1 The calculus of “predicates”
... shall be regarded as being identical if they share the same extension. [Chap.2, Sec. 1.3.1] The symbol ...
... shall be regarded as being identical if they share the same extension. [Chap.2, Sec. 1.3.1] The symbol ...
Design and Analysis of Cryptographic Protocols
... These flaws result from poor implementation of the underlying cryptographic algorithms. Next the authors of the paper classify methods for analyzing security protocols. The authors discuss 2 categories – attack construction tools and inference based methods. I will discuss one such inference method ...
... These flaws result from poor implementation of the underlying cryptographic algorithms. Next the authors of the paper classify methods for analyzing security protocols. The authors discuss 2 categories – attack construction tools and inference based methods. I will discuss one such inference method ...
Propositional/First
... under all interpretations, no matter what the world is actually like or what the semantics is. Example: “It’s raining or it’s not raining.” • An inconsistent sentence or contradiction is a sentence that is False under all interpretations. The world is never like what it describes, as in “It’s rainin ...
... under all interpretations, no matter what the world is actually like or what the semantics is. Example: “It’s raining or it’s not raining.” • An inconsistent sentence or contradiction is a sentence that is False under all interpretations. The world is never like what it describes, as in “It’s rainin ...
Speaking Logic - SRI International
... satisfiability problems in propositional logic (SAT). Define a 1-bit full adder in propositional logic. The Pigeonhole Principle states that if n + 1 pigeons are assigned to n holes, then some hole must contain more than one pigeon. Formalize the pigeonhole principle for four pigeons and three holes ...
... satisfiability problems in propositional logic (SAT). Define a 1-bit full adder in propositional logic. The Pigeonhole Principle states that if n + 1 pigeons are assigned to n holes, then some hole must contain more than one pigeon. Formalize the pigeonhole principle for four pigeons and three holes ...
General Dynamic Dynamic Logic
... Theorem 4.11 in [12], first noted in [20], which states that any dynamic operator whose effect on a model can be described in PDL (without Kleene’s iteration operator ∗) can be reduced to the underlying modal logic using essentially only the standard axioms of PDL. We show how this idea can be used ...
... Theorem 4.11 in [12], first noted in [20], which states that any dynamic operator whose effect on a model can be described in PDL (without Kleene’s iteration operator ∗) can be reduced to the underlying modal logic using essentially only the standard axioms of PDL. We show how this idea can be used ...
Lecture01 - Mathematics
... hand, are seldom useful since a table of values for x 2 y 2 has infinitely many lines, one for each possible value of x and y. (3) Logical variables (p, q, r, etc.) stand for infinitely many propositions, but these propositions can take on only two truth values. Thus there are only finitely many w ...
... hand, are seldom useful since a table of values for x 2 y 2 has infinitely many lines, one for each possible value of x and y. (3) Logical variables (p, q, r, etc.) stand for infinitely many propositions, but these propositions can take on only two truth values. Thus there are only finitely many w ...
On Elkan`s theorems: Clarifying their meaning
... omitted from the first version of Elkan’s theorem. As to the rest of the assumptions, both t~A ∧ B! ⫽ min$t~A!, t~B!% and t~¬A! ⫽ 1 ⫺ t~A! are quite reasonable and, in fact, are often used in applications of fuzzy logic. Let us now concentrate on the last assumption, that is, on t~A! ⫽ t~B! if A and ...
... omitted from the first version of Elkan’s theorem. As to the rest of the assumptions, both t~A ∧ B! ⫽ min$t~A!, t~B!% and t~¬A! ⫽ 1 ⫺ t~A! are quite reasonable and, in fact, are often used in applications of fuzzy logic. Let us now concentrate on the last assumption, that is, on t~A! ⫽ t~B! if A and ...
LOGIC MAY BE SIMPLE Logic, Congruence - Jean
... The structure is factored by the relation of logical equivalence. This is possible because this relation is a congruence. Now in which sense the factor structure K/⊣⊢ = hF /⊣⊢ , ⊢ /⊣⊢ i is a Boolean algebra? (nb the original construction by Tarski was not presented like this; see [Tarski 1935]). If ...
... The structure is factored by the relation of logical equivalence. This is possible because this relation is a congruence. Now in which sense the factor structure K/⊣⊢ = hF /⊣⊢ , ⊢ /⊣⊢ i is a Boolean algebra? (nb the original construction by Tarski was not presented like this; see [Tarski 1935]). If ...
Classical Logic and the Curry–Howard Correspondence
... proceed line by line, with each line derived from those preceding it by means of some inference rule. Nowadays such logics are known as ‘Hilbert systems’. This format can be somewhat cumbersome and inelegant, both because it does not follow the reasoning-patterns of ordinary mathematics and because ...
... proceed line by line, with each line derived from those preceding it by means of some inference rule. Nowadays such logics are known as ‘Hilbert systems’. This format can be somewhat cumbersome and inelegant, both because it does not follow the reasoning-patterns of ordinary mathematics and because ...
Second-Order Logic of Paradox
... (purely) false B. Logics of this general nature had been developed earlier, including in particular the investigations of Asenjo [1, 2], whose logic is essentially just LP. The model-theoretic semantics for a predicate logic of LP is, again, a natural generalization of that familiar from classical l ...
... (purely) false B. Logics of this general nature had been developed earlier, including in particular the investigations of Asenjo [1, 2], whose logic is essentially just LP. The model-theoretic semantics for a predicate logic of LP is, again, a natural generalization of that familiar from classical l ...
Monadic Second Order Logic and Automata on Infinite Words
... words and uses the Büchi acceptance condition (defined below). A Büchi automaton A is a tuple hQ, A, q0 , ∆, F i, where Q is the finite set of states, A is the alphabet, q0 ∈ Q is the start state, ∆ ⊆ Q × A × Q is the transition relation, and F ⊆ Q is the set of accept states. A run of A on w is a ...
... words and uses the Büchi acceptance condition (defined below). A Büchi automaton A is a tuple hQ, A, q0 , ∆, F i, where Q is the finite set of states, A is the alphabet, q0 ∈ Q is the start state, ∆ ⊆ Q × A × Q is the transition relation, and F ⊆ Q is the set of accept states. A run of A on w is a ...
Aristotle, Boole, and Categories
... But while technically correct, the variables S, M, P range over classes and the relations a, e, i, o as binary predicates hold between classes, making this a second-order logic. Moreover it introduces Boolean connectives into a subject that had previously only seen them in translations of individual ...
... But while technically correct, the variables S, M, P range over classes and the relations a, e, i, o as binary predicates hold between classes, making this a second-order logic. Moreover it introduces Boolean connectives into a subject that had previously only seen them in translations of individual ...
Tactics for Separation Logic Abstract Andrew W. Appel INRIA Rocquencourt & Princeton University
... proving an imperative program, much of the reasoning is not about memory cells but concerns the abstract mathematical objects that the program’s data structures represent. Lemmas about those objects are most conveniently proved in a general-purpose higher-order logic, especially when there are large ...
... proving an imperative program, much of the reasoning is not about memory cells but concerns the abstract mathematical objects that the program’s data structures represent. Lemmas about those objects are most conveniently proved in a general-purpose higher-order logic, especially when there are large ...
Multi-Agent Only
... If Alice believes that all that Bob knows is that birds normally fly and that Tweety is a bird, then Alice believes that Bob believes that Tweety flies. But technically things were surprisingly cumbersome! The problem lies in the complexity in what agents consider ...
... If Alice believes that all that Bob knows is that birds normally fly and that Tweety is a bird, then Alice believes that Bob believes that Tweety flies. But technically things were surprisingly cumbersome! The problem lies in the complexity in what agents consider ...
1 Introduction 2 Formal logic
... • A semantics that explains the meaning of statements in our formal language in informal terms. • A deductive system that establishes formal rules of reasoning about logical statements which we can apply without having to constantly consider their informal explanation. It is important to remember th ...
... • A semantics that explains the meaning of statements in our formal language in informal terms. • A deductive system that establishes formal rules of reasoning about logical statements which we can apply without having to constantly consider their informal explanation. It is important to remember th ...
SECOND-ORDER LOGIC, OR - University of Chicago Math
... pleases. It can be any cardinality.2 Call a first-order language with a set K of non-logical symbols L1K. If it has equality, call it L1K =. A set of symbols alone is insufficient for making a meaningful language; we also need to know how we can put those symbols together. Just as we cannot say in E ...
... pleases. It can be any cardinality.2 Call a first-order language with a set K of non-logical symbols L1K. If it has equality, call it L1K =. A set of symbols alone is insufficient for making a meaningful language; we also need to know how we can put those symbols together. Just as we cannot say in E ...
full text (.pdf)
... For this reason, models for one language can be viewed as models for the other. We base S on + instead of ∗ because the resulting deductive system is cleaner—it contains no contraction rule1 . This is perhaps due to the fact that + can be viewed as a more primitive operation than ∗ . A test is eithe ...
... For this reason, models for one language can be viewed as models for the other. We base S on + instead of ∗ because the resulting deductive system is cleaner—it contains no contraction rule1 . This is perhaps due to the fact that + can be viewed as a more primitive operation than ∗ . A test is eithe ...
the theory of form logic - University College Freiburg
... categories and formulate rules which take account of this classification? Such a syntax may result in a more restrictive system than predicate logic, perhaps ruling out the two arguably meaningless sentences from above. On the other extreme, why not shoehorn all terms into a single syntactical categ ...
... categories and formulate rules which take account of this classification? Such a syntax may result in a more restrictive system than predicate logic, perhaps ruling out the two arguably meaningless sentences from above. On the other extreme, why not shoehorn all terms into a single syntactical categ ...
Curry`s Paradox. An Argument for Trivialism
... 2006a, 2006), Priest claims that dialetheism supplies the best solution to the the strengthen liar paradox, a paradox originated from the sentence: (a): (a) is not true by holding that (a) is both true and not true. More generally, he holds that the paradoxical sentences obtained from self-reference ...
... 2006a, 2006), Priest claims that dialetheism supplies the best solution to the the strengthen liar paradox, a paradox originated from the sentence: (a): (a) is not true by holding that (a) is both true and not true. More generally, he holds that the paradoxical sentences obtained from self-reference ...
Definability properties and the congruence closure
... the uniform reduction property for quotients. As a consequence, none of these logics satisfies either A-interpolation or Beth's definability theorem when closed under relativizations. We also show the failure of both properties for any sublogic of L~o~(Th) in which Chang's quantifier or some cardina ...
... the uniform reduction property for quotients. As a consequence, none of these logics satisfies either A-interpolation or Beth's definability theorem when closed under relativizations. We also show the failure of both properties for any sublogic of L~o~(Th) in which Chang's quantifier or some cardina ...
Propositional inquisitive logic: a survey
... We mentioned above that in inquisitive logic, entailments involving questions capture logical dependencies. The relation of dependency is also the focus of recent work in the framework of dependence logic [23]. Dependence logic and inquisitive logic are tightly connected frameworks, as discussed in ...
... We mentioned above that in inquisitive logic, entailments involving questions capture logical dependencies. The relation of dependency is also the focus of recent work in the framework of dependence logic [23]. Dependence logic and inquisitive logic are tightly connected frameworks, as discussed in ...
An Overview of Intuitionistic and Linear Logic
... Γ ` ∆, A A, Γ ` ∆ cut Γ`∆ It may varies from one system to another, but it is present in all logical systems formalised in sequent calculus. It essentially embodies the principle of modus ponens, the core of any formal logical system. However, notice that the formula A has no structural link with Γ ...
... Γ ` ∆, A A, Γ ` ∆ cut Γ`∆ It may varies from one system to another, but it is present in all logical systems formalised in sequent calculus. It essentially embodies the principle of modus ponens, the core of any formal logical system. However, notice that the formula A has no structural link with Γ ...
Dissolving the Scandal of Propositional Logic?
... implication was never introduced in propositional logic to capture merely some connection of dependence between two statements. It was introduced to capture the notion of implication or logical consequence in natural language. So having to rely on an adjunctive interpretation of material implication ...
... implication was never introduced in propositional logic to capture merely some connection of dependence between two statements. It was introduced to capture the notion of implication or logical consequence in natural language. So having to rely on an adjunctive interpretation of material implication ...
Classical BI - UCL Computer Science
... the rules above are not sets or sequences, as in standard sequent calculi, but rather bunches: trees whose leaves are formulas and whose internal nodes are either ‘;’ or ‘,’ denoting respectively additive and multiplicative combinations of assumptions. The crucial difference between the two operatio ...
... the rules above are not sets or sequences, as in standard sequent calculi, but rather bunches: trees whose leaves are formulas and whose internal nodes are either ‘;’ or ‘,’ denoting respectively additive and multiplicative combinations of assumptions. The crucial difference between the two operatio ...
On the specification of sequent systems
... an involutive negation and this makes it difficult to address directly dualities in object-logic proof systems. This lack of dualities is particularly unfortunate when specifying sequent calculus [Gen69] since they play a central role in the theory of such proof systems. Pfenning in [Pfn95,Pfn00] us ...
... an involutive negation and this makes it difficult to address directly dualities in object-logic proof systems. This lack of dualities is particularly unfortunate when specifying sequent calculus [Gen69] since they play a central role in the theory of such proof systems. Pfenning in [Pfn95,Pfn00] us ...