Basic Proof Theory with Applications to Computation

This chapter describes the categorical proof theory of the cut rule, a very basic component of any sequent-style presentation of a logic, assuming a minimum of structural rules and connectives, in fact, starting with none. It is shown how logical features can be added to this basic logic in a modular fashion, at each stage showing the appropriate corresponding categorical semantics of the proof theory, starting with multicategories, and moving to linearly distributive categories and *-autonomous categories. A key tool is the use of graphical representations of proofs (“proof circuits”) to represent formal derivations in these logics. This is a powerful symbolism, which on the one hand is a formal mathematical language, but crucially, at the same time, has an intuitive graphical representation.

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Some Fixpoint Techniques in Algebraic Structures and Applications to Computer Science

Fundamenta Informaticae ◽

10.3233/fi-1987-10405 ◽

1987 ◽

Vol 10 (4) ◽

pp. 387-413

Author(s):

Irène Guessarian

Keyword(s):

Logic Programming ◽

Computer Science ◽

Proof Theory ◽

Algebraic Structures ◽

Data Bases

This paper recalls some fixpoint theorems in ordered algebraic structures and surveys some ways in which these theorems are applied in computer science. We describe via examples three main types of applications: in semantics and proof theory, in logic programming and in deductive data bases.

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Non-well-founded Proof Theory of Transitive Closure Logic

ACM Transactions on Computational Logic ◽

10.1145/3404889 ◽

2020 ◽

Vol 21 (4) ◽

pp. 1-31

Author(s):

Liron Cohen ◽

Reuben N. S. Rowe

Keyword(s):

Proof Theory ◽

Transitive Closure

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Structures with many-valued information and their relational proof theory

Proceedings 30th IEEE International Symposium on Multiple-Valued Logic (ISMVL 2000) ◽

10.1109/ismvl.2000.848635 ◽

2002 ◽

Cited By ~ 2

Author(s):

I. Duntsch ◽

W. MacCaull ◽

E. Orlowska

Keyword(s):

Proof Theory

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Dummetova koncepcija kao teorija značenja za Hintikkin tip semantike teorije igre (III)

Papers on Philosophy Psychology Sociology and Pedagogy ◽

10.15291/radovifpsp.2709 ◽

2018 ◽

Vol 30 (7) ◽

Author(s):

Heda Festini

Keyword(s):

General Theory ◽

Information Seeking ◽

Proof Theory ◽

Winning Strategy ◽

General Information ◽

Language Games ◽

Inductive Probability ◽

Theory Of Meaning ◽

The Difference ◽

Key Terms

With the analysis of the key terms such as truth/use, proof - verification, falsification, inductive probability/semantic probability, winning/losing, winning strategy, it is shown that Dummett’s general theory of meaning does not include Hintikka’s game theory, that it, the conception of the winning strategy. The difference between them arises from the different understanding of Wittgenstein's idea about language games and from their attitudes toward theoretical proof theory. Hintikka’s semantic games about exploration of the world do not reject the bivalence principle but he gives it a different characteristic - one of the two players always has a winning strategy. Looking at Dummett’s philosophical theory of meaning and the most recent Hintikka’s suggestion about general information - seeking through questioning and answering, the author establishes that Dummett’s falsificational and dialogical games as well as Hintikka’s semantic games are subparts of Hintikka’s general information - seeking game Thus Dummett’s statement that Hintikka’s semantic games can be subsumed under Dummett’s conception is rejected and thus the answer is partly given to Saarinen’s suggestion that new affinity should be established. Apart from the comparison of these views with the outline of possible Wittgenstein’s general theory of meaning as rule - testing, together with his treatment (although not always adequate) of verification/falsification, inductive probability and čonfirmation/corroboration, the advantage of Wittgenstein’s view is affirmed.

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On the Decidability of Intuitionistic Tense Logic without Disjunction

Proceedings of the Twenty-Ninth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2020/249 ◽

2020 ◽

Author(s):

Fei Liang ◽

Zhe Lin

Keyword(s):

Proof Theory ◽

Tense Logic ◽

Heyting Algebras ◽

Finite Model ◽

Algebraic Proof ◽

Model Property ◽

Algebraic Models ◽

Finite Axiomatizability ◽

Finite Model Property ◽

Modal Operators

Implicative semi-lattices (also known as Brouwerian semi-lattices) are a generalization of Heyting algebras, and have been already well studied both from a logical and an algebraic perspective. In this paper, we consider the variety ISt of the expansions of implicative semi-lattices with tense modal operators, which are algebraic models of the disjunction-free fragment of intuitionistic tense logic. Using methods from algebraic proof theory, we show that the logic of tense implicative semi-lattices has the finite model property. Combining with the finite axiomatizability of the logic, it follows that the logic is decidable.

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PROOF THEORY OF WEAK COMPACTNESS

Journal of Mathematical Logic ◽

10.1142/s0219061313500037 ◽

2013 ◽

Vol 13 (01) ◽

pp. 1350003 ◽

Cited By ~ 4

Author(s):

TOSHIYASU ARAI

Keyword(s):

Set Theory ◽

Proof Theory ◽

Weak Compactness ◽

Weakly Compact ◽

Compact Cardinal ◽

Weakly Compact Cardinal

We show that the existence of a weakly compact cardinal over the Zermelo–Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations.

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