
?V show A
Subdialogue
Subdialogue
(The challenger must
play formally in the
subdialogue)
A
(The defender
chooses the
subdialogue)
V.3.t5
Notice that upper sections and their subdialogues are sections of just one dialogical game
where one of the argumentation partners wins or loses.
Notice also that the particle rules of the operator V allow a change in the right to introduce
atomic formulae, that is, the Proponent is in this version of dialogical logic the argumentation
partner who stated the thesis which motivated the whole dialogue game, not the
argumentation partner who plays formally. Thus the formal structural rule has to be
reformulated. For our present purposes we will introduce a graphic mark that signalises which
of the argumentation partners has to argue formally – let us call this restriction formal
restriction. We will do this by indicating with the subscript “N” (noir) the player who plays
under the formal restriction – the subscript “B” (blanc) will be used as the dual. By means of
this device both cases of arguing with the operator V (with and without changing the formal
restriction) can be distinguished. In order to keep track of different sections of the dialogue
game we will enumerate them in the following way: the initial dialogue section where the
Proponent stated the thesis which motivated the whole dialogue game carries the number I
and will be called the initial dialogue. The immediate subdialogue of the upper section n
carries the number m=n+1 where n and m are roman numbers..
Case 1:
N
m
XB
YN
...
?V show A
VA
XN
YB
A
...
V.3.t6
Case 2:
N
XN
...
YB
VA
?V show A
m
XN
YB
A
...
V.3.t7
42
43
2. The operator F
The operator F is the dual of V. Thus, in stating the formula FA the argumentation partner X
asserts that A can be attacked successfully under certain conditions. The other argumentation
partner Y challenges FA by asserting that there is no condition under which A can be attacked
successfully. Thus, the challenge of Y compels X to demand Y to defend A (by means of a
counterattack). in a sudialogue opened for that purpose Again, the challenger must play
formally:
Attack
FA
Defence
?F
?show A
Subdialogue
Subdialogue
A
(The challenger must
play formally in the
subdialogue)
(The defender
chooses the
subdialogue)
V.3.t8
Again two cases (with and without changing the formal restriction) should be distinguished
here:
Case 1:
N
XB
YN
...
?F
FA
?show A
m
XN
YB
A
...
V.3.t9
Case 2:
N
XN
...
?F
YB
FA
?show A
m
XN
YB
A
...
V.3.t10
VALIDITY AND THE MODAL STRUCTURAL RULES: PROOFS
44
Let us now prove with dialogues that the validity of the formulae of table X.2 characterize the
the frames as implemented by the modal structural rules of table X.3.
T –intuitionist
From left to right :
If we apply the structural rule for T then there is a winnings strategy for A→A (i.e. is valid)
O
1
1
1 A
3 A
P
0
A→A
A
? «1
(0)
(4)
(2)
1
1
1
P WINS
V.3.f1
From right to left
Let us prove that if there is winning strategy for
(reflexivity) applies.
A→A then structural rule for T
But, as we already mentioned in the model theoretical proof this amounts to show that at least
one counterexample to A→A can be constructed on any not reflexive frame. One way to
carry out our dialogical proof is to let the Proponent assert explicitly this point. Namely by
stating the following
the formula to A→A cannot be proved if the dialogue has to be performed assuming
that the starting context is w and that w has no access to itself.
Making use of the operator F introduced and using above this can be written as
F w: A→A [w has no access to itself]
The first move of the Opponent will challenge this assertion and now he has the
burden of the proof. Indeed, he (the Opponent) must now engage himself in a dialogue where
he states A→A at w as thesis and which will be played with the rules at his choice but under
the assumption that w has no access to itself. The opening of the dialogue, call it II, will be
considered as part of the challenging move and the finishing of II as defence. Moreover, if II
ends up with a win of the Proponent we will write II to close the round opened by the
challenge on F.
45
We describe the corresponding dialogue below and recall that we indicate with the
subscript “N” (noir) the player who plays under the formal restriction. – the subscript “B”
(blanc) is the dual:
I
OB
PN
F i: A→A [i has no access to itself] (0)
II
(2)
(1) ?F II
II
D
ON
[1 has no access to itself]
1
1.1
(0) A→A
PB
[1 has no access to itself]
I
II
D
0
1
1.1
0
(2)? « 1.1
1
A
(1)
A
(3)
P wins the whole dialogue
V.3.f2
The Opponent, attacks the F operator by opening the subdialogue II, where he has accepted to
defend the formula at stake and under the assumption of seriality. Indeed, seriality is in fact
the only move which allows him to continue in the game. However this does not yield a
winning strategy and the proponent can state II in the move I(2) to close the round opened by
the challenge I(1).
EXERCISE: The countermodel constructed is pretty straightforward. Can you write it down?
For the proof in relation to K, we have to show that if there is no specific structural condition
then there is a winnings strategy for (A→B)→( A→ B) (i.e. is valid)
- K -intuitionist:
O
1
1
1
1
1.1
1.1
1.1
(1) (A→B)
(3) A
(5)? « 1.1
(7) A→B
(9)A
(11) B
P
0
2
4
(A→b)→( A→ B)
A→ B
(2)
B
(4)
B
(12)
1 ? « 1.1
(6)
3 ? « 1.1
(8)
7 A
(10
(0)
1
1
1
1.1
1.1
1.1
1.1
P WINS
V.3.f3
EXERCISE
Prove that the validity of the formulae B. D; K4.3, S4 characterize the frames as formulated in
the correspondent structural rules.
46
V.4 Propositional modal dialogic and sequent-calculus
V.5 Soundness and completeness for propositional modal dialogic
Each of the modal logics we have described has been specified in three quite different ways:
using models; using GTS, and using tableaus. We need to know that these three ways agree.
Actually the proof between the GTS and the modeltheoertical has been already accomplished
but we would like to establish a similar result for the dialogical proof system. Moreover, we
will prove then that a given formula is K-valid iff there is a dialogical proof for this formula
using the correspondent structural rules for K. This type of result is about a given proof
system rather than using it to prove formulae and is called a metalogicdal result.
Soundness of K
When we say in metalogic that we prove that a given proof system is “sound” we mean that
with this system we cannot prove any formula it should not. For example; if our dialogical
proof system for K were not sound then we would be able to prove some formula, such as
A→A, which is beyond of the frame validity characterizing K. More precisely, to say that
our dialogical proof system is sound means: if a formula ϕ has a dialogical proof for K; then
this formula is valid in the logic K as described by the modeltheoretical characterisation of
K-validity.
Definition 1:
Let us consider an interpretation I of modal propositional logic MPL in a given model <W,R,v>. Let us
further assume that the notion of formula in MPL has been extended to include signed formulae such as iO A∧B,
i.nP A∨B, j.nP A∨B, etc. (where i, j, n, are positive integers)
•
Thus, Say that I is faithful to a branch B of a given semantic tableau iff there is a map f assigning to
each label i of a formula in the branch some possible world f (i) in W such that:
1. for every signed formula iXA, (where X signalises that the formula is O- or P-signed), on the
branch; v(A)=1 at f(i) in I, that is A is true at the world f(i) of the model I. – where vi (OA)=vi
(A)and vi (PA)= vi (¬A).
2.
if i and i.n is on the branch, then f(i)R f(i.n) in I.
We say here the f shows that I faithful.
Soundness lemma:
If I = <W,R,v> is faithful to a branch of a tableau B, and a tableau-rule is applied to B; then I
is faithful to at least one of the branches generated
PROOF:
The proof is by cases. That is; by the consideration of all the ways to produce a branch by the
application of the tableau-rules to a formula at the end of another branch. Namely by the
47
application of O-conditional-rule, a P-conditional-rule, a O-necessary-rule; a P-necessaryrule and a O-possible rule, and a P-possible rule.
1) Let us start with iP(A→B). And suppose that I is faithful to the branch B1.
If we apply the correspondent rule we will produce the branch B1.1containing the formulae:
iOA
iPA
We have thus to prove that if I is faithful for B1, where v(P(A→B))=1 at f(i) in I., then I is faithful too for B1.1,
where iOA and iPA occur. Hence, we have to prove that under these conditions
v(OA)=1 at f(i) in I., i.e. v(A)=1 at f(i) in I. and
v (P(B)=1 at f(i) in I., i.e. v (¬(B)=1 at f(i) in I.
Now, we assumed that v(P(A→B))=1 at f(i) in I. But as mentioned above this amounts to
v(¬(A→B))=1 at f(i) in I..
If v (¬(A→B))=1 at f(i) in I., then
v(A)=1 at f(i) in I - that is v (OA)=1 at f(i) in I - and
v(B)=0 at f(i) in I
If v(B)=0 at f(i) in I, then
v(¬(B)=1 at f(i) in I - that is v (PA)=1 at f(i) in I
But then; the proof has been achieved. Indeed we proved that assuming v(P(A→B))=1 at B1, then the evaluation
function v is faithful to the branch generated B1.1. Indeed at B1.1 it is the case that v(OA)=1 and v(PB)=1 and
thus v is faithful for this branch.
2) Let us start with iO(A→B). And suppose that iO(A→B)is faithful to the branch B1.
If we apply the correspondent rule we will produce two branches B1.1and B1.2 containing respectively the
formulae:
iPA
iOA
We have thus to prove that if v is faithful for B1, where v(P(A→B))=1 at f(i), then v is faithful too at least one of
the branches. Hence, we have to prove that under these conditions either
v(PA)=1 at f(i), i.e. v(¬A)=1 at f(i) or
v(OA)=1 at f(i), i.e. v(A)=1 at f(i)
Now, we assumed that v(O(A→B))=1 at f(i). But as mentioned above this amounts to
v(A→B)=1 at f(i).
If v((A→B))=1 at f(i), then either
v(A)=0: v(¬A)=1- that is v(PA)=1 at f(i)
or v(B) =1: v(OB) =1 at f(i)
In the first case v is faithful to B1.1 and in the second to B1.2.
Quod erat demonstrandum. Indeed, v will be then faithful to at least one of the branches generated.
3) Let us now assume iO◊A and suppose that iO◊A is faithful to the branch B1.
If we apply the correspondent rule we will produce the branch B1.1 containing i.nOA, where i.n is a new label.
Hence, f(i.n) is not defined but by assumption we know that v(◊A)=1 at f(i). From the latter it follows that
v(◊A)=1 at some possible world wj such that f(i)Rwj.
Define a new mapping f* as follows:
48
•
•
For all the labels occurring in B.1, let f* be the same as f.
Since i.n did not occur in our original branch we can define the new mapping as we please. Let us make
the choice then that f*(i.n)=wj.
Notice that, since the new mapping f* agrees with f on all labels of B1 also f* shows that I is faithful to B1.
Further, f*(i)R f*(i.n), since f*(i)= f(i), f*(i.n)=wj and f(i)Rwj.
It follows then that f* shows that I is faithful to the branch B1 extended to B1.1 with i.nA.
◊
The other cases are left to the reader.
Soundness theorem:
VI The Name of the World: Dialogues and Hybrid languages
VII Non-Normal Dialogics for a Wonderful World
The aim of this chapter is to offer a dialogical interpretation of non-normal modal logic which will suggest
some explorations beyond the concept of non-normality. This interpretation will be connected to the
discussion of two issues, one more philosophical and the second of a more technical nature: namely a
minimalist defence of logical pluralism and the difficulties involved in the application of the so-called
Hintikka strategy and hybrid languages while constructing tableau-systems for non-normal modal logics.
At the end of the 19th century Hugh MacColl (1837-1909), the father of pluralism in formal logic,
attempted in the north of France (Boulogne sur mer) to formulate a modal logic which would
challenge the semantics of material conditional of the post-Boolean wave. It seems that in some of his
various attempts MacColl suggested some systems where the rule of necessitation fails.iii Moreover,
the idea that no logical necessity has universal scope - or that no logic could be applied to any
argumentative context - seems to be akin and perhaps even central to his pluralistic philosophy of
logic.iv Some years later Clarence Irwin Lewis furnished the axiomatics for several of these logics and
since then the critics on the material implication have shown an increasing interest in these modal
logics called "non-normal". When Saul Kripke studied their semantics of "impossible worlds" as a
way to distinguish between "necessity" and "validity" these logics reached a status of some
respectability.v As is well known, around the 70s non-normal logics were associated with the problem
of omniscience in the epistemic interpretation of modal logic, specially in the work of Jaakko Hintikka
and Veikko Rantala.vi Actually impossible worlds received a intensive study and development too in
the context of relevant and paraconsistent logics - specially within the "Saint-Andrews- Australasian
connection" in the work of such people as Graham Priest, Stephen Read, Greg Restall and Richard
Routley-Sylvan. Nowadays, though the association with omniscience seems to have faded out, the
study of non-normal logics has received a new impulse motivated through the study of
counterlogicals.
The aim of this chapter is to offer a dialogical interpretation of non-normal modal dialogics which will
suggest some explorations beyond the concept of non-normality. This interpretation will be connected
to the discussion of two issues, namely:
1) counterlogicals as a minimalist defence of logical pluralism (pluralism for a monist) following
the path prefigured by MacColl and
2) the difficulties involved in the application of the so-called Hintikka strategy and hybrid
languages while constructing tableau systems for non-normal modal logics.
49
PLURALISM FOR A MONIST AND THE CASE OF THE
COUNTERLOGICAL
Convincitur ergo etiam insipiens esse vel in intellectu …
Anselm of Canterbury,
Proslogion, capitulum II, Ps 13, 1, 52, 1
(Thus, even he who knows no better will be convinced that
at least it is in the intellect…)
Would the real logic please stand up?
Conceiving situations in which not every mathematical or logical truth holds is a usual
argumentation practice within formal sciences. However, to formulate the precise conditions
which could render an adequate theory of logical arguments with counterpossibles in formal
sciences is a challenging issue. Hartry Field has felt the need to tackle this challenge in the
context of mathematics. Field writes:
It is doubtless true that nothing sensible can be said about how things would be different if there were no number
17; that is largely because the antecedent of this counterfactual gives us no hints as to what alternative mathematics
is to be regarded as true in the counterfactual situation in question. If one changes the example to 'nothing sensible
can be said about how things would be different if the axiom of choice were false', it seems wrong …: if the axiom
of choice were false, the cardinals wouldn't be linearly ordered, the Banach-Tarski theorem would fail and so forth
(Field [1989]; pp; 237)
These lines actually express the central motivation for a theory of counterpossibles in formal sciences.
Namely, the construction of an alternative system where e.g. the inter-dependence of some axioms of a
given formal system could be studied. If we were able to conceive not only a counterpossible situation
where some axioms fail to be true but also even an alternative system without the axioms in question,
then a lot of information could be won concerning the original "real" system. By the study of the
logical properties of the alternative system we could e.g. learn which theorems of our "real system" are
dependent on axioms missing in the alternative one.vii Moreover, I would like to add that a brief survey
of the history of mathematics would testify that this usage of counterpossibles seems to be a common
practice in formal sciences.
The case of the study of counterpossibles in logic called counterlogicals is an exact analogue of the
case of mathematics and motivates the study of alternative systems in the very same way. We learned
a lot of intuitionistic logics, even the insipiens classical logical monist learned about his system while
discussing with the antirealist. This seems to be a generally accepted fact, but why should we stop
there? From free logics we learned about the ontological commitment of quantifiers, from
paraconsistent logic ways of distinguishing between triviality and inconsistency;viii from connexive
logics the possibility of expressing in the object language that a given atomic proposition is
contingently true; from relevance logics that it is not always wise to distinguish between metalogical
and logical "if, then"; from IF and epistemic dynamic logic we learned about arguments where various
types of flow of information are at stake, for linear how to reason with limited resources, and so forth.
Are these alternative logics "real" or even the "true" logic? Well actually to motivate its study the mere
mental construction of them is enough, the mere being in intellectu, provided such a construction is
fruitful. I would even be prepared to defend t he that as a start it is enough if they teach us something
about the logic we take to be the "real" one. The construction of alternative logics, which in the latter
case is conceived as resulting from changes in the original "real" logic, can be thought of as following
a substructural strategy: changes of logic are structural changes concerning logical consequence.
50
In the next section will offer a dialogical interpretation of non-normal logics which should offer the
first steps towards such a minimalist defence of logical pluralism. In this interpretation the pair
standard-non-standard will be added to the pair "normal"-"non-normal". Furthermore, the adjectives
standard and non-standard will qualify the noun logic rather than world, e.g. I will write "the standard
logic Lk in the argumentative context m ". Normal will qualify those contexts, which do not allow the
choice of a logic other than the standard one. Non-normal contexts do allow the choice of a new logic
underlying the modalities of the chosen context. Before we go into the details let us distinguish
between the following different kinds of counterlogical arguments:
1
Assume an intuitionist logician who puts forward the following conditional:
If tertium non-datur were valid in my logic, then the two sides of de Morgan Laws would hold (in my logic) too.
2. We take here once more our intuitionist
If tertium non-datur were valid in the non-standard logic Lk, then the two sides of de Morgan Laws would hold in Lk too.
In the first case the alternative logic − here classical logic − might be thought of as a conservative
extension of the standard one here intuitionistic logic− i.e. any valid formula of the standard logic will
be valid too in the non-standard logic. In the second case this seems to be less plausible: Lk could be a
logic which is a combination of classical logic with some other properties very different from the
intuitionistic ones. The situation is similar in the following cases where it is assumed that the standard
logic is a classical one and the alternative logic can be a restriction:
3
4
If tertium non-datur were not valid in my logic, then one side of de Morgan Laws would fail (in my logic).
If tertium non-datur were not valid in the non standard logic Lj, then one side of de Morgan Laws would fail (in Lj).
Because of this fact it seems reasonable to implement the change of logics by means of a substructural
strategy (akin to the concept of dialogics) - i.e. a strategy where the change of logics involves a change
of the structural properties.ix
Now in these examples the precise delimitation of a logic is assumed as a local condition. However;
the conditional involved in the counterlogical seems to follow another logic which would work as a
kind of a metalogic that tracks the changes of the local assumption of a given logic while building
arguments with such conditionals. The point here is that in this type of study classical logic has no
privileged status. Classical logic might be "the metalogic" in many cases but certainly not here.
VII.1 Dialogics and non-normal logics
Motivation
Let us call non-standard such argumentation contexts (or "worlds") where a different logic
holds relative to the logic defined as standard. Thus, in this interpretation of non-normal
modal logic the fact that the law of necessitation does not hold is understood as implementing
the idea that no logically valid argument could be proven in such systems to be
unconditionally necessary (or true in any context and logic). Logicians have invented several
logics capable of handling logically arguments that are aware of such a situation. The main
idea of their strategy is simple: logical validity is about standard logics and not about the
imagined construction of non-standard ones; we only have to restrict our arguments to the
notion of validity involved in the standard logic. Actually there is a less conservative strategy:
namely, one in which a formula is said to be valid if it is true in all contexts whether they are
ruled by a standard or a non-standard logic. The result is notoriously pluralistic: no logical
argument could be proven in such systems to be unconditionally necessary.
Anyway if we have a set of contexts, how are we to recognise those underlying a standard
logic? The answer is clear in modal dialogics if we assume that the players can not only
choose contexts but also the (non-modal) logic which is assumed to underlie the chosen
context. In this interpretation the Proponent fixes the standards, i.e. determines which is the
(non-modal) standard logic underlying the modalities of a given context. However under
given circumstances the Opponent might choose a context where he assumes that a (non51
modal) logic different from the standard one is at work. Now, there are some natural
restrictions on the Opponent choices. Assume that in a given context O has explicitly
conceded that P fixes the standards. In other words, the Opponent concedes that the
corresponding formulae are assumed to hold under those structural conditions which define
the standard logic chosen by the Proponent: we call these contexts normal. Thus, O has
conceded that the context is normal − or rather, that the conditions in the context are normal.
In this case O cannot choose the logic: it is P who decides which logic should be used to
evaluate the formulae in question, and as already mentioned, P will always choose the logic
he has fixed as the standard one. That is what the concession means: P has the choice.
Notice once more that "standard" logic does not really simply stand for "normal": normality,
in the usual understanding of non-normal modal logic, is reconstructed here as a condition
which when a context m is being chosen restricts the choice of the logic underlying the
modalities of m.
Dialogics for S.05, S.2 and S3
The major issue here is to determine dynamically – i.e., during the process of a dialogue – in
which of the contexts may the Opponent not have to conceded that it is a non-normal one and
allowing him thus to choose a non-modal propositional logic different of the standard one.
This must be a part of the dialogue's structural rules (unless we are not dealing with dialogues
where the dialogical contexts with their respective underlying propositional logic are
supposed to have been given and classified from the start). I will first discuss the informal
implicit version of the corresponding structural rules and in the following chapter we will
show how to build tableaux which implement these rules while formulating the notion of
validity for the non-normal dialogics. Let us formulate a general rule implementing the
required dynamics but some definitions first:
Definitions:
•
Normality as condition: We will say that t a given context m is normal iff it does not allow to choose a (propositional)
logic underlying the modalities of m other than the standard one. Dually a context is non-normal iff it does allow the
choice of a new logic
•
Standard logic: P fixes the standards, i.e. P fixes the (propositional) logic which should be considered as the standard
logic underlying modalities and relative to which alternatives might be chosen.
•
Closing dialogues:
•
No dialogue can be closed with the moves (P)a and (O)a if these moves correspond to games with different logics
•
Particle rules for non-normal dialogics:
The players may choose not only contexts they may also choose the propositional logic underlying the modalities in the
chosen contexts:
,◊
Am
( A has been stated at
context m underlying a
logic Lk)
◊A m
(◊A m has been stated at
context m underlying a
logic Lk)
Attack
? n Lj m
(at the context m the
challenger attacks by
choosing an accessible
context n and logic Lj)
?◊ m
Or in the more formal notation of state of game (see appendix):
52
Defence
ALj n
ALj n
(the defender chooses the
accessible context n and
the logic Lj)
-particle rule: From A follows <R, σ, A, λ*A L j / n >, responding to the attack ? / Lj n stated by the challenger at m
(underlying the logic Lk) and where λ*A L/n is the assignation of context n (with logic Lj) to the formula A, and n and Lj
are chosen by the challenger.
◊-particle rule: From ◊A follows <R, σ, A, λ*A L j / n >, responding to the attack ? ◊ n stated by the challenger at m
(underlying the logic Lk) and where λ*A L/ n is the assignation of context n (with logic Lj) to the formula A, and n and Lj
are chosen by the defender.
The accessibility relation is defined by appropriate structural rules fixing the global semantics (see appendix). To produce
non-normal modal dialogic we proceed by adding the following (structural) rule:
(SR-ST10.O5) (SO5-rule):
•
O may choose a non-standard logic underlying the modalities while choosing a (new) context n with an attack on a
Proponent's formula of the form A or with a defence of a formula of the form ◊A stated in m if and only if m is nonnormal.
•
P chooses when the context is normal and he will always choose the standard logic but he may not change the logic of a
given context (generated by the Opponent). Furthermore, P may not choose a context where the logic is non-standard.
•
The logic underlying the modalities of the initial context is assumed to be the standard logic.
Three further assumptions will complete this rule:
SO5 assumptions
(i)
The dialogue's initial context has been assumed to be normal.
(ii)
The standard logic chosen by P is classical logic Lc.
(iii)
No other context than the initial one will be considered as been normal.
The dialogic resulting from these rules − combined with the rules for T - is a dialogical
reconstruction of a logic known in the literature as S.O5. In this logic validity is defined
relative to the standard logic being classical and has the constraint that any newly introduced
context could be used by O to change the standards. Certainly (a∨¬a) will be
valid. Indeed, the newly generated context, which has been introduced by the challenger while
attacking the thesis, has been generated from the normal starting context and thus will
underlie the classical structural rule SR-ST2C (see appendix). The formula (a∨¬a) on the
contrary will not be valid. P will lose if O chooses in the second context, e.g., the
intuitionistic structural rule SR-ST2I:
contexts
1{Lc}
1.1{Li}
1.1.1{Li}
1.1.1{Li}
O
1
3
5
<? /1.1>
<? /1.1.1{Li}>
<?∨>
a
P
(a∨¬a)
(a∨¬a)
a∨¬a
¬a

0
2
4
6
0
2
4
6
contexts
1{Lc}
1.1{Lc}
1.1.1{Li}
1.1.1{Li}
1.1.1{Li}
The Proponent loses playing
with intuitionistic rules
O wins by choosing in 3 the structural rule, which changes the standard logic into an intuitionistic logic.
Let us produce a dialogical reconstruction of another logic, known as S2, where we assume
not only that the logic of the first context is normal and in general SR-ST10.O5, but also:
(SR-ST10.2) (S2-rule):
•
If O has stated in a context m a formula of the form A (or if P has stated in m a formula of the form ◊A), then the
context m can be assumed to be normal. Let us call (O) A and (P) ◊A normality formulae.
53
•
•
P will not change the logic of a given context (and he may not choose a context where the logic is non-standard) but
he might induce O to withdraw a choice of a non-standard logic by forcing him to concede that the context at stake is
a normal one. .
A normal context can only be generated from a(nother) normal context. .
The first two points establish that a formula like B could be stated by P under the condition
that another formula, say, A holds. In this case O will be forced to concede that the context
is normal and this normality will justify the proof of B within the standard logic. The third
point of the rule should prevent that this process of justification from becoming trivial:
formulae such as (P) ◊A m, or (O) ◊ A m should not yield normality if m is no normal
themselves: the normality of m should come from "outside" the scope of (P) … m and (O)
◊… m.
This is, for our purposes, a more appealing logic than S.05 because it makes of the status of
the contexts at stake a question to be answered within the dynamics of the dialogue. One can
even obtain certain iterations such as ( (a→b)→( a→ b)) which is not valid in S.05, but is
in S2: the first context underlies the standard classical logic by the second S.05 assumption,
the second context too because O will concede a there. Now, because the second context has
been Ls-conceded by O, he cannot choose a logic different of the classical one, and P will
thus win. Adding transitivity to S2 renders S3.
Dialogics for E.05, E2 and E3
The point of the logics presented in the chapter before was not to ignore the non-standard
logics, but only to take into consideration the standard one while deciding about the validity
of a given argument. We will motivate here a less conservative concept, namely, one in which
a formula is said to be valid if it is true in all contexts whether they are ruled by a standard or
a non-standard logic. These logics are known as E. In no E system will A be valid for any
formula A.
Suppose one modifies S.05 in such a way that no context is assumed to be normal and thus
every modality will induce a change of logic. This logic, called E.05, is unfortunately not of
great interest: a formula will be valid in E iff it is valid in non-modal logics (think of
(a→b)→( a→ b), which in this logic cannot be proven to be valid). Modality seems not be
of interest there, and this logic can be thought of as a kind of a modal lower limit.
Now the elimination of the assumption that the first context is normal in S2 − that is, take SRST10.O5 and SR-ST10.2 but drop the first and third S0.5 assumptions − yields an interesting
dialogic for our purposes. (a→b)→( a→ b) is valid there, signalising a more minimal
structural condition for the validity of this formula than K (for it does not even assume, as K
does, that validity concerns only contexts with the same kind of logic). Similarly one could
produce D versions, etc. Indeed E2 seems to be the appropriate language where the logical
pluralist might explore the way to formulate statements of logical validity which do not
assume a universal scope
In fact, up to this point; this interpretation only offers a way to explore the scope of the
validity of some arguments when confronted with counterlogical situations, where no middle
term is to be conceived between what is to be considered standard and what not. Moreover,
that a central aim of this dialogic is to explore fruitful counterlogicals seems not to have been
implemented yet. In the next chapter I would like to suggest some further possible distinctions
in order to perform this implementation.
54
Beyond non-normality
Let us take once more the following example, where the standard logic is classical logic:
If tertium non-datur were not valid in my logic, then one sense of double negation would fail (in my logic).
One possible formalisation consists of translating not-valid by "non-necessary". Now the
problem with this example is; that, if P does not change the logic; he can win the (negative)
conditional in, say, S2 in a trivial way. Indeed, O will attack the conditional conceding the
protasis, P will answer with the apodosis and after the mutual attacks on the negation P will
win defending tertium non-datur in classical logic. But then the argument seems not to be
terribly interesting. This follows from the fact that in the interpretation displayed above P may
not change the standard logic once it has been fixed. In general this is sensible because
validity should be defined relative to one standard and we cannot leave it just open to just any
change. Moreover, though there is some irrelevance there this irrelevance concerns only the
formula conceded at the object language: in our case double negation. But what is relevant
and is used is the concession that the standard logic is the one where the classical structural
rule applies. Finally why should P change the logic if he can easily win in the one he defined
as standard?
However, in order to implement the dialogic of counterlogicals, one could leave some degree
of freedom while changing the logical standard without too much complexity and inducing a
more overall relevant approach: a given standard logic may change into a restriction of this
logic. In other words, the standard logic may be changed to a weaker logic where any of its
valid formulae are also valid in the stronger one P first defined as standard. True, the problem
remains that it does not seem plausible that P will do it on principle: on principle he wishes to
win, and if the proof is trivial all the better for him. There are two possibilities:
One is to build a dialogue under conditions determining from the start which contexts are
played under the standard logic and which are the ones where the restriction of the standard
logic hold (fix a model).
The other is to leave O to choose a conservative restriction of the logic P first defined as
standard.
(SR-ST10.2) *:
•
If O has stated in a context m a formula of the form A (or if P has stated in m a formula of the form ◊A), then the
context m can be assumed to be normal. In these cases O might choose once a restriction of the standard logic and P
must follow in his choices the restrictions on the standard logic produced by O.
•
A normal context can only be generated from a(nother) normal context.
In our example O will choose intuitionistic logic and there P will need the concession of double
negation if he wants to prove tertium non-datur. One way to see this point is that O actually tests if in
the substructural rules defining the standard logic there are not some redundancies. Perhaps a sublogic
might be enough.
For the example of this chapter this seems enough but one could even allow such restrictions in the
case of the initial context in S.05. Moreover one could even drop the second S.05 assumption and let P
choose an arbitrary standard logic. Take for example the case
If transitivity were not holding in my logic, then a→
a would fail too (in my logic).
Suppose the standard logic is S4. We should use a notation to differentiate the modality which defines
the standard logic and which is normal from the modalities which are used within the corresponding
non-normal logic. Let us use "∆" (or "∇") for necessity (or possibility) in the standard logic.
55
Furthermore let us use Blackburn's hybrid language to "propositionalise" the properties of the
accessibility relation. We could thus write
¬ (∇∇νi→∇ν)i (transitivity) (in my S4 logic) → ¬ (∆a→∆∆a) (in my S4 logic).
If SR-ST10.2* applies then the Opponent will choose, say, the logic K and the Proponent will win. In
these types of dialogue the Opponent functions more constructively than in the sole role of a
destructive challenger. In fact, the Opponent is engaged in finding the minimal conditions to render
the counterlogical conditional. Actually there has already been some work done concerning the
dialogic adequate for seeking the minimal structural conditions for modal logic. The dialogues have
been called structure seeking dialogues (SSD) and have been formulated in Rahman/Keiff [2003]. In
these dialogues the "constructive" role of the Opponent is put into work explicitly.x
Here is another kind of example:
If the principle of non-contradiction were not valid in my logic, then one sense of double negation would fail (in my logic).
One other way to formalise this would be to put the negation inside the scope of the necessity
operator:
If it were necessary that the principle of non-contradiction does not hold, then it would be necessary that one sense of
double negation will fail.
If we assume here too that SR-ST10.2* applies then the Opponent will choose some sort of
paraconsistent logic (such as Sette's P1). Certainly, the Opponent will lose, anyway but other choices
would lead to a trivial winning strategy of the Proponent.
If, instead of using SR-ST10.2*, we leave the choice of the standard logic open, P might choose any
logic as standard and then it would seem that almost anything goes. It is perhaps not the duty of the
logician to prevent this but the application of SR-ST10.2*and the corresponding SSD can help there,
leaving the Opponent to search for the "right" the structural conditions under which the formula should
be tested.
The point may be put in a different way. In the dialogues of the preceding chapters the role of the
Opponent is to test if the thesis assumes surreptitiously that its validity holds beyond the limits of the
standard logic. In this role the Opponent may choose any arbitrary logic without any constraints. Let
us now assume, that the Opponent, still in the role already mentioned, comes to the conclusion that the
thesis of the Proponent holds as it is. The Opponent can then play a slightly different role and explore
the possibilities of another strategy: he might try to check if the standard logic chosen is not too strong
concerning the thesis at stake. The latter is the aim of the structure seeking dialogues.
The preceding considerations hardly settle the matter of the ways the change of logics can be studied
dialogically. There are many other possible variations − one could for example think that the SSD
would be activated when some problematic assumption of the standard logic arises which might not
actually concern the thesis. This will do for the present though.
VII.2 Hybrid languages and non-normal dialogics
The aim of this section is to discuss the failure of the so-called Hintikka strategy concerning
the implementation of the accessibility relation while constructing tableau systems for nonnormal modal logics. This problematic seems to apply too to the "propositionalisation"
techniques of frame conditions such as practised in hybrid languages.
56
Let us first present the tableaux which result from our dialogic.
2.1 Dialogical tableaux for non-normal modal logics
As discussed in the appendix mentioned, the strategy dialogical games introduced above
furnish the elements for building a tableau notion of validity where every branch of the
tableau is a dialogue. Following the seminal idea at the foundation of dialogic, this notion is
attained via the game-theoretical notion of winning strategy. X is said to have a winning
strategy if there is a function, which, for any possible Y-move, gives the correct X-move to
ensure the winning of the game.28
Indeed, it is a well known fact that the usual semantic tableaux in the tree-shaped structure we owe
to Raymond Smullyan are directly connected with the tableaux for strategies generated by dialogue
games, played to test validity in the sense defined by these logics. E.g.
(O)-cases
Σ,(O)A→B
------------------------------Σ,(P)A, ... | Σ,<(P)A> (O)B
(P)-cases
Σ,(P)A→B
------------------Σ,(O)A,
Σ,(P)B
The vertical bar "|" indicates alternative choices for O, P's strategy must have a defence for both possibilities (dialogues). Σ is
a set of dialogically signed expressions. The signs "<" and ">" signalise that the formulae within their scope are moves but
not formulae which could be attacked. The elimination of expressions like <(P)A> and the substitution of P by F(alse) and O
by T(rue) yields the signed standard tableau for the conditional.
However, strictly speaking, as discussed in Rahman/Keiff 2003, the resulting tableaux are not
quite the same. A special feature of dialogue games is the notorious formal rule (SR-ST4)
which is responsible for many of the difficulties of the proof of the equivalence between the
dialogical notion and the truth-functional notion of validity. The role of the formal rule, in this
context, is to induce dialogue games which will generate a tree displaying the (possibly)
winning strategy of P, the branches of which do not contain redundancies. Thus the formal
rule actually works as a filter for redundancies, producing a tableau system with some flavour
of natural deduction. This role can be generalised for all types of tableau generated by the
various dialogics. Once this has been made explicit, the connection between the dialogical and
the truth-functional notion of validity becomes transparent.
Let us see first the dialogical tableaux for normal logic as presented in Rahman/Rückert 1999
and improved in Blackburn 2001, though the notation there diverges slightly from the present
one:
(O)-cases
(P)-cases
(O) «A m
------------------<(P)?« n #>(O)ALs n
the context n does not
need to be new
(P)«A m
-------------------<(O) )?« n >(P) ALi n
the context n is new
(O)¡A m
--------------------<(P)?>(O) A n
the context n is new
(P)¡A m
------------------------<(O)?>(P)A n#
the context n does not
need to be new
"m" and "n " stand for contexts; "#" restricts the choices of P according to the properties of the accessibility relation which
define the corresponding normal modal logics. Dialogical contexts always constitute a set of moves. These contexts may
have a finite number, or a countable infinity of elements, semi-ordered by a relation of succession, obeying the very well
known rules which define a tree. The thesis is assumed to have been stated at a dialogical context which constitutes the origin
57
of the tree. The initial dialogical context is numbered 1. Its n immediate successors are numbered 1.i (for i=1 to n) and so on.
An immediate successor of a context m.n is said to be of rank +1, the immediate predecessor m of m.n is said to be of rank -1,
and so on for arbitrarily higher (lower) degree ranks.
I will leave the discussion of how to specify # for the next section and display now the
tableaux for non-normal dialogics:
(O)-cases
(P)-cases
(O)«A m
------------------<(P)?« n #/ Ls >(O) ALs n
the context n does not need
to be new
the logic at m is the standard
logic Ls
(P)«A m
-------------------<(O) )?« n / Li >(P) ALi n
the context n is new
the logic Li is different from
the standard one Ls iff m is
non-normal
(O)¡A m
--------------------<(P)?>(O) ALi n
the context n is new
the logic Li is different from
the standard Ls iff m is nonnormal
(P)¡A m
------------------------<(O)?>(P) ALs n#
the context n does not need
to be new
the logic at m is the standard
logic Ls
We need the following rule concerning closure:
•
Closing branches:
No branch can be closed with the moves (P)a and (O)a if these moves correspond to games with different logics.
To produce S.05 add to the adequate implementation of the accessibility relations the
following:
SO5 normality conditions:
1. The dialogue's initial context has been assumed to be normal. No other context than the initial one will be considered
as being normal.
2. The standard logic chosen by P is classical logic Lc.
3. The Proponent may not:
•
choose a context where the logic is different of the standard one;
•
change the logic of a given context m if m has been generated from a non-normal context.
To produce S2 add to the SO5-rule the following:
(S2--normality conditions):
•
If O has stated in a context m a formula of the form A (or if P has stated in m a formula of the form◊A), then the
context m can be assumed to be normal.
•
A normal context can only be generated from a(nother) normal context.
The construction of the other tableaux is straightforward.
2.2 On how not to implement the accessibility relations
In dialogics, the properties of the accessibility relation could be implemented in the following
way:
(SR-ST9.2K) (K): P may choose a (given) dialogical context of rank +1 relative to the context he is playing in.
58
(SR-ST9.2T) (T): P may choose either the same dialogical context where he is playing or he may choose a (given) dialogical
context of rank +1 relative to the context he is playing in.
(SR-ST9.2B) (B): P may choose a (given) dialogical context of rank -1 (+1) relative to the context he is playing in, or stay in
the same context.
(SR-ST9.2S4) (S4): P may choose a (given) dialogical context of rank >+1 relative to the context he is playing in, or stay in
the same context.
(SR-ST9.2S5) (S5): P may choose any (given) dialogical context.
Moreover we could e.g. build the transitivity part of the rule for S4 in the tableau rule in the
following way:
(O)« A m
n = m >+1
-----------------------------<(P)?« n >(O)ALs n
Actually, there is another technique to implement this and which is connected with the idea of
finding in the object language formulae which express frame conditions: the idea has been
used by Hintikka for the construction of tableaux and is thus known today as Hintikka's
strategy. The idea is a bold one and captures the spirit of the axiomatic approaches. Let us
formulate the rule in Hintikka's style leaving aside for the moment the choice of the logic:
(O) «A m
n = m >+1
-----------------------------<(P)?« n >(O) «A n
That is, if «A holds at m then it should also hold at the context n provided n is accessible from m. The rule stems from the
idea that transitivity is associated with the validity of the formula: «A→««A.
The "up-wards" transitivity of S5 can be formulated simalry. Actually, the only device one
needs is the one concerning K. Then, as soon as context has been "generated" the rules
defining the other modal logics tells what formulae can be used to fill the opened context Hintikka speaks of "filling rules. The simplicity and conceptual elegance of this strategy had
made it very popularxi and it is connected with a more radical formalisation strategy such as
that of hybrid languages.xii In the latter, the point is to fully translate the properties of
accessibility relations into the object language of propositional modal logic, which has been
extended with a device to "name contexts" such as "@m ". The idea behind the @ operator is
to distinguish the assertion that a given formula A can be defended in the dialogical context m
from the dialogical context n where the assertion has been uttered – which could be different
from m.. Properties of the accessibility relation can in this case be formulated as propositions.
One problem for the general application of Hintikka's strategy is that there are some frame
conditions like irreflexivity, asymmetry, antisymmetry, intransitivity and trichotomy which
are not definable in orthodox modal languages. The aim of hybrid languages is to close this
gap by enriching the modal language and apply then Hintikka's strategy.
The hybrid strategy seems at first sight, very appealing to our interpretation of non-normal
modal logic where the concession of normality actually amounts to the concession of a rule
defining the corresponding standard logic. If the standard logic is a modal one, then the
concession, when formulated in the style of hybrid languages, amounts to add a premise.
Now, if it is indeed a premise (stating frame conditions) then it seems a good idea to have this
premise expressed in the same language as the other premisses. For example in the following
way:
59
(O) «A @m
¡¡n→¡n @m
-+----------------------------<(P)?« n >(O) A @n
However, the application of both the Hintikka and the hybrid strategy in the context of nonnormal logic should be done very carefully. If not we might, say in the S3, convert a nonnormal context into a normal one by the assumption that the accessibility relation is
transitive.xiii Moreover, we would come to the result that every non-normal logic with
transitivity collapses into normality. But normality is a condition qualifying worlds and not
about accessibility. In fact the point of logic as S3 is that we could have transitivity without
having necessitation. Certainly, defenders of Hintikka's and hybrid strategies might fight back
introducing the proviso that their rules apply under the condition that the contexts in question
are normal. In fact, Fitting uses such a strategy in his book of 1983 (274).
Anyway, this loss of generality awakes, at least to the author of the paper, a strange feeling. A
feeling of being cheated: Transitivity talks about accessibility between contexts and not about
necessitation in normal contexts. Hybrid languages seem to be the consequent and thorough
development of a notion akin to Hintikka's strategy and perhaps pay the same price. Indeed, in
the language of dialogics we would say that the propositionalisation of frame conditions
amounts to producing a new (extension of a) logic without really changing either the local or
the global semantics. It is analogue to the idea of producing classical from intuitionistic
dialogic just by adding tertium non datur as a concession (or axiom) determined by the
particular circumstances of a given context. Indeed, with this technique we can produce
classical theorems within the intuitionistic local and global (or structural) semantics. Assume
now that we are in the modal dialogic K and that in a given (dialogical) context the
Opponent has attacked a necessary formula a∨b of the Proponent. Assume further that the
Proponent has at his disposal a filling rule which allows him to "fill" this very context with a
necessary formula of the Opponent, say, b.xiv Then obviously, P will win and strictly
speaking, from the dialogical point of view, he always remains in K. One other way to see this
is to realise that, what the "filling rules" do, is to allow appropriate "axioms" to be added to
some contexts specified by these rules in order to extend the set of theorems of K without
changing its semantics. As already acknowledged, the idea is elegant and perspicuous but it
simply does not work so straightforwardly if non-normal contexts are to be included. Perhaps
we should even learn from all this exercise that converting frame conditions into propositions
drives us to a notion of the relation of accessibility which does not yet seem to have been fully
understood.xv
Appendix:
A. Soundness and completeness of the tableaux systems
The tableau systems for non-normal logics presented above are essentially those of Fitting [1983], Girle [2000] and Priest
[2001] without the use of Hintikka's strategy for the accessibility relation of the first two authors. I will not rewrite the proofs
here and rely on the proofs of Fitting[1983] and Priest [2001]. What I will do is to show how to transform the dialogical
tableaux into the ones of the authors mentioned above. To see this notice that if the Opponent (=T in the signed non
dialogical version of the tableau) is clever enough, on any occasion where he may choose a logic he will choose one, where
he assumes that the Proponent (=F in the signed non dialogical version of the tableau) will lose. In fact, if the tableau systems
are thought as reconstructing the usual notion of validity of non-normal modal logic we must assume that it will be always
the case that if O chooses a logic then P will lose − however, notice that dialogically we must not assume this: O might lack
some information and choose the wrong logic. One way to implement the assumption of the cleverness of the Opponent
slightly more directly is to forbid P to answer to an attack on a necessary formula (or to attack a possible° formula of the
Opponent) stated at a context m unless this context is normal. Moreover, if we are interested in freeing ourselves from the
interpretation of the contexts as representing situations where logic could be different, or more generally from any
60
interpretation concerning the "structural inside" of non-normal contexts, the rules will amount to the following simplified
formulation:
(O=T)-cases
(P=F)-cases
(O=T)¡A m
--------------------<(P)?>(O=T) A n
the context n is new
the rule is activated iff m is
normal
(P=F)«A m
-------------------<(O) )?« n > (P) n
the context n is new
the rule is activated iff m is
normal
Furthermore, if we delete from the tableau the expressions <(P)?>( and <(O) )?« n >, which have only a dialogical
motivation, we have the usual tableau systems mentioned above.
REFERENCES
VII.3 Subnormal logics:
VII.3.1 The logic N
The point of this logic is the following in the dialogical framework is the following
It is a logic about deriving from premises. The premises are treated as if they and only they
were necessarily true and this, with S4-force.
Other O-necessity cannot be challenged. BUT WE P CAN WIN IF HE HAS TO DEFEND A
NECESSARY FORMULA AND THE OPPONENT HAS ALREADY CONCEDED IT – but
this only if it is the same. It is as if necessary formulae which are not premises are to be
considered as atomic formulae. If a given premise contain already a necessity; it is like having
two necessities operators. The first one of the premise and the second one cannot be
challenged
It is a bit like something between the logis S05 and the E-systems. The only world conceded
to be normal is the one where the premises have been established. Not the premises won by a
conditional.
VII.3.2 Conditional logics
VIII Bisimulation and IF-Propositional Modal Logic
VIII.1 Bisimulation and modal Logic
VIII.2 IF-propositional modal Logic
IX First Order Modal Logic
The glory and condemnation of first-order modal logic is directly dependent on the notion
of scope. Indeed this notion regulates the interaction of quantifiers and the modal operators
and determines even the features of the logic one is aiming at. Do we want that the domain of
the individuals of each world is different or the same.? Do we prefer a domain which
increases its members from world to world or rather a decreasing one? Do we want that the
interpretation of the terms of our language is always the same fixed object of the domain or do
we want that the interpretation of the very same term might change from world to world?
What does exactly this difference mean? Do we allow terms to designate nonexisting objects?
61
These and related alternatives are all related to the notion of scope but the phenomena seem to
be quite different and the notion of scope involved less than uniform. As already mentioned in
the first chapter the main critics against modal logic stem from the skepsis towards a sensible
notion of scope.
We will first discuss all these different possibilities and the known and not so well known
solutions to them. Then we will attempt to offer a more uniform instrument to study these all
various phenomena of scope by means of IF-First-order modal logic.
IX.1
IX.2
IX.3
Varying and constant domains
Dialogues and games for varying and constant domains
Existence, equality and Barcan again
IX.3.1 Equality and Frege-Kanger’s challenge
Equality
Let us assume that are two copies of the Finnish epic poem Kalevala on the desk. Given this it
would not be unreasonable for someone to say that both items were the same; they are, after
all; copies of a single work. On the other hand, it is no less reasonable for someone to say that
these items are not the same. They are distinguishable in all sorts of ways from one another,
not at least by being located in different places. Moreover, more generally; when we say that
two students own the same book, we might mean either they own the same work, such as, the
Kalevala or they are joint owners of a single copy of the Finnish epic poem. What all this
shows is that the words the same can be used in somewhat different ways, in the former case,
same means qualitative identity or equivalence; in the latter case, it means quantitative or
numerical identity, usually called equality. Keeping this in mind let us now assume that there
is only one copy of Kalevala on the desk and that some person with logical inclinations
describes this situation using the first order sentence Pkn and another person as Pkm . What do
you think should be said about the relationship between the object designated by kn and km?
The answer is pretty obvious, the objects designated are the same. Furthermore, in this reply,
the expression the same is used in the numerical way (notice that the question was about the
objects designated by kn and km and not about the names “kn” and “km” of these objects).
Despite the difference between the qualitative and the numerical identity and despite the
special philosophical place given to the notion of equality, the latter can be seen as a special
case of the former. Indeed, qualitative identities or equivalences are relations which are
reflexive, symmetric and transitive.
DEFINITION IX.3.d1
A relation ~ on a set of objects is an equivalence relation iff for all x, y ∈Σ
x ~ x, i.e. ~ is reflexive
If x ~ y, then y ~ x, i.e. ~ is symmetric
If (x ~ y and y ~ z), then x ~ z, i.e. ~ is transitive
The use of equivalences relations on a domain serves primarily to structure a domain into
subsets whose members are regarded as indiscernible with respect to that relation. More
precisely, for every equivalence relation there is a natural way to divide the set on which it is
defined into mutually exclusive (disjoint) subsets which are called equivalence classes. By
dividing a set into mutually exclusive and collectively exhaustive nonempty subsets we
produce what is called a partition of that set. The elements of an equivalent class are
62
equivalent to one another (and to none outside) in the way defined by the given equivalence
relation. For example, if the relation ~ is same author and our Σ is the set of books authored
by Latin-American writers (where books by co-authors have not been included in the set) , the
relation ~ partitions Σ into the set of books written by Jorge Luis Borges, Julio Cortázar,
Eduardo Galeano, Gabriel García Márquez, Manuel Scorza, Rodolfo Walsh and so forth.
Every pair of distinct equivalence classes is disjoint; because each book of the set Σ having
being written by only one author, belongs exactly to one equivalence class. This is so, even
when some author, such as Alejandro Lemus, has written only one book, consequently
occupying an equivalence class all by himself. The latter example on an equivalence class
containing only one element is connected with the notion of Equality or numerical identity:
DEFINITION IX.3.d2
Equality or numerical identity is the smallest equivalence relation, so that each one of
the equivalence classes is a singleton, i.e., each contains one element
From the point of view of the semantics of first-order logic a strong sense of equality is
intended here: by kn=km we do not mean that the entities to which both constants refer are
identical in the sense that they resemble each other very closely, like identical twins, for
example. What we mean is that they are the same, so that the valuation of the equation is true
just in case the interpretations of the constants involved in the equation are the same:
vM (kn=km)=1 iff (kn)= (km)
Now, quite frequently; first-order logic is extended to logic with equality by the introduction
of two axiom-schemata. Namely:
Identity:
x=x
→φ(y))
Leibniz’s Law: (x = y) → (φ(x)→
Notice that we expressed the axioms with free variables. The reasons for this choice will be
clear in the next paragraph but to advance a bit the point consider that the axioms regulate
relations between objects and not between names for objects.
The first axiom gives us reflexivity and the second symmetry and transitivity. The problem
with these axioms is that it is questionable if they really capture the strong sense of equality
mentioned above. In the second axiom-schema φ can be substituted by the available
predicates of the theory. But these predicates might not be sufficient to discern distinct
objects.
Suppose, for example, that we have only two mutually exclusive predicates “x is a president”
and “x is a queen”. This theory can distinguish between Queen Elisabeth II and president
Georg Bush Jr., but cannot distinguish between, say, the legitimate Argentinean president
Arturo Illia who governed from 1963 to 1966 and the illegitimate Argentinean president who
ruled as a dictator from 1976 to 1980. Since the theory cannot distinguish between individuals
that belong to the same species (in our case the species of presidents and the species of
queens), it is sensible to interpret x = y as the equivalence relation x is of the same species as
y. Clearly, this is not the notion of equality intended. The problem is that the language of our
theory has not enough predicates. If we want to have equality in the strong sense we would
like to have a stronger formulation of Leibniz’s law. Namely, one which establishes the x is
equal to y iff they have all their properties in common. That is,
63
Leibniz-Peirce Law: (x = y) iff ∀φ (φ(x)↔
↔φ(y))
Unfortunately this formal version of the definition of equality, which by the way has been
already formulated with precision by Charles Saunders Peirce (3.399), cannot be expressed in
first-order logic. Indeed in first-order logic we cannot speak of all properties. Indeed, in firstorder logic we cannot even quantify over predicates.
However the situation is not that hopeless for first-order logic. All the usual principles
governing equality are obtainable without necessarily interpreting = equality in the strong
sense. It is enough to interpret = as an equivalence relation that cannot distinguish between
two objects x and y, in the domain of the model that happen to share all properties definable in
the model. This amounts to the validity of the so-called Leibniz’s Law of the Indiscirnibility of
Identicals which holds in many classical first-order models:
↔φ(y))
Indiscirnibility Law: (x = y) → (φ(x)↔
Furthermore, it can be proved that a formula is true in all models where equality is being
taken in the strong sense (known in the literature as normal models) if and only if it is true in
all models allowing the weaker reading of equality discussed above.
It is precisely the Indiscirnibility Law the one which seems to jeopardize the all enterprise of
first-order modal logic. The point stems, not unexpectedly from Gottlob Frege.
Frege-Kanger’s Challenge:
The question about the relation between equality and sameness is an old and venerable one
and puzzled philosophers at least since Parmenides. Plato discusses it in many occasions
particularly in the aporethic dialogues Sophist and Parmenides. Let us quote as an example
the following puzzling lines of the Parmenides:
If the one exists, the one cannot be many, can it? No, of course not …Then in both cases the one would be many, not
one.” “True.” “Yet it must be not many, but one.” “Yes.” (Plato, Parmenides, 137c-d)
Hegel, who recognised the tension in the notion of identity mentioned by Plato, defends the
idea that the concept of identity, conceived as very law of thinking, comprehends both the
idea of different (based in what we called the qualitative version of the princple) and the idea
of identical (based in what we called the numerical version of the principle).
Der Satz der Identität als das erste Denkgesetz] in seinem positiven Ausdrucke A=A, ist zunächst nichts weiter, als der
Ausdruck der leeren Tautologie. Es ist daher richtig bemerkt worden, daß dieses Denkgesetz ohne Inhalt sey und nicht
weiter führe. So ist die leere Identität, an welcher diejenigen festhangen bleiben, welche sie als solche für etwas Wahres
nehmen und immer vorzubringen pflegen, die Identität sey nicht die Verschiedenheit, sondern die Identität und die
Verschiedenheit seyen verschieden. Sie sehen nicht, daß sie schon hierin selbst sagen, daß die Identität ein Verschiedenes
ist; denn sie sagen, die Identität sey verschieden von der Verschiedenheit; indem dieß zugleich als die Natur der Identität
zugegeben werden muß, so liegt darin, daß die Identität nicht äußerlich, sondern an ihr selbst, in ihrer Natur dieß sey,
verschieden zu seyn. - Ferner aber indem sie an dieser unbewegten Identität festhalten, welche ihren Gegensatz an der
Verschiedenheit hat, so sehen sie nicht, daß sie hiermit dieselbe zu einer einseitigen Bestimmtheit machen, die als solche
keine Wahrheit hat. Es wird zugegeben, daß der Satz der Identität nur eine einseitige Bestimmtheit ausdrücke, daß er nur
die formelle eine abstrakte, unvollständige Wahrheit enthalte. - In diesem richtigen Urtheil liegt aber unmittelbar, daß
die Wahrheit nur in der Einheit der Identität mit der Verschiedenheit vollständig ist, und somit nur in dieser Einheit
bestehe.
64
Georg Wilhelm Friedrich Hegel, Wissenschaft der Logik. Erster Teil. Die objektive Logik. Die Lehre vom Wesen (1813),
Neu herausgegeben von Hans-Jürgen Gawoll Felix Meiner Verlag Hamburg 1999 Zweites Kapitel : Die Wesenheiten
oder die Reflexionsbestimmungen A. Die Identität Anmerkung 2 Seiten 29-30
In the Tractatus Logico-Philosophicus Ludwig Wittgenstein, who could be seen as addressed
against Hegel’s remark, gives a more modern version of the puzzles concerning equality:
5.53
Identity of object I express by identity of sign; and not by using a sign for identity. Difference of objects I express
difference of signs
5.5301 Obviously, identity is not a relation between objects …
5.5303 By the way, to say of two things that they are identical is nonsense, and to say of one thing that it is identical with
itself is to say nothing at all
(Wittgenstein; Tractatus Logico-Philosophicus)
Certainly, Wittgenstein’s proposal is hard to implement. We cannot even express equations in
the language proposed. However, this attempt witnesses the struggles of those times to reflect
on the challenges of the notion of equality seen as a relation.
Perhaps the best formulation of the puzzles on the concept of the relation of equality was the
one formulated 1892 by Gottlob Frege in his celebrated paper Über Sinn und Bedeutung. The
paper starts by asking the very question which Wittgenstein answers negatively. Is identity a
relation? If it is a relation, is it a relation between objects, or between signs of objects. To take
the notorious example of planet Venus, the morning star= morning star is a statement very
different in cognitive value from the morning star = the evening star. The former is
analytically true, while the second records an astronomical discovery. If we were to regard
identity as a relation between what the sign stands for it would seem that if a=b is true, then
a=a would not differ form a=b. A relation would thereby be expressed of a thing to itself, and
indeed one in which each thing stands to itself but to no other thing. (Frege, Über Sinn und
Bedetung, 40-42). On the other hand if every sentence of the forma a=b really signified a
relationship between symbols, it would not express any knowledge about the extra-linguistic
world. The equality morning star = the evening star would record a lexical fact rather than an
astronomical fact, Frege’s solution to this dilemma is the famous difference between the way
of presentation of an object, called its sense (Sinn) and the reference (Bedeutung) of that
object. In the equality morning star = the evening star the reference of the two expressions at
each side of the relation is the same, namely the planet Venus, but the sense of each is
different. This distinction allows Frege the following move: a statement of identity can be
informative only if the difference between signs corresponds to a difference in the mode of
presentation of the object designated (Frege, Über Sinn und Bedetung, 65).
Unfortunately, Frege also noticed that this solution, which seems to provide a reasonable
solution to the puzzle of equality for classical logic, does not work in modal – or more
generally intentional - contexts. This fact has been discussed as early as 1957 by S. Kanger in
a paper that for many has been crucial for the development of quantified modal logic. In a
review of Kanger’s paper “The morning star paradox” Hintikka (1969, 306) says
Kanger’s discussion of the Morning Star Paradox will in the reviewer’s opinion remain a historical
landmark as the first philosophical application of an explicit semantical theory of quantified modal logic
Indeed the sentence
(x = y) → «(x = y)
65
will drive us immediately into a new puzzle. Take a and b as instances of the variable of this
sentence and give an epistemic interpretation of the necessity operator, such as the ancients
knew. Furthermore let interpret a and b as the morning star and the evening star respectively.
Since a=b is factually true, by Modus Ponens we obtain that the ancient knew that the
morning star is the evening star. Frege’s sense/reference approach challenges the notion of
equality in modal logic. Quine and many other logicians and philosophers took this result as
reason enough to reject modal logic.
Actually Frege offered a solution even for the case of modal contexts. Frege held that when a
name occurs inside a modal context; it is used to speak about its meaning; not its reference.
Thus, when we say that the ancients knew that the morning star is the evening star we are not
talking about the planet Venus. We are speaking about the meaning of the words, its sense, or
more precisely about the proposition expressed by the equality a=b. But the price to pay is to
preclude de re readings of the modal operators in first-order modal logic. Frege’s solution
boils down to de dicto reading of the expression. But this, seems to spoil all the fun to first
order modal logic. Quine; once more, takes the core of his argument against first-order modal
logic: it is either paradoxical or redundant: de dicto necessities can be understood as logical
necessities and that was it.
Kripke, as we will see later on, simply precludes an epistemic reading of the modal operator
and claims that there is nothing puzzling about the necessity of a=b because the necessity
involved is an ontological one. Fitting and Mendelsohn give a new twist to the argument of
Kripke. But this is already the start of the next section.
IX.3.2 Fitting and Mendelsohn on necessary equalities
Let us start with a remark of Husserl of 1929 who claimed in his Formal and
Transcendental Logic that mathematicians are not in the least interested in the different ways
objects may be given. For them objects are the same which have been correlated together in
some self-evident manner (Husserl 1929, 147-48).
Fitting and Mendelsohn’s analysis of the equality puzzle seems to share a similar spirit to the
remark of Husserl mentioned above. On their view ; the puzzle about informative identities
arises because of the interplay of modal operators and singular terms. They differentiate
between (x = y) → «(x = y) and (a = b) → «(a = b).On Fitting and Mendelsohn’s view the
first expression containing only variables is not puzzling at all. If x and y are the very same
object, and x has the property of being necessarily identical with x, then y has that very same
property. The puzzle arises, so Fitting and Mendelsohn, when we replace the variables with
singular terms by means of names or definite descriptions. For short, the puzzle arises if we
understand the conditional as related to the mode of presentation of the object. Kripke’s
defence of the ontological reading of the modal operator results then of reading the equality
relation in (x = y) → «(x = y) as a relation between objects. (Fitting & Mendelsohn, 144150 ; 204-219).
66
References
IX.3.3 Equality and dialogues
To implement equality in dialogic is not difficult though different possibilities are available.
Some of these possibilities are related to identity, the others to the rule of substitution of
identicals. Namely:
A. 1 Identity and dialogues in non-modal contexts
A. 1.1) Identity as an atomic formula: Let us assume that we would really introduce identity
as an axiom on atomic formulae. That is, let us assume that we are prepared to accept that
identity, despite being an atomic formula it is one that we declare per fiat that does not need
any formal proof.
Dialogically speaking this amounts to the following
•
ki=ki, cannot be challenged but does not need any justification either. That is,
despite being an atomic formula the Proponent can state it though the Opponent did
not state it before.
This rule, which allows an exception to the formal rule, performs exactly the idea that identity
does not need formal justification
Some of the readers might protest to this interference of ontology in logic and prefer another,
more logical formulation of the axiom. Here a step forwards in this direction
A. 1.2) Identity as an axiom: Here we are prepared to introduce identity as an axiom but not
as an atomic formula:
•
At the very start of a dialogue, the Opponent concedes ∀x(x=x).
The advantage of this rule is that we do not need here to understand identity as an atomic
formula. The problem is that if we would like to prove say k2=k2 the dialogue must start with
an atomic formula.
Let us present two solutions
•
2.1 At the very start of a dialogue, the Opponent concedes ∀x(x=x), and atomic
formulae can be challenged.
This requires two extra rules to challenge and defend an atomic formula like the
following
Assertion
X-!-A
Attack
Y-?-A
67
Defence
X-!-A
sic n
(the defender states the
atomic formula at stake
indicating as justification
that the Opponent has
stated this formula at move
n)
•
Atomic formula defended at m with a sic-move cannot be challenged again at the
same move
The next solution is less complicated but requires the introduction of the turnstyle in the
dialogue
•
2.2 Every thesis of the Proponent has the form ∀x(x=x)├ φ.
This requires an extra rule to challenge and defend a turnstyle
Assertion
X-!- φ1 ,..., φn ├.ψ
Attack
Y-?- φ1
Defence
X-!-ψ
...
Y-?- φn
(the challenger concedes
the premisses)
We will use mainly the version 2.2. Though sometimes too the version 1 which is simpler for
the metalogical proofs
A.2 Substitutivity and dialogues
To implement the rule for the the substitution of identicals seems to engage us in a somehow
strange asymmetry. Though the acceptance of identity as axiom seems to be an import of the
ontological level, the rule for the substitution of identicals has its justification in the fact that
with the exception of identity all other equalities concern the relation of certain language
entities, namely terms, with their reference. Because this is so, these type of atomic formulae
can be challenged:
Let us start with the non-modal case.
Assertion
Attack
Defence
Y-? - ki1...n./kj?,
(the challenger asks X to
replace in φ ki with kj at
the occurrences 1 ... n)
X-!- φ[kj],
X-!- ki=kj
.
X-!- φ[ki],
where ‘[ki]’ indicates
68
(where kj replaces ki at
the at the occurrences 1
...n)
that ki occurs in φ
X states φ[ki] and
conceded before
ki=kj
To avoid unnecessary repetitions we will add the following rule
•
Proponent’s atomic formulae cannot be challenged
Or if we are using a version of identity where equalities can be challenged
•
Once the Proponent responded to an attack on atomic formulae this formula cannot
be challenged again.
The Opponent cannot attack
This formulation of the rule is allows a replacement a left-right but a right-left is also possible
as a derived rule. Let us show that the right-left substitutivity is indeed an abbreviation of a
longer sequence of steps of left-right replacements. In order to highlight the point of the proof
assume that we introduced identity as an atomic axiom as one of the components of the
antecedent of the turnstyle:
O
1.1
1.2
1.3
3
5
ki=ki
ki=kj
φ[kj]
kj=ki
φ[ki]
0
II.2
1.2
1.1
1.3
P
ki=ki, ki=kj, φ[kj]├. φ[ki]
0
φ[ki]
? ki1/kj
? kj/ki
6
2
4
The decisive move is certainly move two. Indeed here the clever Proponent takes the identity
as being the formula φ where ki occurs and consequently asks to replace the first occurrence
of ki with kj.
•
For the sake of simplicity we will actually assume that the substitutivity rule can be
uses from left to right or from right to left
EXERCISE IX.3.3.e1:
Prove
k1=k2, k2=k3├ k1=k3]
B. 1 Identity and dialogues in modal contexts
For modal logic we have mutatis mutandis the same options:
B. 1.1) Identity as an atomic formula:
69
•
ki=ki can be stated at any context i, cannot be challenged and does not need any
justification either. That is, despite being an atomic formula the Proponent can state
ki=ki at any context i though the Opponent did not state it before in this context.
A. 1.2) Identity as an axiom in modal contexts:
•
At the very start of any context of a dialogue i , the Opponent concedes ∀x(x=x).
That is, every dialogue context starts i with
• i «∀x(x=x), where reflexivity for this formula is assumed
We leave the notational variants – with and without explicit turnstyle - for the reader
I B.2 Substitutivity and dialogues in modal contexts
Assertion
Attack
Defence
Y-?- ki1...n./kj?,
(the challenger asks X to
replace in φ ki at the
X-!- j φ[kj],
(where kj replaces ki in
φ at the context j at the
at the occurrences 1 ...n)
X-!- i ki=kj
.
X-!- j φ[ki],
where ‘[ki]’ indicates
that ki occurs in φ
context j with kj at the
occurrences 1 ...n)
X states φ[ki] at j and
conceded
ki=kj at i
It is important to realize here that no specific accessibility is being assumed here: If the
singular terms are to work as rigid designators they denote the same object independently of
the accessibility relation. Kripke’s justification of ki=kj├ «ki=kj is and ontological one. The
necessity should not be read epistemically here: If ki=kj is true; then because both terms
design the same object. Now, because this is so, according to Kripke, in a rigid manner, they
will denote the same object in all the possible worlds of the model and thus «ki=kj becames
true. The awkwardness of the formula, comes, according to Kripke, from an epistemic reading
of the necessity operator. But if we introduce identity as a logical validity, this has an
ontological motivation which determines the reading of the necessity operator.
Let us prove
• ki=kj├ «ki=kj
We will assume here for the sake of brevity that the Opponent introduced identity as soon as
he opened the second context with his challenge and do not bother about identity in the first
context. Certainly we could carry out the prove introducing identity in the first context. We
leave this for the reader.
70
O
context
1
1
1.1
1.1
1.1
1.1
1
3.1
3.2
5
7
9
ki=kj
?«1.1
«∀x(x=x)
∀x(x=x)
ki=ki
ki=kj
P
ki=kj├ «ki=kj
«ki=kj
ki=kj
2
1
3.2
5
7
?«1.1
?∀ki
? ki2/kj
0
2
10
1
1
1.1
4
6
8
1.1
1.1
1.1
As allready mentioned we assume that as soon as the Opponent asks for the context 1.1 at
mover 3.1 he concedes identity for this context at move 3.2. At move 4 the Proponent makes
use of the reflexivity of identity and proceeds at move 6 with an adequate choice to challenge
the universal quantifier. At move 8 the Proponent uses the substitutivity rule asking the
Opponent to replace the second occurrence of ki with kj.
IX.3.4 Existence, Equality and dialogues
IX.4 From arbitrary objects to arbitrary constants
In the times of yore there was the view that in addition of individual objects, there are
arbitrary objects: in addition to individual numbers, arbitrary numbers; in addition to
individual horses, arbitrary horses. With each arbitrary object an appropriate range of
individual objects, the so called values of this object, is associated. So, with each arbitrary
number; the range of individual numbers; with each arbitrary horse, the range of individual
horses. An arbitrary object has those properties common to the individual objects in its range.
Thus, an arbitrary number is odd or even and an arbitrary horse is mortal since each individual
number is odd or even and each individual horse mortal. On the other hand, an arbitrary
number fails to be prime, an arbitrary horse fails to be brown since some individual number is
not prime and some individual horse is not brown.
Such a view was quite common between mathematicians and philosophers until Frege.
More precisely, after Frege it came into disrepute, and in the context of the awkward theory of
variable numbers of those days it was good that this happened. The time is ripe now to
recover a bit of this forgotten theory. Our point is that the rejection of this view could be
related to the scepticism of Frege and followers towards modal logic and to Frege’s notions of
scope.
Since Frege, we clearly distinguish in first order logic between variables and
constants. Variables may have different values not constants. Moreover, in this standard
conception we are taught to distinguish between the logical term having as values objects of
the domain and these objects which are elements of the domain. For short, there is no such
thing as an arbitrary object representing generality. Generality is achieved by means of a
variable ( a linguistic entity) having arbitrary many values. Indeed, variable-letters of the
logical language do no designate variable objects neither. Frege makes this point explicitly
([70]), 109):
71
This way of speaking is certainly employed but these letters are not proper names of variable numbers
in the way that ‘2’ and ‘3’ are proper names names of constant numbers; for the numbers 2 and 3 differ
in specified way, but what is the difference between the variables [numbers] that are said to be
designated by ‘x’ and ‘y’?
Husserl made once the remark that when a child says to his father that he would like to
have any of the toys of a certain box, he does not want a “general toy”. What the child wants
is an individual toy, whatever it is the father picks up of the box (QUOTE°). It is the
(intentional) act of choosing which can be qualified as determined (bestimmt) or not
determined (unbestimmt). Indeed, when we say that we can choose any of the objects of a
given domain, we mean that there is no restriction underlying our choice of an object of that
domain. If we combine this remark with the notion of quantifiers of a logical language we
have almost the dialogical and game theoretical concept of quantifiers as choice functions.
So far so good but in modal logic things look quite different and the idea of constant
letters not designating (constantly) at each world the same object is not absurd anymore.10 In
fact, as already mentioned, in quantified modal logic we could take the constant ki to
designate the same element of the domain in all worlds or to designate different elements of
the domain. In the first case, since Kripke, we talk of rigid designators, in the latter of
individual concepts. We will later see that these notion will give us a lot of stuff for
discussion, particularly in relation to the deep insights suggested by Jaakko Hintikka who
contested the very grounds of the irreducibility of these two types of terms (Hintikka [2003]).
For the time being, however late us start with Kripke’s view which is based on an
attack on Frege’s and Russell’s notion of definite descriptions. Indeed, in Frege’s view
constants are constant and design always the same individual. Suppose we pick a constant ki,
with the informal intention to mean the “the president of USA”. Of course ki will designate
different people at different times. In the Frege-Russell conception the expression “the
president of the united states” is considered to be incomplete. Only when completing it to
something like “the president of USA on March 20, 2003” the expression, called definite
description (we will come to them in detail on chapter ???), could be used as meaning of the
constant ki. This strategy eliminates the distinction between definite descriptions and proper
names, if both are used as terms in our logical language. But in modal contexts we would like
not only to have constants as rigid designators, which Kripke associates with proper names
but too as individual concepts. Moreover, sometimes even the ambiguity of the constant
symbols is desirable. For instance consider the sentence,
We hope that someday, there will be a president of USA who will be wiser than the
president of USA at the present time.
A natural way to read this is
We hope that the person that ki (:the president of the USA) designates at some future
will be wiser than the person that ki designates at the present time.
Here we read the constant expression “the president of USA” in the original sentence as nonrigid. Indeed; ki is taken here as meaning “the president of USA now”, so that it can designate
different persons at different times.
10
Hugh MacColl was perhaps the first to attempt to formulate this in a formal setting for modal logic of 1906,
though unfortunately modal logic was not yet ripe enough.
72
Fitting and Mendelsohn ([1998], 188-200), from whom stems the theory on individual
concepts described below, present even more interesting cases of ambiguity than just allowing
constant symbols to designate different things at different worlds. Let us reformulate a bit our
example:
We hope that someday, the president of USA will be wiser than the president of USA
is now.
There are two ways of reading this sentence. Namely:
We hope that the person that ki (:the president of the USA) designates at some future
will be wiser than the person that ki designates at the present time.
We hope that the person that ki (:the president of the USA) designates now will be
someday a wiser than he (the same very person) is now.
The ambiguity involved is an ambiguity related to the scope of modal operator. Let us have a
closer look at this kind of ambiguity by means of the following formal example.
Let the model M= <W, R, v, D, ι> constant two-world model defined as follows:
W: {w1, w2, }
R: {<w1, w2>
ι(k) at w1 is δ1
ι(k) at w2 is δ2
ι(P) at w1 is {δ1}
ι(P) at w2 is {δ1}
More explicitly, we have two worlds where the second is accessible from the first; both with
the same domain δ1 and δ2, and the relation symbol P is interpreted to be δ1 at both worlds.
Furthermore the constant has been interpreted non rigidly; taking it to designate δ1 in the first
world and δ2 in the second.
DW1 ={δ1, δ2} DW2 ={δ1, δ2}
W1• •W2
ι(P)={ δ1} ι(P)={ δ1}
ι(k)={ δ1}
ι(k)={ δ2}
IX.4.f1
Now, what can we say about the truth of following?
M, vw1(◊Pk)
Well, we might want to capture the two readings mentioned above and in this case there is an
ambiguity of scope here.
First, we could take the formula to express that the interpretation of k at w1 is an element of
the interpretation of P in one possible world. That is, we regard the interpretation of k as
primary in relation to it’s interrelation with the possibility operator. In this case
M, vw1(◊Pk)=1
73
because ι(k) at w1 = δ1∈ι(P) at w2
Second, we could take as principal not the constant but the possibility operator. In this case
M, vw1(◊Pk)=0
because ι(k) at w2 = δ2∉ι(P) at w2
This ambiguity, where we can take either the constant symbol as primary or the modal
operator as primary seems to come up every day. Examples of non-rigid interpretations of
constant symbols are definite descriptions. Like in:
The ancients did not know that the morning star was the evening star
Here we mean that, while the definite description morning star and evening star designate in
fact the same object, there are alternate situations compatible with the knowledge of the
ancients (accessible) in which the objects designated by both definite descriptions are
different. Moreover here we consider the alternate situations before we consider what is
designated.
We will came back to definite descriptions later but let us fix here that the syntactic ambiguity
of a non-rigid constant can produce a semantic ambiguity.
Here one last example based in a one given by Kai Wehmeier (2003) and which will give us
later on some stuff for discussion, because it relates to the problem of scope but with another
nuance:
Under certain counterfactual circumstances every USA president who actually has
flown to the moon would not have flown to the moon.
Let us here to highlight our point take a restricted version of this sentence. We will first
consider that that k is the definite description president of the USA and that we have the
following model:
Intuitively,
k will designate in the first world the object “George Bush Senior” and in the second
world “George Bush Junior”.
Furthermore, we will take P to be the predicate flown to the moon and we will assume
that Bush Senior flew to the moon in the first world but only Bush Junior did it in the
second.
Let us advance some notation which will be introduced in the next chapter and write
M, vw1[λx Px]k→[λx ◊¬Px]k
(where “λx Px” in M,w1 reads: “the set of all those x of whom the model assumes that
they flew to the moon in the first world”)
Now, the antecedent is true at the first world if the Px turns to be true at w1 when we assign
to x whatever it is what k designates at the first world - in our case, Bush Senior. And indeed
this is true in our model.
74
Similarly, the consequent is true at the first world if the predicate Px turns to be false in at
least one possible world when we assign to x whatever it is what k designates at the first
world. Since, k designates in the first world Bush Senior and by assumption of the model he
did not flow to the moon in the second world, the conditional is true.
Let us now take the slighter simpler
Under certain counterfactual circumstances every USA president who actually has
flown to the moon would not have flown to the moon.
The standard formulation
∀x (Px→◊¬Px)
is incorrect. The formula says that for everyone who actually has flown to the moon in this
(actual) world there have to be circumstances under which he would not have flown to the
moon, and these circumstances need not to be the same for everyone. The problem is the
interaction between the scope of the universal quantifier and the possibility operator. As
already mentioned, we will come to these type of examples later but let us turn now our
attention to the possibilities offered by the predicate abstraction as introduced in first order
modal logic by Fitting and Mendelsohn
IX.5 The syntax and semantics of predicate abstraction
Predicate abstraction was introduced into modal logic by Stalnaker and Thomason (1968),
Bressan (1972), Fitting (1972a, 1973, 1975, 1991, 1993 and 1996b). This notion has been
widely applied in Lambda-Calculus and linguistics, more precisely in Montague Grammars
and their successor Discourse Representation Theory of Hans Kamp and collaborators. We
will discuss here the last version of predicate abstraction for first order modal logic as
presented in chapters 9-12 by Fitting and Mendelsohn (1998).
The reader is probably familiar with the way sets are sometimes specified by the so-called
predicate-notation, as in the following examples:
{x: x is an Argentinian football team winner of the worldcup}
{x: 711<x<1492}
The first example names a set containing the Argentinian teams of 1978 and 1986, while the
second example names the set of numbers greater than 711 but less than 1492.
The only requirements for this way of naming sets are
•
•
that we have a sentence containing a variable in a language we understand clearly
(English in the first example, the informal language of arithmetic in the second) and
that we have a convention for marking which variable (in the case there might be more
than one in the sentence) is for naming the corresponding set; in the above notation the
75
variable written to the left of “:” indicates this (in other notations a vertical line
replaces “:”).
Of course, there is no reason why we could not name sets in this way using formulae of a
formalized artificial language in place of sentences of English, as long as we have already
supplied an adequate interpretation of that language. In fact, we will do just this by adding the
lambda operator, following Alonzo Church 1940, where a formal calculus containing lambda
operators was first developed. Instead of the notation above, we will specify a set with the
expression
λx ϕ
If ϕ is a formula, then λxϕ will denote the set specified by ϕ with respect to the
variable x. (notice that this expression is not a formula but a predicate abstract by
means of which formulae can be build up!)
It is perhaps worth to make the point that the predicate abstraction involves determining a
name for the object (the corresponding set) specified by the formula ϕ. This becomes crucial
when we are dealing with functions. Actually, the historical origin of abstract predicates is
connected to the distinction between the name of the function and the output of the function
(for the concept of function see appendix A). In Frege’s time, there was a widespread
confusion in mathematics about expressions like x2 +2. This expression was considered on the
one hand to be the name for the number obtained by squaring x and adding 2 to the result.
Since this function is not the same thing as any one of its outputs, the ambiguity of the
expression produced considerable confusion quite frequently. Eventually, Alonzo Church
proposed the notation of predicate abstraction in order to distinguish between the number x2
+2 and the function λx (x2 +2) which maps any x to precisely the number x2 +2.
In first order-logic the difference between the formula and predicate extracted from this
formula is a subtlety which might be spared. 11 In linguistics, higher-order logic, logic of
functions and in general in type theory this difference can make all the difference.
Furthermore, in a modal setting, as the one of this book; formulae can no longer be thought of
as representing predicates, pure and simple. The examples of the preceding chapter are aimed
to show that. Because of the ambiguity of scope mentioned above we need to distinguish
between a formula and the predicate abstracted from that formula. In fact one way to
disambiguate the formula
◊Pk
11
What we mean is that in general and under certain precise conditions sentences which are build up using
predicate abstracts can be converted into standard first-order sentences. Let us take, the formula [λx Px] k to say
that Mohandas is one of the x such that x is president of USA. Intuitively, this ought to be an equivalent but more
complicated way of saying Pk. The rule that a formula of the form [λx …x...] ki may be converted to a logically
equivalent formula of the form […ki...] (or vice versa) is called principle of lambda conversion (Church 1940)
must be understood as the result of replacing all free occurrences of the variable x in the first formula […x...]
with ki. Actually, in order to handle lambda conversion safely the notion of free occurrence has to be made more
accurate (see details in GAMUT, II, 109-111).
76
where k is non-rigid is to abstract the two predicates represented by the two formulae.
Namely,
the possible predicate ◊P applied to k:
[λx ◊Px] k
the predicate P applied to k in a possible in a possible world
◊[λx Px] k
IX.5.1 Syntax
We assume we have available an infinite list of n-ary place function symbols ƒ1n, ƒ2n, ƒ3n, …,.
(for the notion of function see appendix A). Quite often we will be informal and use ƒ and g,
or something similar for a function symbol, with its arity determined from context. here and
use We also assume we have an infinite list of constant symbols k1, k2, k3, … Constant
symbols are sometimes taken to be function symbols of arity 0.
Term
• Every variable (including parameters) is a term. Recall from chapter II that parameters
were introduced as a special kind of free variables which can never be quantified nor
bound by a lamda operator.
• Every constant symbol is a term
• If ƒ is n-place function symbol; and t1, …tn are terms, then ƒ( t1, …tn) is a term
The definition of atomic formula and formula is exactly as it was in chapter IX.1 but with the
addition of the following:
Predicate abstract formula:
• Predicate abstract: If ϕ is formula and v a variable then [λvϕ] is a predicate abstract;
the free variable occurrences of [λvϕ] are those of ϕ except for occurrences of v.
• Predicate abstract formula: If [λvϕ] is a predicate abstract and t is a term, [λvϕ] (t) is
a formula; the free variable occurrences of [λvϕ] t are those of [λvϕ] (t) together with
all variable occurrences in t.
IX.5.2 Semantics
First let us explicitly fix the non-rigidity of constant symbols and functions symbols,
where the first should designate objects non rigidly and the second designate, not surprisingly,
functions. For the sake of simplicity we assume here that the domain is constant.
DEFINITION: Non -Rigid Interpretation:
The interpretation is a non-rigid interpretation in the frame F: <W, R, D, > if meets the
conditions of chapter IX.1 and also meets the following:
• To each constant symbol k, and to each wi∈WF, assigns some member of the
domain: iwi,F(k) ∈ DF. We say that the constant symbols have been interpreted as
individual concepts.
77
•
To each n-place function symbol ƒ and to each w1∈WF, assigns some n-ary function
from the domain to itself: iwi,F(ƒ): DnF d DF. We say that the function t symbols have
been interpreted as function-concepts.
DEFINITION: Non -Rigid Model:
• A model M: <W, R, v, D, > is a non-rigid model if is a non-rigid interpretation
Remark: Our present notion of non-rigidity allows that the reference of a term at a world of a
model might not be something in the domain of that world. We allow thus that a term to
designate nonexistent objects; though the objects designated must exist in some world.
Consider the definite description “The Mahatma” given by Rabindranath Tagore to Mohandas
Karamchand Gandhi in a modal context. In the temporal reading of the modality, the definite
description designates Gandhi now, even though he, unfortunately, does not now exist.
Actually this is a direct consequence to the possibilist (or constant domain) approach to firstorder modal logic assumed in this section. Since, as far as terms are concerned the essential
difference between a constants domain semantics and the varying domain (or actualist)
semantics is that, for constant domains terms always designate existent objects, while for
varying domains this is not so.
TERMS AND ASSIGNMENTS
In our setting, terms can be free variables, non-rigid constants and non-rigid functions. Now,
as already mentioned in IX.1, the standard way to fix the meaning of variable in a given
model is by means of an assignment, while fixing the meaning for non-rigid constants and
non-rigid functions is done by means of an interpretation. Thus, to specify the meaning of
term in uniform way, both must be taken into account, assignments and interpretations. What
we will do is here is to enlarge the notion of assignment:
DEFINITION: ( -i)-assignment
Let M: <W, R, D, v> be a non rigid model, let w∈WM, and let be an assignment in M. Then
the notion of ( -i)-assignment of each term t in w is formulated as iM,w(t) and defined as
follows
•
•
•
If x is a free variable (possible a parameter) the ( -i)--assignment boils down to the
standard notion of assignment. Namely, iM,w(x) = M,w(x).
If k is a constant symbol, then the ( -i)-assignment is the non-rigid interpretation of
this constant: iM,w(k)= iM,w (k).
If ƒ is n-place function symbol then the non-rigid interpretation at a given world w at
stake assigns to this symbol a n-place function. The ( -i)-assignment of ƒ for the terms
t1, …, tn at precisely this world results from applying this function to the designations
of these terms at w,
i
i
i
M,w(ƒ(t1, …, tn)) = iM,w (ƒ)(( M,w (t1), …,. M,w (tn))
DEFINITION: Truth in a Non-rigid model
Let M: <W, R, v, D, > be a non-rigid model. The definition of truth is that same as in the
preceding chapters with the addition of the following:
78
The formula [λxϕ] (t) is true world w of the model if ϕ turns out to be true at w when we
assign x whatever it is that t designates at that world:
M, vw, ([λxϕ] (t))=1 iff M, vw, *(ϕ))=1, where γ* is a x-variant of γ such that γ*(x) =
i
M,w(t)
EXAMPLE: Let us come back to the example of Fitting and Mendelsohn and repeat figure IX.4.f1
DW1 ={δ1, δ2} DW2 ={δ1, δ2}
W1• •W2
ι(P)={ δ1} ι(P)={ δ1}
ι(k)={ δ1}
ι(k)={ δ2}
IX.5.f1
and study there once more the truth of
M, vw1(◊Pk)
First we will disambiguate the expression abstracting two predicates represented by the two formulae. Namely,
the possible predicate ◊P applied to k:
[λx ◊Px] k
the predicate P applied to k in a possible in a possible world
◊ [λx Px] k
On one hand, according to our definition of truth [λx ◊Px] k is true at w1 when ◊Px turns out to be true at w1
when we assign x whatever it is that k designates at that world:
M, vw1, ([λx ◊Px] k)= 1 iff M, vw1, *(◊Px)=1, where γ* is a x-variant of γ such that γ*(x) =
i
M,w1(k)
But at that world k designates the object δ1. That is
γ*(x) = iM,w1(k) = δ1
Thus,
M, vw1, *(◊Px)=1 iff M, vw1, *(◊P iM,w1(k))=1
Since w2 is the only world accessible from w1, we have to calculate M, vw2, *(Px) which is indeed true. Indeed,
γ*(x) = iM,w1(k) = δ1 ∈ M,w2 (P)
On the other hand ◊[λx Px] k is true at w1 when Px turns out to be true at w2 when we assign x whatever it is that
k designates at w2:. But at that world k designates the object δ2. That is
γ*(x) = iM,w2(k) = δ2
Thus, we have to calculate M, vw2, *(Px) which is false. Indeed,
γ*(x) = iM,w2(k) = δ2 ∉ M,w2 (P)
IX.5.3 Equality, abstraction and rigidity
IX.5.4 To know and not to know and predicate abstracts
Suppose that the American basketball fan Bill knows that in the game between the Dallas
Mavericks and the Portland Trailblazers Dirk Nowitzki was the top-scorer. In fact, Dirk
Nowitzki is the best German player in the NBA. But, Bill does not know that (let’s say, Bill
79
thinks that Nowitzki is a Russian, because of the name, and thus still considers Detlef
Schrempf to be the only German NBA-player). What about Bill’s knowledge that the best
German NBA-player scored best in that game? In one sense of the word he has that
knowledge because he knows that Nowitzki has been the best scorer and Nowitzki is in fact
identical with the best German NBA-player. But in another sense of the word, Bill does not
have the knowledge in question (ask him, whether the best German player has scored most,
and his answer will be ”No”), because he does not know that the proper name ”Dirk
Nowitzki” and the definite description ”the best German NBA-player” denote the same
person.
Knowledge in the first sense is called de re knowledge, knowledge in the second sense de
dicto knowledge. If we apply this terminology to our example, it follows that Bill has the de
dicto knowledge that Dirk Nowitzki was the top-scorer but he has not the de dicto knowledge
that the best German NBA-player was the top-scorer. But, Bill has de re knowledge of the
best NBA-player (who is Dirk Nowitzki) to have been the top-scorer.
In order to determine more exactly the notions of de dicto and de re knowledge, let’s
consider first only knowledge concerning elementary propositions. Take an elementary
proposition of the form Fa1,a2,...,an containing a predicate F and n singular terms a1, a2,..., an.
We will apply some terminological proposals by John Perry. He defines the subject matter of
such a proposition as follows:
The subject matter of a sentence. This is the objects (or conditions) designated
by the terms in the sentence, and the condition designated by the condition word
in the sentence. ((Perry 2001b), p. 8)12
Thus, the subject matter of Fa1,a2,...,an is the set containing the relation designated by F and
the objects named by a1, a2,... an. Now, we can develop the concept of subject matter content.
The subject matter content of a proposition says what has to be the case with respect to the
subject matter so that the proposition is true. Namely in our case, that the objects named by a1,
a2,... an stand to each other in the relation designated by F. De re knowledge is nothing else as
knowledge of the subject matter content, and this means knowledge of the existence of certain
states of affairs or facts.13
The subject matter content is distinguished from the reflexive content of a proposition by
Perry.14 While it is irrelevant with concern to the subject matter content how for example the
objects of the subject matter are named (the only thing that matters is which objects are
named), this relation between language and world is decisive for the reflexive content.
Because it determines the truth-conditions concerning the relation between language and
world that have to be fulfilled so that the proposition is true. The reflexive content of for
example Fa1,a2,...,an reads as follows: ”The expression ‘a1’ names an object d1, the expression
‘a2’ names an object d2,..., the expression ‘an’ names an object dn, and ‘F’ designates a relation
R in such a way that the objects d1, d2,..., dn stand in the relation R.”
De re knowledge can be conceived of as knowledge of the subject matter content of a
proposition and de dicto knowledge can be conceived of as knowledge of the reflexive
12
I have to thank Albert Newen (Bonn) for sending me Perry’s yet unpublished paper.
13
Instead of applying Perry’s conception of subject matter content in order to specify de re knowledge it would
have been possible to work with so-called Russellian propositions, too (cf. for example (Barwise/Etchemendy
1987)).
14
The concept of reflexive content was introduced in (Barwise/Perry 1983). (Perry 2001a) makes use of it to
refute several counter-arguments against the thesis of the identity between the physical and the mental.
80
content of a proposition. But what about the relation between de re and de dicto knowledge?
A person has de re knowledge that α if and only if there is a β having the same subject matter
content as α and the person has the de dicto knowledge that β. A person has no de re that α
knowledge if and only if there is no such β.
It remains to be said a word about de re and de dicto knowledge concerning complex
propositions. In order to avoid problematic entities like complex states of affairs we simply
define de re knowledge as above via de dicto knowledge, and the subject matter content of
complex propositions is determined in the usual way via the subject matter contents of their
contained elementary propositions. For example, the subject matter content of a conjunction α
∧ β says that concerning the subject matter of α it has to be as α says and concerning the
subject matter of β it is to be as β says for the whole conjunction to be true.
Lamda Conversion
INCLUDE DISCUSSION OF GAMUTS COUNTEREXAMPLES
UNDERSTANDING TERMS AS INDIVIDUAL CONCEPTS
PUT IT IN RELATION TO THE EXERCISES OF PAGES 164 and 224
TO
THE
IX.6 Reasoning about nothing
IX.7 Modal scope and the subjunctive marker
IX.8 Dialogues and sequent calculi for first order modal logic
X IF and first-order modal logic
XI Higher order modal logic
In fact predicate abstraction can be seen as directly equivalent to comprehension axioms of
Higher-order logic. In the latter, speaking informally, one wants to make sure that every
formula specifies a class. Comprehension axioms ensure that to each formula corresponds
such and object and they have usually the following general form:
∃X [X(x1,… , xn) ≡ ϕ(x1,… , xn)]
Let us think of the atomic formula Px as telling us that P is true at world x, and R(x,y) as
saying y is a world accessible from x. Then
∀yR(x,y) →Py corresponds to P being true at every world accessible from x, and hence to P
being true at world x. Thus P→P is true at the world x corresponds to (∀yR(x,y) →Py)→Px
P ∀P (∀x(∀yR(x,y) →Py)→Px) →∀xR(x,x)
0
O ∀P (∀x(∀yR(x,y) →Py)→Px)
1
P ∀xR(x,x)
O ki?
2
3
P R(ki,ki)
12
P? λzR(ki,z)
4
81
O ∀x(∀yR(x,y) →[λzR(ki,z)]y)→[λzR(ki,z)]x
P ki?
5
6
O (∀yR(ki,y) →[λzR(ki,z)]y)→[λzR(ki,z)]ki
P (∀yR(k,y) →[λzR(k,z)]y)
7
8
now to subdialogues are opened
If the Opponent responds
O [λzR(ki,z)]ki
9
The Proponent will ask to instantiate λz with ki and win with move 12:
P λz /ki?
10
Recall: Actually because of the particle rule of lamda at the first order level the player has no real
option when he challenges a lamda operator. He has to attack with the constant the name of the
predicate applies to
O R(ki,ki)
13
If the Opponent responds with
O kj?
11*
the Proponent wins with the following sequence of moves:
P R(ki,kj) →[λzR(ki,z)]kj
12*
O R(ki,kj)
P [λzR(ki,z)]kj
13*
14*
O λz /kj?
15*
P R(ki,kj)
16*
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481-487, 1998.
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ii
Gabbay (1987) used this operator for modal logic. Hoepelmann and van Hoof (1988) applied this idea of
Gabbay’s to non-monotonic logics. Finally Rahman (1997, chapter II(A).4.2), introduced the F-Operator in the
formulation of semantic tableaux and dialogical strategies for connexive logic.
NOTES
iii
Unfortunately he does not seem to have succeeded. Read [1998], differs from Storrs MacCall's ([1963] and
[1967]) argues that the reconstruction of MacColl's modal logic yields T and not one of the non-normal logics.
iv
Cf. Grattan-Guinness [1998], Rahman [1997], [1997], [1998] [2000], Read [1998] and Wolenski [1998].
v
Cf. Kripke [1965].
vi
Cf. Hintikka [1975] and Rantala [1975]. See too Cresswell [1972] and Girle [1973].
vii
See too Read [1994], 90-91 and Priest [1998], 482.
viii
Already Aristoteles used counterlogical arguments while studying the principle of non-contradiction, which
he saw as the principal axiom of logic.
ix
This strategy, as developed in Rahman/Keiff [2003], could be implemented either implicitly or explicitly. The
implicit formulation presupposes that the structural rules are expressed at a different level than the level of the
rules for the logical constants which are part of the object language. The explicit formulation renders a
propositionalisation of the structural rules using either the language of the linear logicians or hybrid languages in
the way of Blackburn [2001].
x
In the context of the SSD with the thesis; say, A, the Proponent's claims that he assumes that a determined
element δi (of a given set ∆ of structural rules) is the minimal structural condition for the validity of A.
Informally, the idea is that structural statements can be attacked by the challenger in two distinct ways. First, by
conceding the condition δi, claimed by the player X to be minimal, and asking X to prove the thesis. Second, by
(counter)claiming that the thesis could be won with a (subset of) condition(s) of lesser rank in ∆. In that case, the
game proceeds in a subdialogue, started by the challenger who now will claim that the formula in question can
be won under the hypothesis δj, where δj is different from δi and has a lesser rank as δi. Since the challenger (Y)
starts the subdialogue he now has to play formally. See details in Rahman/Keiff [2003].
xi
See for example Fitting [1983], 37; Fitting/Mendelsohn [1998], 52, Girle [2000], 32-34.
xii
Cf. Blackburn [2001] and Blackburn/de Rijke/Venema [2002].
xiii
Cf. Girle [2000], 187 where the exercise 3.3.1. 2(a) shows how such a mistake slipped into his system.
83
xiv
. Moreover, if the thesis were «b→««(a∨b) it would be valid.
It could be even be fruitful to relate this problematic with tonk. From the dialogical point of view , tonk
produces an extension into triviality because it has been introduced without semantic support (see Rahman/Keiff
[2003]°. Here, if the semantics concerning the accessibility relation is not changed according to the classification
of worlds into two disjoint, the logic will collapse into another normal modal logic.
xv
II.1 GTS-truth and the dialogical proof
II.2 Dialogues and sequent-calculus
II.3 History
III IF-Logic
84