Download Basic Metatheory for Propositional, Predicate, and Modal Logic

Basic Metatheory for Propositional,
Predicate, and Modal Logic
Christopher Menzel
1
Propositional Logic
A formal system S consists of a formal language, a formal semantics, or model
theory, that defines a notion of meaning for the language, and a proof theory,
i.e., a set of syntactic rules for constructing arguments — sequences of
formulas — deemed valid by the semantics.1
In this section, we define a formal system of propositional logic (a.k.a.
sentential logic or sentence logic). Propositional logic is so-called because
the basic meaningful units are propositions, or sentences, such as “John is
happy”, or “CSIRO is Australia’s leading research organization”. Propositional logic is the logic of the propositional connectives: and, or, if...then,
if and only if, and not, or it is not the case that, which can be used to be
build logically complex propositions from basic, logically simple propositions. The language of propositional logic introduces symbols to represent
propositions and connectives, the semantics assigns the connectives precise meanings that reflect their informal meanings, and the proof theory
captures the logic of the connectives axiomatically. We begin with the language.
1.1
1.1.1
The Language L P of Propositional Logic
The Lexicon
The lexicon of L P consists of the following elements.
1 Formal systems are are,
strictly speaking, be purely syntactic, and hence are typically
defined without any essential reference to semantics.
• Propositional constants: p0 , q0 , r0 , p1 , q1 , r1 , ...
• Boolean operators: ¬, →
• Punctuation: (, )
An expression of L P is any finite string of elements of the lexicon of L P .
The metavariables ϕ, ψ, and θ will range over expressions of L P , and the
metavariables p, q, and r (possibly with primes 0 , 00 , etc.) will range over
the propositional constants. (Note that, because the propositional constants are themselves expressions, the Greek metavariables range over
them as well.)
1.1.2
The Grammar of L P
The grammar of L P is defined recursively as follows.
1. Every propositional constant of L P is an (atomic) formula of L P .
2. If ϕ and ψ are formulas of L P , so are ¬ ϕ and ( ϕ → ψ).
3. Nothing is a formula of L P except those expressions generated directly by rules 1 and 2.
A little more exactly, the set of formulas of L P is the smallest set FLP
containing the propositional constants of L P and the expressions ¬ ϕ and
( ϕ → ψ) whenever it contains ϕ and ψ. Formulas of the form ¬ ϕ are
known as negations, and those of the form ϕ → ψ are known as conditionals.
Outermost parentheses can be dropped, so that, e.g., ( ϕ → ψ) can be
written simply as ϕ → ψ. Note that “outermost” here means that the
left parenthesis occurs as the leftmost symbol of the formula and the right
parenthesis as the rightmost. Thus, one may not drop the parentheses in,
e.g., the formula ¬( ϕ → ψ).
(Immediate) constituents; main connectives
ϕ is known as the immediate constituent of ¬ ϕ, and ¬ as its main connective;
likewise, ϕ and ψ are the immediate constituents of ( ϕ → ψ), and → its
main connective. ψ is a constituent of ϕ iff ψ is either identical to ϕ or is an
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immediate constituent of a constituent of ϕ (i.e., an immediate constituent
of an immediate constituent ... of an immediate constituent of ϕ).
We will use the metavariables Γ and ∆ to range over sets of formulas
of L P .
1.1.3
Other Boolean Operators
All other standard boolean operators can defined in terms of ¬ and →.
Specifically,
• Conjunction: ( ϕ ∧ ψ) =df ¬( ϕ → ¬ψ)
• Disjunction: ( ϕ ∨ ψ) =df (¬ ϕ → ψ)
• Biconditional: ( ϕ ↔ ψ) =df (( ϕ → ψ) ∧ (ψ → ϕ)).
1.2
1.2.1
Model Theory (Semantics) for L P
Interpretations
An interpretation I of L P is a simply a function V, known as a valuation
function, that maps the constants p0 , q0 , r0 , ... of L P into the set {T, F}. V,
that is, is simply an arbitrary assignment of truth values to the propositional constants. (Despite the fact that an interpretation just is a valuation
function in basic propositional logic, in general the notion of an interpretation is broader than that of an evaluation function (see the next section on
modal propositional logic, for example) and so we keep them conceptually distinct.) Every valuation function V determines a unique extension V
that maps each formula of L P into the set {T, F}. V is defined as follows.
Let ϕ and ψ be formulas of L P .
1. V ( p) = V ( p), for propositional constants p.2
2. V (¬ ϕ) = T iff V ( ϕ) = F.
3. V ( ϕ → ψ) = T iff V ( ϕ) = T only if V (ψ) = T (equivalently, if
V ( ϕ) = F or V (ψ) = T).
2 That
is, V and V agree on the truth values of the propositional constants. This is why
V is said to be an extension of V: it tells us everything that V does and more besides.
3
In essence, an extension spells out how the truth value assigned to a complex (i.e., nonatomic) formula depends on the truth values assigned to its
immediate constituents and, ultimately, on the truth values assigned to its
atomic parts by V. Note that it does this in a way that assigns negation
as the meaning ¬ and the conditional “if ... then” (i.e., “only if”) as the
meaning of →.
1.2.2
Truth Tables
The dependence of a complex formula’s truth value on the truth values
of its constituents can be displayed graphically by means of a truth table.
Truth tables come in two varieties: truth tables for connectives, and truth
tables for formulas. A truth table for a connective ∗ displays in a general
fashion how a complex formula in which ∗ is the main connective depends
for its truth value on the formula’s immediate constituent(s). The truth
tables for ¬ and → are as follows:
ϕ
T
T
F
F
ϕ
T
F
¬ϕ
F
T
ψ
T
F
T
F
ϕ→ψ
T
F
T
T
The rows of these truth tables represent all possible combinations of truth
values that might be assigned to the immediate constituents of a complex
formula of the form in question, and the columns the truth values assigned
to the formulas themselves given the truth values assigned to their immediate constituents.
A truth table for a complex formula ϕ applies the truth tables for connectives explicitly to each of ϕ’s complex constituents to display explicitly
how its truth value depends on the truth values of its constituents — and
ultimately its atomic constituents — under any interpretation. The following table shows this dependence for formulas of the form ( p → ¬q) →
¬(q → p), for distinct p and q. Each row represents a way in which truth
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values can be assigned to the atomic constituents p and q, i.e., more exactly,
it represents the class of interpretations that assign those truth values to p
and q.
p
T
T
F
F
q
T
F
T
F
¬q
F
T
F
T
p → ¬q
F
T
T
T
q→p
T
T
F
T
¬(q → p)
F
F
T
F
( p → ¬q) → ¬(q → p)
T
T
F
F
In general, for a formula with n distinct atomic constituents there will be
2n distinct ways of assigning truth values to those constituents, and hence
2n rows in its truth table.
It is useful to give the truth tables for the defined connectives to see
that they behave as expected; complete truth tables for the formulas of L P
used to define these connectives yield the following (we combine three
tables into one to save space):
ϕ
T
T
F
F
1.2.3
ψ
T
F
T
F
ϕ∧ψ
T
F
F
F
ϕ∨ψ
T
T
T
F
ϕ↔ψ
T
F
F
T
Truth, Logical Truth, and Related Notions
Let ϕ be a formula of L P and Γ any set of formulas of L P .
Truth in an interpretation
ϕ is true in an interpretation I = V of L P — symbolically, |=I ϕ — iff V ( ϕ) =
T. ϕ is false in I iff V ( ϕ) = F.
Logical truth
ϕ is logically true — symbolically, |= ϕ — iff |=I ϕ, for all interpretations I ,
i.e., iff ϕ is true in all interpretations.
• E.g., ( p → p) and ( p ∨ ¬ p) → q are logically true.
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Logical falsity
ϕ is logically false, or unsatisfiable, iff not-|=I ϕ, for all interpretations I , i.e.,
iff ϕ is false in all interpretations.
Satisfiability
ϕ is satisfiable iff |=I ϕ, for some interpretation I , i.e., iff ϕ is true in at least
one interpretation.
• Logical truth, of course, implies satisfiability, but not vice versa; e.g.,
p → q is satisfiable but not logically true.
Logical equivalence
ϕ and ψ are logically equivalent iff, for all I , |=I ϕ iff |=I ψ, i.e., iff ϕ and ψ are
true in the same interpretations.
• E.g., p → q and ¬q → ¬ p are logically equivalent.
Models
I is a model of Γ — symbolically |=I Γ — iff |=I ϕ, for all ϕ ∈ Γ, i.e., iff every
formula of Γ is true in I . Γ is said to be satisfiable if it has a model, and
unsatisfiable, otherwise.
• E.g., Let V ( p) = T, for all propositional constants p, and let Γ be the
set of all formulas of L P that do not contain the negation operator ¬.
Then the interpretation h{T, F}, V i is a model of Γ. (Why?)
• The set { p, p → q, ¬q} is unsatisfiable.
Entailment
Γ entails ϕ — symbolically, Γ |= ϕ — iff, for all interpretations I , |=I Γ
implies |=I ϕ, i.e., iff ϕ is true in every model of Γ.
• E.g., { p → (q → r ), p, ¬r } |= ¬q
• An unsatisfiable set entails every formula. (Why?)
Note that logical truth and satisfiability (hence also logical falsehood and
unsatisfiability) are interdefinable.
T HEOREM : There is an effective procedure for determining whether or not
a given formula is logically true (equivalently: logically false, satisfiable).
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Proof: Construct appropriate truth tables.
Because the size of a formula’s truth table is an exponential function of
the number of its atomic constituents, this procedure is highly intractable.
And while there are more efficient procedures, the general problem of determining logical truth (logical falsehood, satisfiability) is intractable, i.e.,
there is no guarantee that the logical truth of an arbitrary sentence can be
calculated in a “reasonable” amount of time.3
1.2.4
Truth Functions
An (n-place) truth function is a function from {T, F}n into {T, F}. That is,
an n-place truth function takes n-tuples of truth values as arguments and
returns a truth value. There are four 1-place truth functions:
x
f1 (x)
f2 (x)
f3 (x)
f4 (x)
T
F
T
T
T
F
F
T
F
F
There are sixteen 2-place truth functions:
x
y
g1 ( x, y)
g2 ( x, y)
g3 ( x, y)
g4 ( x, y)
g5 ( x, y)
...
g16 ( x, y)
T
T
F
F
T
F
T
F
T
T
T
T
F
T
T
T
T
F
T
T
T
T
F
T
T
T
T
F
...
...
...
...
F
F
F
F
n
In general, there are 22 n-place truth functions.
The formula ¬ p0 can be said to express the truth function f 3 in the following sense: Given any interpretation I = h D, V i, V (¬ p0 ) = f 3 (V ( p0 )).
Similarly, the formula p0 → p1 can be said to express the truth function g3
in the sense that, for any I = h D, V i, V ( p0 → p1 ) = g3 (V ( p0 ), V ( p1 )).
This idea can be extended to all formulas as follows. Let ϕ be a formula of L P , and suppose that ψ1 , ..., ψn are the propositional constants occurring in ϕ ordered by their indices. (So, e.g., p2 would be “before” p5 in
3 More
exactly, the problem of determining validity is said to be “NP-complete.” For
more on the subject of tractability, see, e.g., Hopcroft and Ullman, Introduction to Automata
Theory, Languages, and Computation, Addison Wesley, 1979, Chs. 12-13.
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the ordering.) Then ϕ expresses the n-place truth function f iff, for all interpretations I = h D, V i, V ( ϕ) = f (V ( p1 ), ..., V ( pn )). The truth function
expressed by a formula, of course, can be identified simply by constructing
the formula’s truth table.
• The truth functions g1 , g2 , g4 , g5 , g6 , and g16 are expressed by the formulas p0 → ( p1 → p0 ), p0 → ¬ p1 , ¬ p0 → ¬ p1 , ¬ p0 → p1 , p0 →
¬( p1 → p1 ), and ¬( p0 → ( p1 → p0 )), respectively.
1.2.5
Adequate Sets of Connectives
That every formula of L P expresses a truth function raises the issue of
whether every truth function is expressed by some formula of L P . The
issue here hinges on the connectives of L P . A set of connectives in an interpreted language (i.e., a language together with its semantics) for propositional logic is said to be adequate iff every truth function can be expressed
by some formula of the language. The question, then, is whether the set
{¬, →} is adequate. In fact it is. Indeed, if we take the connectives ∧ and
∨ as primitive (i.e., we don’t define them, but simply include them in the
language with the truth tables above), the following can be shown:
T HEOREM : The sets {¬, →}, {¬, ∧}, and {¬, ∨} are all adequate.
Proof sketch: Given an explicit truth function, it is easy to construct a formula containing only the connectives ∧, ∨, and ¬ that expresses it. Because ϕ ∧ ψ and ¬(¬ ϕ ∨ ¬ψ) are logically equivalent, this means that
any formula containing no connectives other than ∧, ∨, and ¬ is logically
equivalent to one containing only ∨ and ¬. By the same equivalence, any
such formula is equivalent to one containing only ∧ and ¬, which is equivalent to one containing only → and ¬, by the equivalence of ϕ ∧ ψ and
¬( ϕ → ¬ψ).
T HEOREM : No subset of {→, ∧, ∨, ↔} is adequate.
Proof idea: No formula containing only a single constant and the connectives in the above set can express negation (i.e., the truth function f 3 ).
T HEOREM : {¬, ↔} is not adequate.
Proof sketch: By induction it is straightforward to show that no formula
containing only two propositional constants and these connectives can express a 2-place truth function that has an odd number of Ts or Fs among
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its values. So in particular, no such formula will be able to express a conditional, conjunction, or disjunction.
1.3
1.3.1
Proof Theory: The System P
Central notions of proof theory
Proofs and provability
A proof in a formal system S in a language L is a sequence ϕ1 , ..., ϕn of
formulas of L such that every element ϕi of the sequence is either (i) an
axiom of S or (ii) follows from formulas occurring earlier in the sequence
by a rule of inference. If ϕ is the last element of a proof ϕ1 , ..., ϕn (so ϕn =
ϕ) the proof is also said to be a proof of ϕ in S , written `S ϕ. ϕ is said to be
provable in S , or a theorem of S , if there is a proof of ϕ in S .
The Consequence relation
Let Γ be a set of formulas of L. Then ϕ1 , ..., ϕn is a proof from Γ iff each ϕi
is either (i) an axiom of S , (ii) a member of Γ, or (iii) follows from formulas occurring earlier in the sequence by a rule of inference. In that case,
ϕ1 , ..., ϕn (= ϕ) is a proof of ϕ from Γ in S , written Γ `S ϕ. ϕ is said to be
provable from, or a consequence of, Γ in S if there is a proof of ϕ from Γ in S .
Some simple properties of the consequence relation are the following:
• If Γ ⊆ ∆ and Γ `S ϕ, then ∆ `S ϕ.
• Γ `S ϕ iff ∆ `S ϕ, for some finite subset ∆ of Γ.
• If ∆ `S ϕ and Γ `S ψ, for all ψ ∈ ∆, then Γ `S ϕ.
Consistency
A set Γ of formulas of L P is consistent iff for no formula ϕ do we have both
Γ `P ϕ and Γ `P ¬ ϕ. Γ is inconsistent iff it is not consistent, i.e., iff Γ `P ϕ and
Γ `P ¬ ϕ, for some formula ϕ.
1.3.2
The System
Let ϕ, ψ, and θ be formulas of L P . Then every instance of any of the following schemas is an axiom of P.
P1
ϕ → (ψ → ϕ)
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P2
P3
( ϕ → (ψ → θ )) → (( ϕ → ψ) → ( ϕ → θ ))
(¬ ϕ → ψ) → ((¬ ϕ → ¬ψ) → ϕ)
Rule of Inference for P
MP
ψ follows from ϕ and ϕ → ψ.
A simple but useful consequence of MP and the notion of proof is the
following:
(∗) If Γ `P ϕ and Γ `P ϕ → ψ, then Γ `P ψ.
Example
T HEOREM : `P p → p
The following sequence of formulas constitutes a proof of p → p in P.
1. ( p → (( p → p) → p)) →
(( p → ( p → p)) → ( p → p))
Axiom schema P2
2. p → (( p → p) → p)
Axiom schema P1
3. ( p → ( p → p)) → ( p → p)
From 1 and 2, by MP
4. p → ( p → p)
Axiom schema P1
5. p → p
From 3 and 4, by MP
The Deduction Theorem
The Deduction Theorem states that our system captures an important intuitive property of deduction, viz., that if we can prove one statement ψ by
assuming another ϕ, then the conditional statement ϕ → ψ must be true:
T HEOREM: If Γ ∪ { ϕ} `P ψ, then Γ `P ϕ → ψ.
The proof of the theorem is a straightforward induction on the number of
connectives. The Deduction Theorem enables one to shorten the lengths
many proofs considerably.
T HEOREM : Every formula is a consequence of an inconsistent set.
Proof: Suppose Γ is inconsistent. Then for some formula ϕ, Γ `P ϕ and
Γ `P ¬ ϕ. Given this we reason as follows. Let ψ be any formula of L P .
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1.
2.
3.
4.
5.
6.
7.
8.
9.
Γ
Γ
Γ
Γ
Γ
Γ
Γ
Γ
Γ
`P ϕ
`P ¬ ϕ
`P ϕ → (¬ψ → ϕ)
`P ¬ψ → ϕ
`P ¬ ϕ → (¬ψ → ¬ ϕ)
`P ¬ψ → ¬ ϕ
`P (¬ψ → ϕ) → ((¬ψ → ¬ ϕ) → ψ)
`P (¬ψ → ¬ ϕ) → ψ
`P ψ
Given
Given
Axiom schema P1
From 1 and 3, by (∗)
Axiom schema P1
From 2 and 5, by (∗)
Axiom schema P3
From 4 and 7, by (∗)
From 6 and 8, by (∗)
Note the obvious parallel between the semantic notion of unsatisfiability
and the syntactic, proof theoretic notion inconsistency: every formula is
entailed by an unsatisfiable set, and every formula is a consequence of an
inconsistent set. This raises the question of whether the unsatisfiable sets
are exactly the inconsistent sets, or equivalently, whether a set is satisfiable
if and only if it is consistent. This question is answered by the soundness
and completeness theorems.
1.4
Soundness and Completeness
The central purpose of a proof theory in a formal system (on one view, at
least) is to capture in a mechanical way the semantics of the system. More
exactly, a proof theory aims to have as theorems all and only the logical
truths of the system. Proof theories with this property are said to be sound
and complete with respect to the semantics in question. More specifically,
with respect to the system at hand:
Soundness
P is sound if and only if Γ `P ϕ implies Γ |= ϕ, i.e., if and only if every
consequence of a set Γ of formulas is entailed by Γ. In the case where
Γ = ∅, soundness says that every theorem of P is a logical truth.
Completeness
P is complete if and only if Γ |= ϕ implies Γ `P ϕ, i.e., if and only if formula
Γ entails is a consequence of Γ. Again, where Γ = ∅, completeness says
that every logical truth is a theorem of P.
T HEOREM : P is sound and complete.
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Soundness is equivalent to the claim that every satisfiable set of formulas
is consistent, and Completeness is equivalent to the converse, i.e., that every consistent set of formulas is satisfiable. Hence, relative to the system
P, a set is satisfiable if and only if it is consistent.
2
Predicate Logic
We turn now to predicate logic. Predicate logic is vastly more expressive
than propositional logic, as it enables us to represent the much of the subsentential components of sentences — names, verb phrases, quantifiers,
etc. This makes it possible to formally capture the validity of a huge number of arguments whose validity depends on the logical properties of these
components, rather than simply their boolean components.
2.1
The Languages of Predicate Logic
Whereas there is essentially only one (countable) language of propositional logic, there are infinitely many languages of predicate logic. Some
of these have as few as one nonlogical lexical item, others many more. To
deal with this variability, we introduce the idea of the lexical stock of predicate logic, on which a given language will draw.
2.1.1
The Lexical Stock
The lexical stock of predicate logic consists of the following denumerable
classes:
• The propositional constants of L P .
• Individual constants: a0 , b0 , c0 , a1 , b1 , c1 , ....
• n-place predicates: P0n , Q0n , R0n , P1n , Q1n , R1n , ..., for all n > 0.
2.1.2
Languages LQ for Predicate Logic
A language LQ of predicate logic consists of a lexicon and a grammar. The
lexicon of LQ consists of
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• Propositional constants: Some (possibly empty) subset the propositional constants from the lexical stock.
• Individual constants: Some (possibly empty) subset the individual
constants from the lexical stock.
• n-place predicates: Some nonempty subset the n-place predicates
from the lexical stock.
• Individual variables: x0 , y0 , z0 , x1 , y1 , z1 , ...
• Universal quantifier symbol: ∀
Individual constants and variables are known collectively as terms. As
above, an expression of LQ is any finite string of elements of the lexicon
of LQ . The metavariables ϕ, ψ, and θ will range over expressions of LQ ,
and the metavariables p, q, and r (possibly with numerical subscripts) will
range over the propositional constants, a, b, and c will range over individual constants, x, y, and z will range over individual variables, and Pn , Qn ,
and Rn will range over n-place predicates, though numerical subscripts
will usually be suppressed. An expression of the form ∀ x is called a universal quantifier.
2.1.3
The Grammar of LQ
The grammar of LQ is defined recursively as follows.
1. Every propositional constant is an (atomic) formula (of LQ ).
2. If P is an n-place predicate and t1 , ..., tn are any terms, then P(t1 , ..., tn )
is an (atomic) formula.
3. If ϕ and ψ are formulas, so are ¬ ϕ and ( ϕ → ψ).
4. If ϕ is a formula and x is any variable, then ∀ xϕ is a formula.
5. Nothing is a formula of LQ except those expressions generated by
rules 1 through 5.
13
As always, outermost parentheses can be dropped. We extend the notion of immediate constituency by stipulating that ϕ is the immediate constituent of ∀ xϕ, and ∀ x is its main logical operator. We also say that the main
connective of a boolean formula is its main logical operator. A formula
whose main connective is a universal quantifier is said to be universally
quantified.
Scope, Freedom, and Bondage
The scope of an occurrence of a universal quantifier ∀ x in a formula ϕ is that
constituent of ϕ of which that occurrence of ∀ x is the main connective. So,
for example, the scope of the occurrence of ∀y in (∀ x )( Px → ¬∀y( Qxy →
¬ Rzyx )) is ∀y( Qxy → ¬ Rzyx )). An occurrence of a variable x in ϕ is
said to be free if it does not occur in the scope of any occurrence of the
quantifier ∀ x. Otherwise the occurrence is said to be bound. A variable y
is free for a variable x in ϕ if no free occurrence of x in ϕ is in the scope of
an occurrence of the quantifier ∀y. (The intuitive idea here is that y won’t
become bound by any quantifier in ϕ when it is substituted for any free
occurrence of x in ϕ.)
We define the existential quantifier ∃ in the usual fashion, viz., (∃ x ) ϕ =d f
¬∀ x ¬ ϕ.
2.2
2.2.1
Model Theory for LQ
Interpretations
An interpretation I of LQ is a pair h D, V i, where D — the domain of I —
is an arbitrary nonempty set and V is a valuation function that assigns
meanings to elements of the lexicon as follows:
1. If p is a propositional constant, then V ( p) ∈ {T, F};
2. If t is a term, then V (t) ∈ D.
3. If P is an n-place predicate, then V ( P) ⊆ D n .
Clause 3 here deserves a little attention. By ‘D n ’ we mean the nth cartesian
product of D, i.e., the set of all n-tuples of elements of D — so V ( P) is a set
of n-tuples of members of D. In the case where n = 1, we define D n just to
be D, so that V ( P) in that case will just be a subset of D.
14
Just as a valuation function in an interpretation of propositional logic
determines the truth value of every complex formula of L P , so the function
V in an interpretation of LQ determines the truth value of every complex
formula of LQ . More exactly, V determines a unique function V mapping
each formula into a truth value. (We don’t call vbar an extension of V in
this case because V is not defined on terms and predicates. It is, however, an extension of that portion of V that is defined on propositional
constants.)
First, a definition. Let I = h D, V i be an interpretation for LQ , and let e
be any element of D. Then we define V [ x/e] to be the valuation function
just like V, except that V [ x/e] assigns e to x in place of what V assigns to
x. (So, in particular, if it happens that V ( x ) = e, then V [ x/e] just is V.)
So let ϕ and ψ be formulas of LQ . We define V to be a total function on
the formulas of LQ into the set {T, F} as follows.
1. V ( p) = V ( p), for any propositional constant p.
2. V ( P(t1 , ..., tn )) = T iff hV (t1 ), ..., V (tn )i ∈ V ( P).4
3. V (¬ ϕ) = T iff V ( ϕ) = F.
4. V ( ϕ → ψ) = T iff V ( ϕ) = T only if V (ψ) = T.
5. V (∀ xϕ) = T iff, for all e ∈ D, V [ x/e]( ϕ) = T.
There are two new clauses to take note of here – Clauses 2 and 5 — that
stipulate how the truth values of sentences with subsentential components
(terms, predicates, and quantifiers) are determined in terms of the meanings of their component parts. Clause 2 captures the semantical intuition
that a simple atomic sentence like “John loves Mary” is true if and only
if Austin and Barbara, respectively, stand in the loving relation, that is, if
and only of the referent of ‘Austin’ and the referent of ‘Barbara’, respectively, stand in the relation expressed by the predicate ‘loves’. Relations, in
turn, are represented formally in basic predicate logic as sets of n-tuples. ntuples are used to capture the fact that, in general, for each n-place relation
there is a set of n roles that are played whenever some things stand in the
relation. Thus, in the case of the loving relation, there are the roles of lover
4 In
the case n = 1, we define hV (t)i just to be V (t) so that this clause works as it
should with the definition of ‘V ( P)’ just given above.
15
and beloved, and it is critical that the two are not confused. For basic formal purposes it is enough simply to represent these roles simply by identifying each one with one of the slots in an n-tuple. Thus, for most formal
purposes we can identify the loving relation with the set L of ordered pairs
(i.e., ordered 2-tuples) such that the first element of the pair loves the second. Hence, the fact that Austin stands in the loving relation with Barbara
is represented by the formal fact that the ordered pair hAustin, Barbarai is
a member of the set L, i.e., hAustin, Barbarai ∈ L. If we let V ( a0 ) = Austin,
V (b0 ) = Barbara, and V ( P20 ) = the set of pairs h x, yi such that x loves
y, then we have V ( P20 ( a0 , b0 )) = T iff hAustin, Barbarai ∈ L — just as in
Clause 2 of our semantics.
Clause 5, of course, is the clause for quantified sentences. A simple
example might be useful to illustrate how it works. Let our interpretation
I = hD , V i be such that D is the set of humans, and let V ( P) be the set of
people residing in College Station and V ( Q) be the set of people residing
in Texas. Then we have that V (∀ x ( P( x ) → Q( x ))) = T iff for all e ∈ D,
V [ x/e](( P( x ) → Q( x )) = T iff for all e ∈ D, V [ x/e]( P( x )) = T only if
V [ x/e]( Q( x )) = T (by Clause 4) iff for all e ∈ D, V [ x/e]( x ) ∈ V [ x/e]( P)
only if V [ x/e]( x ) ∈ V [ x/e]( Q) (by Clause 1) iff for all e ∈ D, e ∈ V ( P)
only if e ∈ V ( Q) (by the definition of V [ x/e]. But given our definitions of
D and V, this latter assertion holds only if, for all humans e, e resides in
College Station only if e resides in Texas, i.e., more idiomatically, iff every
College Station resident is a Texas resident.
2.3
Proof Theory: The System Q
Generalization
A generalization of a formula ϕ is the result of prefixing ϕ with one or more
universal quantifiers.
The system Q of predicate logic consists of all instances of the following
schemas along with their generalizations:
P1
ϕ → (ψ → ϕ)
P2
( ϕ → (ψ → θ )) → (( ϕ → ψ) → ( ϕ → θ ))
(¬ ϕ → ψ) → ((¬ ϕ → ¬ψ) → ϕ)
∀ x ( ϕ → ψ) → (∀ xϕ → ∀ xψ)
ϕ → ∀ xϕ, if x is not free in ϕ
P3
Qu1
Qu2
16
UI
∀ xϕ → ϕtx
The rule of inference for Q is MP.
Note that P1-P3 are not exactly the same schemas as in the system P,
since the metavariables here now range over formulas of LQ .
The idea behind allowing the generalization of every axiom to be an
axiom is this. Suppose that an axiom ϕ contains a free variable. Then
surely what qualifies ϕ as an axiom has nothing to do with the particular
entity that x happens to refer to — let ϕ be, for example, a formula of the
form ∀yPy → Px, i.e., an instance of UI. Surely the validity of the this
formula has nothing to do with whatever x happens to denote in a model;
indeed, one can let it denote whatever one wishes in any model and it will
still be true in the model. Consequently, the generalization of the formula
— ∀ x (∀yPy → Px ) will surely be true as well, and hence (since the same
goes for any model) valid.
The system Q is sound and complete with respect to its semantics.
2.4
Identity
It is a simple matter to add the important relation of identity to predicate
logic. On the syntactic side, first, one designates one of the 2-place predicates to be “special” or “distinguished” — we will stipulate that this is to
be the predicate P12 . Next, we introduce a convention in the grammar: we
allow the predicate P12 to be written in the more familiar manner as =. Finally we allow atomic formulas involving the predicate = to be written in
“infix” notation, in which the predicate is placed in between its arguments
instead of to the left, to accord with common usage. Thus, we will write
t1 = t2 instead of =(t1 , t2 ).
Semantically, we have to alter the notion of a model to ensure that the
identity predicate is always interpreted in the same general way. Specifically, given an interpretation I = hD , V i, we stipulate that V (=) = {he, ei | e ∈
D }. This ensures that the identity predicate is always interpreted in a way
that really means identity.
Finally, we need to supplement the axioms of predicate logic in order
to capture the intended meaning of the identity predicate. The following
two schemas are standard:
Id
x=x
17
Ind
x = y → ( ϕ → ϕ0 ), where y is free for x in ϕ, and ϕ0 is the result
of replacing one or more free occurrences of x in ϕ with y.
Call this system Q= . Like Q, Q= is sound and complete with respect
to its semantics.
3
Modal Propositional Logic
3.1
The Languages of Modal Propositional Logic
To make L P into a modal language L P2 we simply add a single element 2
to the lexicon of L P and we alter grammatical clause 2 of Section 1.1.2 as
follows:
2’. If ϕ and ψ are formulas of L P , so are 2ϕ, ¬ ϕ, and ( ϕ → ψ).
Again, outermost parentheses can be dropped. The notion of immediate constituency is extended by stipulating that ϕ is the immediate constituent of 2ϕ.
Just as various other boolean operators can be defined in terms of ¬
and →, other modal operators can be defined in terms of 2. The most
common definition is that of the operator ‘it is possible that’ as follows:
3ϕ =df ¬2¬ ϕ
Note the similarity of this definition to the definition of the existential
quantifier ∃. We will return to this in the next section.
3.2
3.2.1
Model Theory for L P2
Interpretations
The problem of interpreting the modal operators 2 and 3 faced by the
early twentieth century modal logicians is that their intended interpretations — ‘necessarily’ (or, ‘it is necessary that’) and ‘possibly’ (or, ‘it is
possible that’) — are not truth functional. Thus, for some true sentences ϕ
(‘There are nine planets’, say) ‘Necessarily, ϕ’ is false, whereas for others
(‘Everything that is red is colored’, say) ‘Necessarily ϕ’ is true. Similarly,
for some false sentences ϕ (‘There are seven planets’, say) ‘Possibly, ϕ’ is
18
true, whereas for others (‘Some bachelors are married’, say) ‘Possibly ϕ’ is
false. Thus, unlike sentences containing only boolean operators, the truth
values of sentences of the form 2ϕ and 3ϕ are not in general determined
by the truth value of ϕ alone. How, then, can the meaning of the modal operators be represented formally by the sorts of clear mathematical models
available in nonmodal propositional and predicate logic?
The insight that led to the solution of this problem built upon the Leibnizian intuition that a proposition is necessary if and only if it is true in all
possible worlds, possible if and only if true in at least one. These equivalences enable one to give the modal operators a sort of “generalized” truth
functional semantics: while the truth value of 2ϕ or 3ϕ is not in general determined solely by the truth value of ϕ — i.e., its truth value here
in the actual world — it is determined by ϕ’s truth value across some or
all possible worlds. The semantics of a modal language is thus a natural
generalization of the semantics for nonmodal languages.
Formally, then, the interpretation of the modal operators requires some
representation of the notion of a possible worlds, along with a representation of truth at such a world. The former is achieved by simply introducing a nonempty set W into the notion of an interpretation. The latter
is achieved by extending the notion of a valuation function so that it assigns truth values to propositional constants with respect to members of
W, rather than “absolutely”.
To capture these ideas formally, we define an interpretation I for L P2
to be a triple hW, wo , V i where W is a nonempty set, w0 is a distinguished
elements of W, and V is a function that takes a propositional constant p
and a world w into the set of truth values {T, F}. That is, for all such p and
w, V ( p, w) ∈ {T, F}. V ( p, w) is said to be the truth value of p at w in I .
As in nonmodal propositional logic, every valuation function V in an
interpretation I = hW, w0 , V i of L P2 determines a unique extension V
that assigns a truth value to each formula of L P2 at each “possible world”
as follows. Let ϕ and ψ be formulas of L P2 , and let w ∈ W.
1. V ( p, w) = V ( p, w), for propositional constants p.
2. V (¬ ϕ, w) = T iff V ( ϕ, w) = F.
3. V ( ϕ → ψ, w) = T iff V ( ϕ, w) = T only if V ( ϕ, w) = T (equivalently,
if V ( ϕ, w) = F or V ( ϕ, w) = T).
19
So far, of course, except for the addition of the world parameter, nothing
distinguishes the notions of a valuation function V and its extension V for
L P2 from the corresponding notions for L P . But L P2 also contains the
modal operator ‘2’, and so a corresponding semantic clause needs to be
specified for its interpretation.
4. V (2ϕ, w) = T if, for all w0 ∈ W, V ( ϕ, w0 ) = T; otherwise, V (2ϕ, w) =
F.
Clause 4, of course, formally represents the idea that a proposition is necessary if and only if it is true in all possible worlds.
Note that, given the definition of the possibility operator 3 above, the
corresponding semantic clause for 3 follows:
(1)
If ϕ is 3ψ, then V ( ϕ, w) = T if, for some w0 ∈ W, V (ψ, w0 ) = T;
otherwise, V ( ϕ, w) = F.
That is, intuitively, a proposition is possible if and only if it is true in some
possible world.
In effect, then, modal operators are quantifiers over possible worlds. This
fact explains the structural similarity of the definitions for the possibility
operator and the existential quantifier: the possibility operator just is (semantically) a restricted existential quantifier, viz., an existential quantifier
over possible worlds. So of course the definition of the 3 should be structurally similar to the definition of ∃.
The notion of truth simpliciter (in an interpretation I ) can now be defined straightforwardly as truth at the actual world. That is, formally,
Truth in an interpretation
ϕ is true in an interpretation I = hW, w0 , V i of L P — symbolically, |=I ϕ —
iff V ( ϕ, w0 ) = T. ϕ is false in I iff V ( ϕ, w0 ) = F.
That this semantics is a natural generalization of our nonmodal semantics for propositional languages can be seen in the fact that, if there are
only finitely many “worlds” in the set W of an interpretation I of L P2 ,
we can construct a sort of modal truth table for an arbitrary sentence of
L P2 that illustrates how the semantics for L P2 works. Thus, suppose
I = hW, V i, where W = {w1 , w2 , w3 }, and where V assigns the propositional constant p the values T, F, T, q the values T, T, T, and r the values
20
F, F, T at the worlds w1 , w2 , and w3 , respectively. Then we can construct
the following table to show how the truth value of the complex sentence
2( p → (2q → r ) of L P2 at each world is determined by the truth values
of its immediate constituents at that and, in the case of sentences of the
form 2ϕ, other worlds:
w1
w2
w3
p
T
F
T
q
T
T
T
r
F
F
T
2q
T
T
T
2q → r
F
F
T
p → (2q → r )
F
T
T
2( p → (2q → r ))
F
F
F
Note in particular how the truth value at a world of each constituent of
2( p → (2q → r )) of the form ϕ → ψ is determined by the truth values
of its immediate constituents at that world (since → is truth functional),
while the truth values of those constituents of the form 2ϕ is determined
by the truth value of its immediate constituent at (in general) all worlds.
Note further (i.e., convince yourself) that, in this semantics, if a sentence
of the form 2ϕ is true (false) at any world, it is true (false) at all worlds.
3.2.2
Refinement: Generalized Modalities
In fact, the property of our modal semantics just noted in the last sentence
of the previous paragraph leads to difficulties. An important way in which
the modal operators differ from the boolean operators is the impact that
different metaphysical and semantical views can have on our choice of
logical principles. Recognition of this diversity suggests that the notion of
an interpretation just introduced is not sufficiently general. Suppose, for
instance, that we agree that the proposition (expressed by) ϕ is necessary.
Does it follow that ϕ had to be necessary; that is, is ϕ necessarily necessary?
More formally, for any well-formed formula ϕ, does 22ϕ follow from 2ϕ?
Equivalently, given the interdefinability of possibility and necessity, from
the fact that p could be possible, does it follow that it is possible? That is,
more formally again, does 3ϕ follow from 33ϕ?
If you answered yes, then you believe that the necessary truths of our
world, the actual world, are necessary in every world; or equivalently,
that the possibilities of any given world are possibilities in our world as
well. But, although reasonably intuitive, depending on certain metaphysical choices we might make, these principles can become dubious. For
21
instance, there is a strong intuition that artifacts maintain their identity
through small changes. Consider, a large wooden table t.5 Intuitively, t
could have differed in a small number of its constituent atoms, or in some
miniscule way in its shape, and it still would been the same table. To help
us think about this,
P0 be the property that t has in virtue of being exactly as it is here in the
actual world w0 — its exact atomic make-up and its exact geometry — and
let p0 express the proposition that t has P0 . Let P1 be the analogous property that t has in a possible world w1 in which its shape and constituent
atoms differ ever so slightly from its shape and constituents in w0 , and let
p1 express the proposition that t has P1 . Intuitively, then, it is possible that
t have property P1 . Now let P2 indicate a slight difference in shape and
constituent atoms from P1 , though with even fewer common atoms than
those involved in P0 , and let p2 express that t has P2 . Intuitively, just as t
as it is in w0 could be in state P1 , t as it would be in w1 could be in state
P2 . That is, it is possible that t be in such a state — notably, P1 — that it is
possible that it be in state P2 ; otherwise put, it is possibly possible that t be
in state P2 , i.e., 3p2 .
Now, it remains intuitive that, after only two such small changes, it
is, not merely possibly possible, but possible straightaway that t, as it is
here in the actual world, be in state P2 . Eventually, however, after enough
incremental changes, we will have a world wn , for some very large number
n, such that the corresponding state Pn is so remote from t’s initial state P0
that it can’t be considered a state that t, as it is in the actual world, could
possibly be in — after so many incremental changes, Pn could, for instance,
be constituted by completely different atoms that jointly form an artifact
with a completely different shape, e.g., a cabinet.
To simplify matters, let’s idealize a bit (well, a lot) and suppose that
n = 2, i.e., that, while it is possible that t, as it is (i.e., in state P0 ), be in
state P1 , and while it is possible that t as it would be in state P1 could
retain its identity by undergoing a change into state P2 , it is not possible
that t, as it is, in state P0 , be in state P2 . Then it appears that our reasoning
has us acknowledging all of the following as true:
5 This
example comes from Nathan Salmon, “The Logic of What Could have Been,”
[REF].
22
p0
3p1
33p2
¬3p2
(t is in state P0 )
(t could be in state P1 )
(t could be in a such a state — notably, P1 — that it would be
possible for it to be in state P2 )
(t, as it actually is, can’t possibly be in state P2 )
But notice now that if we agree that 3ϕ follows from 33ϕ, then, from the
third of our propositions above, it follows that 3p2 . But that contradicts
our fourth proposition.
This example uncovers a problem with our notion of an interpretation
that we defined in Section 3.2.1. For, given the the notion of truth in an
interpretation defined there, the inference from 33ϕ to 3ϕ is valid! The
reason for this is that possibility as characterized there there is represented
simply to be truth in some possible world or other. This effectively puts
every possible world on an equal footing relative to every other world,
thus rendering the possibly possible and the actually possible equivalent.
To see this explicitly, we note first that, by clause 4 in the definition of truth
in a world — specifically, by its corrollary (1) — 33ϕ is true in a world w
iff
(2)
There is some possible world w0 such that 3ϕ is true in w0 .
Unpacking “3ϕ is true in w0 ” it follows that (2) is equivalent to
(3)
There is some possible world w0 such that, for some possible world
w00 , ϕ is true in w00 .
But the initial quantifier “There is some possible world w0 such that” is
now vacuous; that is, (3) is equivalent to
(4)
There is some possible world w00 such that ϕ is true in w00 .
But (4) holds, by (1) again, iff 3ϕ is true in w. A subtler picture is needed
if we are to be able to a representation of modality that is flexible enough
to capture the intuitive reasoning above about the modal properties of artifacts like tables.
3.2.3
Interpretations Again: Accessibility
The required subtlety is introduced by means of an accessibility relation
on possible worlds. Intuitively, the idea is that not every world is possible
23
relative to every other world. A proposition is thus defined to be necessary
at a world w if it is true, not at all worlds simpliciter, but rather if it is true
in all the worlds that are possible relative to w, that is, all the worlds that
are accessible from w. Hence, for a proposition p (the one expressed ‘P2 t’,
say) to be possibly possible in a world w means that it is true in some world
w00 that is accessible from some world w0 that is accessible from w. But,
without further conditions on the accessibility relation, it will not follow
that w00 is accessible from w, and hence that p is possible in w. In particular,
it will not follow from the truth of 33P2 t at w0 that 3P2 t is true there as
well.
Formally, then, the notion of an interpretation will be just as before
except for the presence of an accessibility relation. Thus, an interpretation
I for L P2 is now a 4-tuple hW, wo , R, V i where W is a nonempty set, w0 is
a distinguished elements of W, R is a binary relation on W (that is, a set of
ordered pairs consisting of members of W), and V is a function that takes a
propositional constant p and a world w into the set of truth values {T, F}.
That is, for all such p and w, V ( p, w) ∈ {T, F}. V ( p, w) is said to be the
truth value of p at w in I .
As before, an interepretation I = hW, w0 , R, V i of L P2 determines a
unique extension V of V that assigns a truth value the formulas of L P2 at
each member of W. The only difference with our first-cut notion of truth
in an interpretation is in the modal clause:
40 . V (2ϕ, w) = T if, for all w0 ∈ W such that Rww0 ,6 V ( ϕ, w0 ) = T;
otherwise, V (2ϕ, w) = F.
Given our definition of 3, we can derive a corresponding clause for sentences of the form 3ϕ:
400 . V (3ϕ, w) = T if, for some w0 ∈ W such that Rww0 , V ( ϕ, w0 ) = T;
otherwise, V (3ϕ, w) = F.
Now, return to our example in Section 3.2.2, and consider the interpretation I = hW, w0 , R, V i such that W = {w0 , w1 , w2 }, R = {hw0 , w1 i, hw1 , w2 i
(i.e., Rw0 w1 and Rw1 w2 ) and such that V ( p0 , w0 ) = 1, V ( p1 , w1 ) = 1, and
V ( p2 , w2 ) = 1. Since p2 is true at w2 , 3p1 is true at w1 , because w2 is accessible from w1 , e.g., Rw1 w2 . Hence, because Rw0 w1 , 33p2 is true at w0 .
However, because w2 is not accessible from w0 , 3p2 is not true at w0 .
6 That
is, more exactly, such that hw, w0 i ∈ R.
24
3.2.4
Validity With Respect to a Class of Interpretations
The preceding example shows that the addition of an accessibility relation
gives one a greater degree of flexibility regarding the modal propositions
one admits as valid. Note, in particular, that if restrict our attention to
interpretations in which the accessibility relation is transitive — i.e., those
interpretations in which the worlds accessible from those accessible from
a given world w are also accessible to w — then 3p2 is true at any world
w if 33p2 is
Specifically, by fine-tuning the accessibility relation one way or another
and restricting one’s attention to interpretations that are so tuned, one can
3.3
Proof Theory: The Systems K, S4, and S5
3.3.1
The System K
The previous section showed that our modal intuitions concerning the validity of certain modal principles can vary depending on our metaphysical
presuppositions. Consequently, there are different systems of modal logic,
depending on the principles that we take to be valid and the rules of inference that we take to be sound. The simplest of all systems is systems
known as K, which extends the axiom schemas of propositional logic by
adding a rule of inference and a simple axiom schema. Specifically, K consists of the axiom schema
P1
ϕ → (ψ → ϕ)
P2
( ϕ → (ψ → θ )) → (( ϕ → ψ) → ( ϕ → θ ))
(¬ ϕ → ψ) → ((¬ ϕ → ¬ψ) → ϕ)
P3
(where, of course, the metavariables now range over formulas of L P2 ), the
single modal schema
2( ϕ → ψ) → (2ϕ → 2ψ)
K
and, in addition to MP, the rule of Necessitation:
Necessitation (Nec)
2ϕ follows from ϕ, if ϕ is a theorem of K.7
7 The
notion of theoremhood as defined in the section on propositional logic remains
unchanged.
25
Modal systems that include the system K are known as normal modal logics.
3.3.2
The System T
The system T is the result of adding the following schema to K:
2ϕ → ϕ
T
3.3.3
The System S4
The system S4 is the result of adding the following schema to T:
2ϕ → 22ϕ
4
3.3.4
The System S5
The system S5 is the result of adding the following schema to T:
3ϕ → 23ϕ
5
3.4
Soundness and Completeness of the Systems
It is easy to see that every instance of P1, P2, or P3 is valid, i.e., true in every
interpretation of L P2 , for the same reason that any instance of L P with the
same propositional form would be valid with respect to all interpretations
of L P . For the truth value of an instance of P1-3 is determined entirely by
Clauses 1, 2, and 3 in the definition of truth in an interpretation, and those
clauses are identical to their nonmodal counterparts, but for the presence
of the irrelevant (for P1-3) world parameter ’w’. To see that (every instance
of) K is valid, suppose that p2( ϕ → ψ)q and 2ϕ are true at w. Then p( ϕ →
ψ)q and ϕ are true at every world w0 accessible from w. By the soundness
of MP, it follows that ψ is true at all such w0 as well, and hence 2ψ is true at
w. Note that no particular properties of the accessibility relation R needed
to be assumed in this argument.
The idea behind Necessitation is that every axiom is itself a necessary
truth, and hence that we can always prefix a theorem with a necessity
operator. The rule can be demonstrated to be sound by a simple mathematical induction on the lengths of proofs.8
8 It
might be instructive to see the proof in detail. A theorem of a given system like K,
26
4
Modal Predicate Logic
In this section we will look at several systems of modal predicate logic.
The language of modal predicate logic will be constant across all of these
systems. What separates them, of course, are their model theory and concomitant axiomatizations. All the systems will build upon the propositional axioms of S5. Additionally, because identity is so critical to the
underlying philosophical issues, we will build only upon systems Q= of
predicate logic with identity.
4.1
The Languages LQ2 of Modal Predicate Logic
Let LQ be a language of predicate logic that includes at least the designated identity predicate P12 . As with the move from our propositional
language L P to the language to our modal propositional language L P2 ,
a language LQ2 of modal predicate logic is the result of adding a single
element 2 to the lexicon of our language LQ and altering grammatical
clause 3 of Section 2.1.3 as follows:
3’. If ϕ and ψ are formulas, so are 2ϕ, ¬ ϕ, and ( ϕ → ψ).
As in ordinary predicate logic, we will write the designated identity predicate P12 in the usual way as = and will permit the use of infix notation. We
will also continue to assume the definition of 3 in terms of 2 from modal
propositional logic.
Note that modal propositional axioms for all the languages we will
study in this section will be those of S5. Because the conditions on the
recall, is a formula that can be proved from the axioms of the system. To establish the
soundness of Nec, it will be enough to show that all theorems are true in all worlds, from
which it follows that the necessitation of any theorem (being itself a theorem) is true in
all worlds. The validity of the axioms of K establish the soundness of Nec for theorems
that have proofs of length 1. So let us then suppose, as our induction hypothesis, that
theorems with proofs of length n are true in all worlds, and let us suppose that ϕ has a
proof of length n + 1. If ϕ follows by MP, then there must be formulas ψ and pψ ⊃ ϕq
earlier in the proof from which it follows. By our induction hypothesis, both ψ and pψ ⊃
ϕq are true in all worlds, and so ϕ is as well, by the soundness of MP. If, on the other
hand, ϕ follows by Nec, then ϕ is of the form 2ψ, where ψ occurs earlier in the proof.
By the induction hypothesis once again, ψ is true in all worlds, and hence in all worlds
accessible from any given world, and so 2ψ — i.e., ϕ — is true in all worlds as well.
Hence, for any theorem ϕ and world w, 2ϕ is true at w, i.e., Nec is sound.
27
accessibility relation in S5 interpretations effectively render the relation
otiose, we will henceforth, in the interpretations below, not include an explicit accessibility relation in our interpretations. In all of the following
sections, let L P2 be a language for predicate logic and let LQ2 by the language of modal predicate logic that results from L P2 by modifying it as
just noted.
4.2
Model Theory I: QS5 Interpretations
The simplest model theory for LQ2 arises from simply combining the
model theory for L P2 with the possible world semantics for the system
S5. Accordingly, we call the resulting logic QS5.9 We therefore define an
QS5-interpretation to be a 4-tuple I = hW, w0 , D, V i such that W and D
are nonempty sets, w0 (the “actual” world) is a distinguished member of
W, and V is a function that assigns meanings to elements of the lexicon of
LQ2 as follows:
1. If p is a propositional constant, then V ( p, w) ∈ {T, F};
2. If t is a term, then V (t) ∈ D.
3. If P is an n-place predicate, then V ( P, w) ⊆ D n . In particular, if P is
the identity predicate =, then V ( P, w) = {he, ei | e ∈ D }.
Intuitively, of course, the idea is that the truth values of propositional constants and the extensions of n-place predicates are assigned relative to a
possible world. Thus, the predicate being a politician, say, is true of Hillary
Clinton relative to the actual world but not relative to some other worlds.
Notice in particular that, unlike propositional constants and predicates,
terms are not assigned world-relative semantic values. Note also that, although it is technically world relative, the interpretation of the identity
predicate is constant across all worlds.
As in all of the logics we have examined to this point, every valuation function V in an interpretation determines a unique function V that
assigns truth values to formulas. Accordingly, let LQ2 be a language of
modal predicate logic and I = hW, w0 , D, V i an QS5-interpretation of
9 QS5
is exacly what Linsky and Zalta call the Simplest Quantified Modal Logic, or
SQML; see [1].
28
LQ2 , and let ϕ and ψ be formulas of LQ2 , and let w ∈ W. We define
vbar as follows:
1. V ( p, w) = V ( p, w), for propositional constants p.
2. V ( P(t1 , ..., tn ), w) = T iff hV (t1 ), ..., V (tn )i ∈ V ( P, w).10
3. V (¬ ϕ, w) = T iff V ( ϕ, w) = F.
4. V ( ϕ → ψ, w) = T iff V ( ϕ, w) = T only if V ( ϕ, w) = T.
5. V (∀ xϕ, w) = T iff, for all e ∈ D, V [ x/e]( ϕ, w) = T.
6. V (2ϕ, w) = T if, for all w0 ∈ W, V ( ϕ, w0 ) = T; otherwise, V (2ϕ, w) =
F.
V ( ϕ w) = T(F) can be read as “ϕ is true (false) in w”. A sentence ϕ is said
to be true, simpliciter, just in case it is true in the actual world w0 ; more
formally:
Truth in a QS5-interpretation
ϕ is true in an QS5-interpretation I = hW, w0 , D, V i of LQ2 — symbolically, I |=
ϕ — iff V ( ϕ, w0 ) = T. ϕ is false in I iff V ( ϕ, w0 ) = F.
QS5
4.2.1
Proof Theory: The System QS5
As noted, the idea behind QS5 is just to conjoin the modal propositional
logic S5 with predicate logic. Metavariables, of course, now range over
formulas of LQ2 :
P1
ϕ → (ψ → ϕ)
P2
( ϕ → (ψ → θ )) → (( ϕ → ψ) → ( ϕ → θ ))
(¬ ϕ → ψ) → ((¬ ϕ → ¬ψ) → ϕ)
∀ x ( ϕ → ψ) → (∀ xϕ → ∀ xψ)
ϕ → ∀ xϕ, if x is not free in ϕ
∀ xϕ → ϕtx
P3
Qu1
Qu2
UI
10 As
in ordinary predicate logic, in the case n = 1, we define hV (t)i just to be V (t) so
that this clause works as it should with the definition of ‘V ( P, w)’ just given above.
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K
2( ϕ → ψ) → (2ϕ → 2ψ)
T
2ϕ → ϕ
5
3ϕ → 23ϕ
Our rules of inference are MP, Generalization, and the rule of Necessitation. This system if sound and complete with respect to the class of QS5
interpretations.
4.2.2
Controversial Consequences of QS5
The following are logical truths with respect to the class of QS5 interpretations and, hence, given completeness, they are theorems of QS5:
NE
∀ x2∃y y = x
NNE
2∀ x2∃y y = x
BF
3∃ xϕ → ∃ x3ϕ
CBF
∃ x3ϕ → 3∃ xϕ(2∀ xϕ → ∀ x2ϕ)
References
[1] Bernard Linsky and Edward Zalta. In defense of the simplest quantified modal logic. In James Tomberlin, editor, Philosophical Perspectives
8: Logic and Language, pages 431–458. Atascadero: Ridgeview Publishing Co., 1994.
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