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

Masazumi Hanazawa
1966 Proceedings of the Japan Academy  
Let be a 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.  ... 
doi:10.3792/pja/1195521795 fatcat:nd72kdyihjbq3lvkjlonllsnly

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

Fernando Ferreira, Gilda Ferreira
2015 Studia Logica: An International Journal for Symbolic Logic  
It is known that there is a sound and faithful translation 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 .  ... 
doi:10.1007/s11225-015-9620-5 fatcat:vjheqmu6anbbdmpvtficnabwga

On Herbrand's Theorem for Intuitionistic Logic [chapter]

Alexander Lyaletski, Boris Konev
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,  ... 
doi:10.1007/11853886_25 fatcat:pgcsqj3455e4rh2ypuzxffpxem

On the separation theorem of intermediate propositional calculi

Tsutomu Hosoi
1966 Proceedings of the Japan Academy  
The algebraic proof 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*.)  ... 
doi:10.3792/pja/1195521924 fatcat:cokwr4zeobf4pfqrtpnq6ixly4

Decision Problems for Second-Order Linear Logic [chapter]

P. D. Lincoln, A. Scedrov, N. Shankar
1997 Logic and Scientific Methods  
Among the consequences are the undecidability 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  ... 
doi:10.1007/978-94-017-0487-8_7 fatcat:sp5afdrfa5hythn66hgfja5nkq

Propositional Calculus and Realizability

Gene F. Rose
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  ... 
doi:10.2307/1990776 fatcat:jdjm2wf3gfaoveblt5onfza2eq

Propositional calculus and realizability

Gene F. Rose
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  ... 
doi:10.1090/s0002-9947-1953-0055952-4 fatcat:pp3njj5qkraj5oqsbgoqt5eomu

A General Theory of Completeness Proofs

Sh^|^ocirc;ji MAEHARA
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.  ... 
doi:10.4288/jafpos1956.3.242 fatcat:szgmwmcf5vhhjewqc3n6t6jare

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

В.М. Попов
2013 Logical Investigations  
This short paper presents a new domain 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).  ... 
doi:10.21146/2074-1472-2013-19-0-197-199 fatcat:43b4eekftraerbfye6tzhu6grq

On the computational content of intuitionistic propositional proofs

Samuel R Buss, Pavel Pudlák
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.  ... 
doi:10.1016/s0168-0072(01)00040-9 fatcat:opphbkqdj5g37eswodpmobtdnm

Propositional mixed logic: its syntax and semantics

Karim Nour, Abir Nour
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?"  ... 
doi:10.3166/jancl.13.377-390 fatcat:5ayrje3lezgttas3n4ofudubka

On an intuitionistic logic for pragmatics

Gianluigi Bellin, Massimiliano Carrara, Daniele Chiffi
2015 Journal of Logic and Computation  
Work 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.  ... 
doi:10.1093/logcom/exv036 fatcat:kiqprtcp6vfrhnutwqpyadmpyy

Formula Transformers and Combinatorial Test Generators for Propositional Intuitionistic Theorem Provers [article]

Paul Tarau
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.  ... 
arXiv:1910.01775v1 fatcat:bbbtx56isjdxnlsetphluha4xy

Propositional Mixed Logic: Its Syntax and Semantics [article]

Karim Nour, Abir Nour
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  ... 
arXiv:0905.0369v1 fatcat:rz44kldnjvfinduzyr2scgtyie
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