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Least and greatest fixed points in Ludics, CSL 2015, Berlin.
... 1⊕(A⊗(S ( S)) ` S ( S the concantenation function. More gen(µL ) L`S(S erally, it is hard to tell what such proofs compute, when two proofs compute the same function, etc. It requires to step back from the finite, syntactic proof system under consideration and to start considering its semantics; thi ...
... 1⊕(A⊗(S ( S)) ` S ( S the concantenation function. More gen(µL ) L`S(S erally, it is hard to tell what such proofs compute, when two proofs compute the same function, etc. It requires to step back from the finite, syntactic proof system under consideration and to start considering its semantics; thi ...
Linear Contextual Modal Type Theory
... a linear contextual modality [Γ; ∆] that restores the modal box implication A [Γ; ∆]→ B = [Γ; ∆]A ( B. In addition, this discussion justifies why linear contextual modal logic is in fact a modal logic. In Section 5 we introduce a Curry-Howard correspondence. Every proof rule is endowed with a proof ...
... a linear contextual modality [Γ; ∆] that restores the modal box implication A [Γ; ∆]→ B = [Γ; ∆]A ( B. In addition, this discussion justifies why linear contextual modal logic is in fact a modal logic. In Section 5 we introduce a Curry-Howard correspondence. Every proof rule is endowed with a proof ...
A Propositional Modal Logic for the Liar Paradox Martin Dowd
... easy to deal with. The concern of this paper is systems simpler than Kripke’s which contain both self-reference and a truth modality. These systems are propositional modal systems [Le77]. Smullyan [Sm57] considers such a system, although his is extremely simple. Solovay’s [So76] propositional modal ...
... easy to deal with. The concern of this paper is systems simpler than Kripke’s which contain both self-reference and a truth modality. These systems are propositional modal systems [Le77]. Smullyan [Sm57] considers such a system, although his is extremely simple. Solovay’s [So76] propositional modal ...
Inference and Proofs - Dartmouth Math Home
... We concluded our last section with a proof that the sum of two even numbers is even. That proof contained several crucial ingredients. First, we introduced symbols for members of the universe of integers. In other words, rather than saying “suppose we have two integers,” we introduced symbols for th ...
... We concluded our last section with a proof that the sum of two even numbers is even. That proof contained several crucial ingredients. First, we introduced symbols for members of the universe of integers. In other words, rather than saying “suppose we have two integers,” we introduced symbols for th ...
The Pure Calculus of Entailment Author(s): Alan Ross Anderson and
... "entailment", or "the converse of deducibility" (Moore 1920), expressed in such logical locutions as "if ... then -," "implies," "entails," etc., and answering to such conclusion-signalling logical phrases as "therefore," "it follows that," "hence," "consequently," and the like. There have been nume ...
... "entailment", or "the converse of deducibility" (Moore 1920), expressed in such logical locutions as "if ... then -," "implies," "entails," etc., and answering to such conclusion-signalling logical phrases as "therefore," "it follows that," "hence," "consequently," and the like. There have been nume ...
The equational theory of N, 0, 1, +, ×, ↑ is decidable, but not finitely
... As often happens in number theory, these last results use far more complex tools than simple arithmetic reasoning, as in the case of [HR84], where Nevanlinna theory is used to identify a subclass of numerical expressions for which the usual axioms for +, ×, ↑ and 1 are complete. 1.2 Connections with ...
... As often happens in number theory, these last results use far more complex tools than simple arithmetic reasoning, as in the case of [HR84], where Nevanlinna theory is used to identify a subclass of numerical expressions for which the usual axioms for +, ×, ↑ and 1 are complete. 1.2 Connections with ...
Subintuitionistic Logics with Kripke Semantics
... completeness of BPC for finite, transitive, irreflexive Kripke models. Then in 1997, Suzuki and Ono [6] introduced a Hilbert style proof system for BPC as an extension of Corsi’s system [1]. They proved a weak completeness theorem. The structure of this paper is as follows. In Section 2 we introduce ...
... completeness of BPC for finite, transitive, irreflexive Kripke models. Then in 1997, Suzuki and Ono [6] introduced a Hilbert style proof system for BPC as an extension of Corsi’s system [1]. They proved a weak completeness theorem. The structure of this paper is as follows. In Section 2 we introduce ...
Adding the Everywhere Operator to Propositional Logic (pdf file)
... Let VF be a new set of formula variables. We use upper-case letters P, Q, R, . . . for formula variables. Formulas of C are defined as in (2), except that a formula variable is also a formula. For example, p ∨ q , P ∨ Q , and p ∨ Q are formulas of C . A formula of C is concrete if it does not cont ...
... Let VF be a new set of formula variables. We use upper-case letters P, Q, R, . . . for formula variables. Formulas of C are defined as in (2), except that a formula variable is also a formula. For example, p ∨ q , P ∨ Q , and p ∨ Q are formulas of C . A formula of C is concrete if it does not cont ...
Kripke completeness revisited
... predicate calculus with equality, with the addition of the following axiom schemes and rules of inference: A1: 2A ⊃ A A2: ∼ 2A ⊃ 2 ∼ 2A A3: 2(A ⊃ B) ⊃ (2A ⊃ 2B) R1: If ` A and ` A ⊃ B then ` B R2: If ` A then ` 2A Given a non-empty domain D and a formula A, a complete assignment for A in D is a func ...
... predicate calculus with equality, with the addition of the following axiom schemes and rules of inference: A1: 2A ⊃ A A2: ∼ 2A ⊃ 2 ∼ 2A A3: 2(A ⊃ B) ⊃ (2A ⊃ 2B) R1: If ` A and ` A ⊃ B then ` B R2: If ` A then ` 2A Given a non-empty domain D and a formula A, a complete assignment for A in D is a func ...
Chapter 9: Initial Theorems about Axiom System AS1
... metalinguistic ‘if…then’ connective. Similarly, ‘&’ is the metalinguistic ‘and’ connective. Translating this into English, we have: if „α, and „α→β, then „β. The remaining occurrence of ‘→’ is under the predicate ‘„’; it is accordingly a proper noun referring to the conditional connective of the obj ...
... metalinguistic ‘if…then’ connective. Similarly, ‘&’ is the metalinguistic ‘and’ connective. Translating this into English, we have: if „α, and „α→β, then „β. The remaining occurrence of ‘→’ is under the predicate ‘„’; it is accordingly a proper noun referring to the conditional connective of the obj ...
Constructing Cut Free Sequent Systems With Context Restrictions
... rules which works uniformly for classical and intuitionistic logics. The rules so constructed are by construction sound and complete (in the presence of cut) and give rise to unlabelled sequent systems that are amenable to saturation under cuts between rules. In case the resulting rules fulfil our c ...
... rules which works uniformly for classical and intuitionistic logics. The rules so constructed are by construction sound and complete (in the presence of cut) and give rise to unlabelled sequent systems that are amenable to saturation under cuts between rules. In case the resulting rules fulfil our c ...