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A characterization of axiom schema playing the rôle of tertium non datur in intuitionistic logic

1966
*
Proceedings of the Japan Academy
*

Let be a

doi:10.3792/pja/1195521795
fatcat:nd72kdyihjbq3lvkjlonllsnly
*formula*containing only*one**propositional**variable*A. Then*one**of*the*formulas*and A--(OIA) is provable*intuitionistically*. ... By every rule*of*inference*in*the*intuitionistic**propositional**calculus*(e.g. modus ponens), from*one*or more -*formulas*we infer a -*formula*. ... Le$ be a $au$ology containing only*one**propositional**variable*A and be no$ a t-*formula*. ...##
###
Page 2 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 75, Issue 1
[page]

1953
*
American Mathematical Society. Transactions of the American Mathematical Society
*

Kleene reports(’) that his current work

*on*realizability*of**formulas*with function*variables*, as proposed*in*[16, end §2], may cast new light*on*the*intuitionistic*interpretation*of*number-theoretic*formulas*... Moreover,*one*might seek to establish completeness by allowing the sub- stitution*of**formulas*from the enlarged class containing function*variables*. ...##
###
The Faithfulness of Fat: A Proof-Theoretic Proof

2015
*
Studia Logica: An International Journal for Symbolic Logic
*

It is known that there is a sound and faithful translation

doi:10.1007/s11225-015-9620-5
fatcat:vjheqmu6anbbdmpvtficnabwga
*of*the full*intuitionistic**propositional**calculus*into the atomic polymorphic system Fat, a predicative*calculus*with only two connectives: the ... As an application, we give a purely proof-theoretic proof*of*the disjunction property*of*the*intuitionistic**propositional*logic*in*which commuting conversions are not needed. ... The latter reference is a study*on*the translation*of*the commuting conversions*of*the*intuitionistic**propositional**calculus*into F at . ...##
###
On Herbrand's Theorem for Intuitionistic Logic
[chapter]

2006
*
Lecture Notes in Computer Science
*

*in*a

*propositional*

*intuitionistic*tableaux

*calculus*, provided that the

*propositional*proof is compatible with the way how the instances were generated. ...

*In*this paper we reduce the question

*of*validity

*of*a first-order

*intuitionistic*

*formula*without equality to generating ground instances

*of*this

*formula*and then checking whether the instances are deducible ... For any

*propositional*tableau

*intuitionistic*

*calculus*pC it is possible to formulate the notion

*of*compatibility with a substitution such that any

*formula*φ is

*intuitionistically*valid if, and only if, ...

##
###
On the separation theorem of intermediate propositional calculi

1966
*
Proceedings of the Japan Academy
*

The algebraic proof

doi:10.3792/pja/1195521924
fatcat:cokwr4zeobf4pfqrtpnq6ixly4
*of*this corollary will appear*in*near future. ... And the rule*of*substitution is*in*it. (2) The*calculus*is I complete. (3) There exist I*formulas*f (a, b)(i=1, ..., n)whose*propositional**variables*are only a and b such that*formulas**of*the f oorb8*Of*... (iii) A sub*formula**of*the form B is replaced by B c, where c is a*propositional**variable*which does not appear*in*A. (We put the so changed*formula*as A*. And negation does not appear*in*A*.) ...##
###
Decision Problems for Second-Order Linear Logic
[chapter]

1997
*
Logic and Scientific Methods
*

Among the consequences are the undecidability

doi:10.1007/978-94-017-0487-8_7
fatcat:sp5afdrfa5hythn66hgfja5nkq
*of*the*intuitionistic*second-order fragment*of**propositional*multiplicative linear logic and the undecidability*of*multiplicative linear logic with rst-order ... It is shown that the structural rules*of*contraction and weakening may be simulated by second-order*propositional*quanti ers and the multiplicative connectives. ...*In*particular, let us consider the presentation*of*second-order*intuitionistic**propositional**calculus*, lj2,*in*which the*formulas*are built from*propositional**variables*and the constant True by the con-nectives ...##
###
Propositional Calculus and Realizability

1953
*
Transactions of the American Mathematical Society
*

*In*particular, then, every

*formula*

*in*the symbolism

*of*the predicate

*calculus*(with

*proposition*and predicate

*variables*) which is provable

*in*the

*intuitionistic*predicate

*calculus*has the property that ... David Nelson (2) showed that every number-theoretic

*formula*deducible

*in*the

*intuitionistic*predicate

*calculus*(3) (stated by means

*of*schemata, without

*proposition*or predicate

*variables*) from realizable ...

*In*particular, then, every

*formula*

*in*the symbolism

*of*the predicate

*calculus*(with

*proposition*and predicate

*variables*) which is provable

*in*the

*intuitionistic*predicate

*calculus*has the property that ...

##
###
Propositional calculus and realizability

1953
*
Transactions of the American Mathematical Society
*

*In*particular, then, every

*formula*

*in*the symbolism

*of*the predicate

*calculus*(with

*proposition*and predicate

*variables*) which is provable

*in*the

*intuitionistic*predicate

*calculus*has the property that ... David Nelson (2) showed that every number-theoretic

*formula*deducible

*in*the

*intuitionistic*predicate

*calculus*(3) (stated by means

*of*schemata, without

*proposition*or predicate

*variables*) from realizable ...

*In*particular, then, every

*formula*

*in*the symbolism

*of*the predicate

*calculus*(with

*proposition*and predicate

*variables*) which is provable

*in*the

*intuitionistic*predicate

*calculus*has the property that ...

##
###
A General Theory of Completeness Proofs

1970
*
Annals of the Japan Association for Philosophy of Science
*

*In*this treatise, we comment

*on*the classical and the

*intuitionistic*logic only. ... Especially

*in*treatments

*on*the predicate logic, the second E-Theorem

*in*the sense

*of*Hilbert-Bernays [2] is used effectively. ... The

*calculus*LK or LJ based

*on*those

*formulas*gives a system

*of*the classical or

*intuitionistic*

*propositional*

*calculus*, respectively. ...

##
###
Between $Int_{\omega , \omega}$ and intuitionistic propositional logic

2013
*
Logical Investigations
*

This short paper presents a new domain

doi:10.21146/2074-1472-2013-19-0-197-199
fatcat:43b4eekftraerbfye6tzhu6grq
*of*logical investigations. ... A logic is said to be a nonempty set*of**formulas*closed under the rule*of*modus ponens*in*L and the rule*of*substitution*of*a*formula*into a*formula*instead*of*a*propositional**variable**in*L. ... It is proved that Int <0,0> is the set*of**intuitionistic*tautologies*in*L (that is, the*intuitionistic**propositional*logic*in*L). ...##
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On the computational content of intuitionistic propositional proofs

2001
*
Annals of Pure and Applied Logic
*

*In*this paper we further generalize the two results

*on*the

*intuitionistic*

*propositional*

*calculus*. ... The sequent

*calculus*for

*intuitionistic*logic is formulated

*in*the usual way. Each sequent has at most

*one*

*formula*

*in*the succedent. ...

##
###
Propositional mixed logic: its syntax and semantics

2003
*
Journal of Applied Non-Classical Logics
*

*In*this paper, we present a

*propositional*logic (called mixed logic) containing disjoint copies

*of*minimal,

*intuitionistic*and classical logics. ... We present at the end a sequent

*calculus*version for this logic. 1 coherent with what we already know

*on*sequent

*calculus*: classical logic comes from the possibility to put several

*formulas*

*on*the right ... Nour for his help

*in*the writing

*of*this paper. Open question: "Is it possible to eliminate cuts

*in*such a system?" ...

##
###
On an intuitionistic logic for pragmatics

2015
*
Journal of Logic and Computation
*

Work

doi:10.1093/logcom/exv036
fatcat:kiqprtcp6vfrhnutwqpyadmpyy
*on*linear co-intuitionism is then recalled, a linear*calculus**of*co-*intuitionistic*coroutines is defined and a probabilistic interpretation*of*linear co-intuitionism is given as*in*[9] . ... We focus*on*the extension to co-*intuitionistic*logic, seen as a logic*of*hypotheses [8, 9, 13 ] and*on*polarized bi-*intuitionistic*logic as a logic*of*assertions and conjectures: looking at the S4 modal ...*In*all cases the*proposition*is proved simply by checking that the resulting set*of*terms contains exactly*one*free*variable*. ...##
###
Formula Transformers and Combinatorial Test Generators for Propositional Intuitionistic Theorem Provers
[article]

2019
*
arXiv
*
pre-print

*propositional*logic. ... We develop combinatorial test generation algorithms for progressively more powerful theorem provers, covering

*formula*languages ranging from the implicational fragment

*of*

*intuitionistic*logic to full

*intuitionistic*... Catching Bugs Using more General

*Intuitionistic*

*Propositional*

*Calculus*

*Formulas*We can catch a bug if the "suspect" disagrees with itself

*on*the small easy

*formula*and its hard transformed

*formula*. ...

##
###
Propositional Mixed Logic: Its Syntax and Semantics
[article]

2009
*
arXiv
*
pre-print

*In*this paper, we present a

*propositional*logic (called mixed logic) containing disjoint copies

*of*minimal,

*intuitionistic*and classical logics. ... We establish some relations between mixed logic and minimal,

*intuitionistic*and classical logics. We present at the end a sequent

*calculus*version for this logic. ... We also thank Jamil Nour for his help

*in*the writing

*of*this paper. References ...

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