a-logic - Digital [email protected] State University
... Whitehead’s great book, Principia Mathematica (1913) of Quine’s Mathematical Logic (1940) and
Methods of Logic (4th ed.,1982) and of hundreds of other textbooks and treatises which have the same
set of theorems, the same semantical foundations, and use the same concepts of validity and logical truth ...
Announcement as effort on topological spaces
... must be true for any y ∈ Ui ; but y may not be in Uj , in which case (y, Ui , Uj )
is not well-defined: we cannot interpret K̂j Ki p. Our solution to this dilemma
is to consider neighbourhoods that are not only relative to each agent, as usual
in multi-agent subset space logics, but that are also re ...
Independence logic and tuple existence atoms
R relation, ~x , ~y , ~z tuples of attributes.
Then R |= ~x ~y | ~z if and only if, for all r , r 0 ∈ R such that
r (~x ) = r 0 (~x ) there exists a r 00 ∈ R such that
r 00 (~x ~y ) = r (~x ~y ) and r 00 (~x ~z ) = r (~x ~z ).
Huge literature on the topic;
If ~x ~y ~z contains all attri ...
Structural Proof Theory
... what are known as contraction-free systems of sequent calculus. Each of these
logicians added some essential discovery, until a gem emerged. What it is can be
only intimated at this stage: There is a way of organizing the principles of proof
so that one can start from the theorem to be proved, then ...
Color - Alex Kocurek
... To show that no formula (or set of formulas) of a modal language L can express a certain
formula α of LTS , one must generally construct two models such that (a) they agree in L
on all L-formulas (i.e., they are L-equivalent), and (b) they disagree in LTS on α. To make
showing that such models are L ...
Model Theory of Modal Logic, Chapter in: Handbook of Modal Logic
... between the (ﬁrst-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic ﬂavours, each with their own tradition
in the model theory of modal logic. Still, these two semantics meet through the notion
of a general frame (closely related to ...
How to Go Nonmonotonic Contents David Makinson
... The text does not pretend to cover everything in the area of nonmonotonic reasoning. There
are logical constructions exhibiting nonmonotonic behaviour, but which are based in different
principles and which we leave aside. In particular, we will not be considering autoepistemic
logics. They are forme ...
Labeled Natural Deduction for Temporal Logics
... 1.1 Background and motivation
The history of the philosophical and logical reasoning about time goes back at least
to ancient Greece, with the works of Aristotle and Diodorus Cronus. However, the
birth of modern (symbolic) temporal logic is mainly connected to the name of
Prior, who in the late 1950 ...
Goal-directed Proof Theory
... S and check by the theorem prover whether ; ^ S ! A. Again, no matter how it is ecient our
theorem-prover, it is obvious that there are better methods of performing this task, for instance one
attempts a proof of A from ; and determine as far as the proof proceeds what should be added to ;
(i.e. a ...
Logic in Nonmonotonic Reasoning
... knowledge bases, when such hierarchies have been allowed to have exceptions.
The theory of reasoning in such taxonomies has been called nonmonotonic inheritance (see [Horty, 1994] for an overview). The guiding principle in resolving
potential conflicts in such hierarchies was a specificity principl ...
A joint logic of problems and propositions, a modified BHK
... of problems will become a permanent part of [a standard] course of logic.
Creation of a unified logical apparatus dealing with objects of two types —
propositions and problems — was intended.”
We will now describe such a formal system, QHC, which is
a conservative extension of both the intuitionisti ...
a PDF file of the textbook - U of L Class Index
... difference between fact and opinion. Assertions will often express things that would count as
facts (such as “Pierre Trudeau was born in Quebec” or “Pierre Trudeau liked almonds”), but
they can also express things that you might think of as matters of opinion (such as “almonds
Abella: A System for Reasoning about Relational Specifications
... The first version of the Abella theorem prover was developed by Andrew Gacek as
part of his doctoral work carried out at the University of Minnesota . Kaustuv
Chaudhuri and Yuting Wang have subsequently designed and implemented extensions
to the system, resulting in an updated release. The vario ...
A Unified View of Induction Reasoning for First-Order Logic
... for example, a positive clause φ is an inductive consequence of a set of universally quantified
Horn clauses with equality iff φ is valid in their initial model .
A formula φ is false, denoted by 6|=M φ, if it is not valid. Any false formula has (or contains)
at least one false ground instance, ...
Elements of Finite Model Theory
... Finite model theory is an area of mathematical logic that grew out of computer
The main sources of motivational examples for ﬁnite model theory are
found in database theory, computational complexity, and formal languages,
although in recent years connections with other areas, s ...
... Other Quantifiers
The most important quantifiers are ∀ and ∃, but we could define
many different quantifiers: “there is a unique”, “there are exactly
two”, “there are no more than three”, “there are at least 100”, etc.
A common one is the uniqueness quantifier, denoted by ∃!.
∃!xP (x) states “There ...
Introduction to Modal and Temporal Logic
... Base Case l = 1: If ϕ is an atomic formula p, either ϑ(w, p) = t or
ϑ(w, p) = f by definition of ϑ. So the lemma holds for all atomic formulae.
Ind. Hyp. : Lemma holds for all formulae of length less than some n > 0.
Induction Step: If ϕ is of length n, then consider the shape of ϕ.
ϕ = hiψ: If w ha ...
5 model theory of modal logic
... between the (first-order) Kripke structure semantics and the (second-order) frame semantics, give rise to very distinct model theoretic flavours, each with their own tradition
in the model theory of modal logic. Still, these two semantics meet through the notion
of a general frame (closely related t ...
a semantic perspective - Institute for Logic, Language and
... language (that is, if MOD has only one element) we simply write 3 and 2 for its diamond and
box forms. We often tacitly assume that some signature has been fixed, and say things like “the
basic modal language”, or “the basic modal language with one diamond”. We won’t need many
syntactic concepts in ...
... The other sentences are all inappropriate, since there is no
King of France at present and since Bill Clinton never had an
affair with Margaret Thatcher.
Logic and Proof
... Although the patterns of language addressed by Aristotle’s theory of reasoning are
limited, we have him to thank for a crucial insight: we can classify valid patterns of inference by their logical form, while abstracting away specific content. It is this fundamental
observation that underlies the en ...
SEQUENT SYSTEMS FOR MODAL LOGICS
... are referred to [Gabbay, 1996], [Goré, 1999] and [Pliuškeviene, 1998]. Also
Orlowska’s [1988; 1996] Rasiowa-Sikorski-style relational proof systems for
normal modal logics will not be considered in the present chapter. In relational proof systems the logical object language is associated with a la ...
First-order logic is a formal system used in mathematics, philosophy, linguistics, and computer science. It is also known as first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic. First-order logic uses quantified variables over (non-logical) objects. This distinguishes it from propositional logic which does not use quantifiers.A theory about some topic is usually first-order logic together with a specified domain of discourse over which the quantified variables range, finitely many functions which map from that domain into it, finitely many predicates defined on that domain, and a recursive set of axioms which are believed to hold for those things. Sometimes ""theory"" is understood in a more formal sense, which is just a set of sentences in first-order logic.The adjective ""first-order"" distinguishes first-order logic from higher-order logic in which there are predicates having predicates or functions as arguments, or in which one or both of predicate quantifiers or function quantifiers are permitted. In first-order theories, predicates are often associated with sets. In interpreted higher-order theories, predicates may be interpreted as sets of sets.There are many deductive systems for first-order logic that are sound (all provable statements are true in all models) and complete (all statements which are true in all models are provable). Although the logical consequence relation is only semidecidable, much progress has been made in automated theorem proving in first-order logic. First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem.First-order logic is the standard for the formalization of mathematics into axioms and is studied in the foundations of mathematics. Mathematical theories, such as number theory and set theory, have been formalized into first-order axiom schemas such as Peano arithmetic and Zermelo–Fraenkel set theory (ZF) respectively.No first-order theory, however, has the strength to describe uniquely a structure with an infinite domain, such as the natural numbers or the real line. A uniquely describing, i.e. categorical, axiom system for such a structure can be obtained in stronger logics such as second-order logic.For a history of first-order logic and how it came to dominate formal logic, see José Ferreirós (2001).