a-logic - Digital Commons@Wayne 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 ...
... 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 ...
First-Order Theorem Proving and Vampire
... In TPTP: Variable names start with upper-case letters. ...
... In TPTP: Variable names start with upper-case letters. ...
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 ...
... 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
... Definition 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 ...
... Definition 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 ...
... 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 ...
... 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 ...
... 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 ...
... 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 ...
... 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 ...
... 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 ...
... 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 ...
... 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 are delicious”). Throu ...
... 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 are delicious”). Throu ...
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 [19]. Kaustuv Chaudhuri and Yuting Wang have subsequently designed and implemented extensions to the system, resulting in an updated release. The vario ...
... 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 [19]. 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 [24]. 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, ...
... 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 [24]. 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, ...
Logical Methods in Computer Science Vol. 8(4:19)2012, pp. 1–28 Submitted Oct. 27, 2011
... the set of states satisfying ¬φ. Hence, for CML formulas, no positive normal forms can be defined. Obviously, these differences are also reflected in the complete axiomatizations that we present both for CML and for its fragment without Mra -operators. Many axioms of PML, such as ⊢ Lar ⊤ or ⊢ Lar φ ...
... the set of states satisfying ¬φ. Hence, for CML formulas, no positive normal forms can be defined. Obviously, these differences are also reflected in the complete axiomatizations that we present both for CML and for its fragment without Mra -operators. Many axioms of PML, such as ⊢ Lar ⊤ or ⊢ Lar φ ...
Elements of Finite Model Theory
... Finite model theory is an area of mathematical logic that grew out of computer science applications. 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 ...
... Finite model theory is an area of mathematical logic that grew out of computer science applications. 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 ...
Predicate Logic
... 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 ...
... 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 ...
... 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 ...
... 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 ...
Relevant Logic A Philosophical Examination of Inference Stephen Read February 21, 2012
... fools who misguidedly ascribe strange meanings to common symbols. To see the error in this dismissal, we must consider counterfactuals in general, and the relationship between truth and meaning. When people question certain properties of implication, and propose a different theory, they propose it a ...
... fools who misguidedly ascribe strange meanings to common symbols. To see the error in this dismissal, we must consider counterfactuals in general, and the relationship between truth and meaning. When people question certain properties of implication, and propose a different theory, they propose it a ...
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 ...
... 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 ...
Many-Valued Logic
... 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. ...
... 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 ...
... 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 ...
... 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 ...