Introducing Quantified Cuts in Logic with Equality
... The method [5] for propositional logic is shown to never increase the size of proofs more than polynomially. Another approach to the compression of firstorder proofs by introduction of definitions for abbreviating terms is [19]. There is a large body of work on the generation of non-analytic formulas ...
... The method [5] for propositional logic is shown to never increase the size of proofs more than polynomially. Another approach to the compression of firstorder proofs by introduction of definitions for abbreviating terms is [19]. There is a large body of work on the generation of non-analytic formulas ...
Supervaluationism and Classical Logic
... Epistemicists in vagueness want to retain classical logic and they endorse the somewhat surprising claim that there’s actually such an n (they claim we know the existential generalization ‘there is an n that such and such’ even if there is no particular n of which we know that such and such). Many p ...
... Epistemicists in vagueness want to retain classical logic and they endorse the somewhat surprising claim that there’s actually such an n (they claim we know the existential generalization ‘there is an n that such and such’ even if there is no particular n of which we know that such and such). Many p ...
Sequent calculus for predicate logic
... cut rule, then we define the cut rank of π to be the rank of any cut formula in π which has greatest possible rank. Lemma 1.2. (Weakening) If Γ ⇒ ∆ is the endsequent of a derivation π and Γ ⊆ Γ0 and ∆ ⊆ ∆0 , then Γ0 ⇒ ∆0 is derivable as well. In fact, the latter has a derivation π 0 with a cut rank ...
... cut rule, then we define the cut rank of π to be the rank of any cut formula in π which has greatest possible rank. Lemma 1.2. (Weakening) If Γ ⇒ ∆ is the endsequent of a derivation π and Γ ⊆ Γ0 and ∆ ⊆ ∆0 , then Γ0 ⇒ ∆0 is derivable as well. In fact, the latter has a derivation π 0 with a cut rank ...
Let me begin by reminding you of a number of passages ranging
... example, we find Frege arguing, among other things, (i) that attaching the words “is true” to a sentential clause as predicate—as, for example, in “It is true that seawater is salty”—adds nothing to the thought expressed by the sentence—“Seawater is salty”—alone, (ii) that the relation of truth to a ...
... example, we find Frege arguing, among other things, (i) that attaching the words “is true” to a sentential clause as predicate—as, for example, in “It is true that seawater is salty”—adds nothing to the thought expressed by the sentence—“Seawater is salty”—alone, (ii) that the relation of truth to a ...
Identity and Harmony revisited ∗ Stephen Read University of St Andrews
... Is identity a logical operator? The rules for identity in a natural deduction setting are usually given in the form of Reflexivity and Congruence (see, e.g., [9] p. 77): a=b p Congr Refl a=a p(b/a) Here, p(b/a) denotes the result of replacing one or more occurrences of the term a in p by b. Refl wou ...
... Is identity a logical operator? The rules for identity in a natural deduction setting are usually given in the form of Reflexivity and Congruence (see, e.g., [9] p. 77): a=b p Congr Refl a=a p(b/a) Here, p(b/a) denotes the result of replacing one or more occurrences of the term a in p by b. Refl wou ...
Verification and Specification of Concurrent Programs
... In an operational approach, a specification consists of an abstract program written in some form of abstract programming language. This approach was advocated in the early ’80s by Lam and Shankar [24] and others [26]. More recent instances include the I/O automaton approach of Lynch and Tuttle [32] a ...
... In an operational approach, a specification consists of an abstract program written in some form of abstract programming language. This approach was advocated in the early ’80s by Lam and Shankar [24] and others [26]. More recent instances include the I/O automaton approach of Lynch and Tuttle [32] a ...
Chapter 4. Logical Notions This chapter introduces various logical
... of the assumptions will be explicit, but others may be implicit. For example in the argument from All men are mortal to Socrates is mortal, there is an implicit premiss to the effect that Socrates is a man. We shall assume, in the application of the notion of validity, that all implicit premisses ha ...
... of the assumptions will be explicit, but others may be implicit. For example in the argument from All men are mortal to Socrates is mortal, there is an implicit premiss to the effect that Socrates is a man. We shall assume, in the application of the notion of validity, that all implicit premisses ha ...
On Decidability of Intuitionistic Modal Logics
... result in [6] and uses a translation into the two variable monadic guarded fragment of first order logic. Unfortunately, the decidability proof does not give a good decision procedure since it proceeds by reduction to satisfiability of formulas of SkS (monadic second-order theory of trees with const ...
... result in [6] and uses a translation into the two variable monadic guarded fragment of first order logic. Unfortunately, the decidability proof does not give a good decision procedure since it proceeds by reduction to satisfiability of formulas of SkS (monadic second-order theory of trees with const ...
Basic Metatheory for Propositional, Predicate, and Modal Logic
... 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 lo ...
... 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 lo ...
Nelson`s Strong Negation, Safe Beliefs and the - CEUR
... is the Gelfond-Lifschitz reduct, the original definition of the semantics [2]. The extensions to wider families of programs that followed were also defined as reductions à la Gelfond-Lifschitz : from the introduction of strong negation [3] to nested programs [7], a rather wide range of such reducts ...
... is the Gelfond-Lifschitz reduct, the original definition of the semantics [2]. The extensions to wider families of programs that followed were also defined as reductions à la Gelfond-Lifschitz : from the introduction of strong negation [3] to nested programs [7], a rather wide range of such reducts ...
Belief Revision in non
... standard construction mechanism of formulae. For the finite case, a Belnap theory can be seen as a single formula given by the conjunction of all wffs in the set. The formula :p ^ (:q _ r) ^ :r is an example of a finite Belnap theory. In proof theoretical terms, Belnap’s four-valued logic is charact ...
... standard construction mechanism of formulae. For the finite case, a Belnap theory can be seen as a single formula given by the conjunction of all wffs in the set. The formula :p ^ (:q _ r) ^ :r is an example of a finite Belnap theory. In proof theoretical terms, Belnap’s four-valued logic is charact ...
An admissible second order frame rule in region logic
... which together embody some discipline for encapsulation— through the part of a program proof in which client code is verified against assumed specifications that do not mention the module invariant. Our main result implies the soundness of this logic with respect to a standard denotational semantics ...
... which together embody some discipline for encapsulation— through the part of a program proof in which client code is verified against assumed specifications that do not mention the module invariant. Our main result implies the soundness of this logic with respect to a standard denotational semantics ...
Propositional Logic - faculty.cs.tamu.edu
... denote deg p, as the number of occurrences of logical connectives in p. In other words, the degree function satisfies the following properties: D1. An element in S has degree 0. D2. If a in Prop has degree n, then ¬a has degree n + 1. D3. If a and b in Prop are respectively of degree na and nb , the ...
... denote deg p, as the number of occurrences of logical connectives in p. In other words, the degree function satisfies the following properties: D1. An element in S has degree 0. D2. If a in Prop has degree n, then ¬a has degree n + 1. D3. If a and b in Prop are respectively of degree na and nb , the ...
34-2.pdf
... First of all, I’m happy to report that the “story” told by this book is easy to follow. In addition, this book is self-contained in the sense that every theorem presented in this book is proved. Also, a fair number of examples are worked out in great detail using a variety of different methods. That ...
... First of all, I’m happy to report that the “story” told by this book is easy to follow. In addition, this book is self-contained in the sense that every theorem presented in this book is proved. Also, a fair number of examples are worked out in great detail using a variety of different methods. That ...
ch1_Logic_and_proofs
... equal and so are the corresponding angles. Two angles are supplementary if the sum of their measures is 180 degrees. ...
... equal and so are the corresponding angles. Two angles are supplementary if the sum of their measures is 180 degrees. ...
A Nonstandard Approach to the. Logical Omniscience Problem
... nonstandard world possible, we (the logicians who get to examine the situation from t h e outside) know that the "real world" must obey the laws of standard logic. If we consider validity and logical implication with respect to standard worlds, then it is easy to show that logical omniscience fails ...
... nonstandard world possible, we (the logicians who get to examine the situation from t h e outside) know that the "real world" must obey the laws of standard logic. If we consider validity and logical implication with respect to standard worlds, then it is easy to show that logical omniscience fails ...
07.1-Reasoning
... language and how they can be used together. • Semantics: Gives meaning to the syntax. Defines how the symbols in the syntax relate to in the real world. ...
... language and how they can be used together. • Semantics: Gives meaning to the syntax. Defines how the symbols in the syntax relate to in the real world. ...
A Resolution-Based Proof Method for Temporal Logics of
... This paper presents two logics, called KLn and BLn respectively, and gives resolutionbased proof methods for both. The logic KLn is a temporal logic of knowledge. That is, in addition to the usual connectives of linear discrete temporal logic [4], KLn contains an indexed set of unary modal connectiv ...
... This paper presents two logics, called KLn and BLn respectively, and gives resolutionbased proof methods for both. The logic KLn is a temporal logic of knowledge. That is, in addition to the usual connectives of linear discrete temporal logic [4], KLn contains an indexed set of unary modal connectiv ...
Modal Logic - Web Services Overview
... 2. The starting point, once again, is Aristotle, who was the first to study the relationship between modal statements and their validity. 3. However, the great discussion it enjoyed in the Middle Ages. 4. The official birth date of modal logic is 1921, when Clarence Irving Lewis wrote a famous essay ...
... 2. The starting point, once again, is Aristotle, who was the first to study the relationship between modal statements and their validity. 3. However, the great discussion it enjoyed in the Middle Ages. 4. The official birth date of modal logic is 1921, when Clarence Irving Lewis wrote a famous essay ...
Linear Contextual Modal Type Theory
... The central idea in linear logic [Gir87] is that of a resource. Linear assumptions play the role of a fixed set of available resources that must be consumed (exactly once) in a derivation. Therefore, available resources form the philosophical foundation of linear contextual modal logic. The idea of ...
... The central idea in linear logic [Gir87] is that of a resource. Linear assumptions play the role of a fixed set of available resources that must be consumed (exactly once) in a derivation. Therefore, available resources form the philosophical foundation of linear contextual modal logic. The idea of ...
Document
... literals. A P P B A B - In the propositional case, two literals are the same if they have the same proposition (negated in exactly one of them). - In first order logic, the situation is more complex, due to the existence of variables. We may assume that variables are universally quantified. - ...
... literals. A P P B A B - In the propositional case, two literals are the same if they have the same proposition (negated in exactly one of them). - In first order logic, the situation is more complex, due to the existence of variables. We may assume that variables are universally quantified. - ...
slides
... If there are infinitely many possible values for X the meaning of this expression cannot be represented using a propositional formula. In AG, the meaning of aggregate expressions is captured using an infinitary propositional formula. The definition is based on the semantics for propositional aggrega ...
... If there are infinitely many possible values for X the meaning of this expression cannot be represented using a propositional formula. In AG, the meaning of aggregate expressions is captured using an infinitary propositional formula. The definition is based on the semantics for propositional aggrega ...
De Jongh`s characterization of intuitionistic propositional calculus
... We say that a frame F = (W, R) is of depth n < ω, and write d(F) = n if there is a chain of n points in F and no other chain in F contains more than n points. If for every n ∈ ω, F contains a chain consisting of n points, then F is said to be of infinite depth. The depth of a point w ∈ W is the dept ...
... We say that a frame F = (W, R) is of depth n < ω, and write d(F) = n if there is a chain of n points in F and no other chain in F contains more than n points. If for every n ∈ ω, F contains a chain consisting of n points, then F is said to be of infinite depth. The depth of a point w ∈ W is the dept ...
Chapter 2 - Part 1 - PPT - Mano & Kime
... Terms of Use © 2004 by Pearson Education,Inc. All rights reserved. The following terms of use apply in addition to the standard Pearson Education Legal Notice. Permission is given to incorporate these materials into classroom presentations and handouts only to instructors adopting Logic and C ...
... Terms of Use © 2004 by Pearson Education,Inc. All rights reserved. The following terms of use apply in addition to the standard Pearson Education Legal Notice. Permission is given to incorporate these materials into classroom presentations and handouts only to instructors adopting Logic and C ...