Life-World Constitution of Propositional Logic and Elementary Predicate Logic

1972 ◽  
Vol 32 (3) ◽  
pp. 322 ◽  
Author(s):  
L. Eley
Keyword(s):  
Propositional Logic ◽  
Predicate Logic ◽  
Life World

UNIFORM INTERPOLATION IN SUBSTRUCTURAL LOGICS

2014 ◽  
Vol 7 (3) ◽  
pp. 455-483 ◽  
Author(s):  
MAJID ALIZADEH ◽  
FARZANEH DERAKHSHAN ◽  
HIROAKIRA ONO
Keyword(s):  
Propositional Logic ◽  
Predicate Logic ◽  
Interpolation Property ◽  
Substructural Logics ◽  
Substructural Logic ◽  
Sequent Calculi ◽  
Structural Rule ◽  
Modal Propositional Logic ◽  
Uniform Interpolation ◽  
Classical Predicate

AbstractUniform interpolation property of a given logic is a stronger form of Craig’s interpolation property where both pre-interpolant and post-interpolant always exist uniformly for any provable implication in the logic. It is known that there exist logics, e.g., modal propositional logic S4, which have Craig’s interpolation property but do not have uniform interpolation property. The situation is even worse for predicate logics, as classical predicate logic does not have uniform interpolation property as pointed out by L. Henkin.In this paper, uniform interpolation property of basic substructural logics is studied by applying the proof-theoretic method introduced by A. Pitts (Pitts, 1992). It is shown that uniform interpolation property holds even for their predicate extensions, as long as they can be formalized by sequent calculi without contraction rules. For instance, uniform interpolation property of full Lambek predicate calculus, i.e., the substructural logic without any structural rule, and of both linear and affine predicate logics without exponentials are proved.

scholarly journals The natural logic of language and cognition

Pragmatics ◽  
2006 ◽  
Vol 16 (1) ◽  
pp. 103-138 ◽  
Author(s):  
Pieter A.M. Seuren
Keyword(s):  
Set Theory ◽  
Propositional Logic ◽  
Predicate Logic ◽  
Human Cognition ◽  
Natural Logic ◽  
Analogous Argument ◽  
Standard Set ◽  
Language And Cognition ◽  
The Way

This paper aims at an explanation of the discrepancies between natural intuitions and standard logic in terms of a distinction between NATURAL and CONSTRUCTED levels of cognition, applied to the way human cognition deals with sets. NATURAL SET THEORY (NST) restricts standard set theory cutting it down to naturalness. The restrictions are then translated into a theory of natural logic. The predicate logic resulting from these restrictions turns out to be that proposed in Hamilton (1860) and Jespersen (1917). Since, in this logic, NO is a quantifier in its own right, different from NOT-SOME, and given the assumption that natural lexicalization processes occur at the level of basic naturalness, single-morpheme lexicalizations for NOT-ALL should not occur, just as there is no single-morpheme lexicalization for NOT-SOME at that level. An analogous argument is developed for the systematic absence of lexicalizations for NOT-AND in propositional logic.

On logics intermediate between intuitionistic and classical predicate logic

1959 ◽  
Vol 24 (2) ◽  
pp. 141-153 ◽  
Author(s):  
Toshio Umezawa
Keyword(s):  
Propositional Logic ◽  
Predicate Logic ◽  
Classical Propositional Logic ◽  
Intermediate Predicate Logics ◽  
Study Inclusion ◽  
Classical Predicate

In [1] and [2] I investigated logics intermediate between intuitionistic and classical propositional logic. In the present paper I shall study inclusion and non-inclusion between certain intermediate predicate logics. All the logics considered result from intuitionistic predicate logic by addition of classically valid axiom schemes.

UNIFICATION IN SUPERINTUITIONISTIC PREDICATE LOGICS AND ITS APPLICATIONS

2018 ◽  
Vol 12 (1) ◽  
pp. 37-61 ◽  
Author(s):  
WOJCIECH DZIK ◽  
PIOTR WOJTYLAK
Keyword(s):  
Propositional Logic ◽  
Predicate Logic ◽  
Order Logic ◽  
Existential Quantifier ◽  
First Order ◽  
Admissible Rules ◽  
Structural Completeness ◽  
Individual Variables ◽  
Definition Of ◽  
Finitely Presented

AbstractWe introduce unification in first-order logic. In propositional logic, unification was introduced by S. Ghilardi, see Ghilardi (1997, 1999, 2000). He successfully applied it in solving systematically the problem of admissibility of inference rules in intuitionistic and transitive modal propositional logics. Here we focus on superintuitionistic predicate logics and apply unification to some old and new problems: definability of disjunction and existential quantifier, disjunction and existential quantifier under implication, admissible rules, a basis for the passive rules, (almost) structural completeness, etc. For this aim we apply modified specific notions, introduced in propositional logic by Ghilardi, such as projective formulas, projective unifiers, etc.Unification in predicate logic seems to be harder than in the propositional case. Any definition of the key concept of substitution for predicate variables must take care of individual variables. We allow adding new free individual variables by substitutions (contrary to Pogorzelski & Prucnal (1975)). Moreover, since predicate logic is not as close to algebra as propositional logic, direct application of useful algebraic notions of finitely presented algebras, projective algebras, etc., is not possible.

scholarly journals Learnability

1985 ◽  
Vol 14 (199) ◽  
Author(s):  
Gudmund Skovbjerg Frandsen
Keyword(s):  
Mathematical Model ◽  
Probability Distribution ◽  
Finite Number ◽  
Propositional Logic ◽  
Formal Language ◽  
Predicate Logic ◽  
Recognition Algorithm ◽  
Specific Class ◽  
Finite Sets ◽  
Specific Concept

<p>Les Valiant has recently conceived a remarkable mathematical model of learnability. The originality appears through several facets of the model. Objects belonging to a specific concept are given a measure of naturalness in the form of a probability distribution. The learning of a concept takes place by means of a protocol that among other tools allows the use of a source of natural examples. A concept is learnable if a recognition algorithm can be synthesized within a polynomial number of steps. The recognition algorithm is allowed to be incorrect for an adjustable fraction of inputs measured with respect to naturalness.</p><p>Technically the model is based on the propositional logic over a finite number of Boolean variables. However, the underlying ideas are quite universal and can be realised by means of an almost arbitrary formal language, which we will demonstrate in this note. A single concept may include infinitely many objects within a formal language frame. Fortunately we can learn such concepts from finite sets of examples only. We shall prove a specific class of concepts to be learnable within the nontrivial formal language of predicate logic.</p>

scholarly journals Term Logic

Axioms ◽  
2020 ◽  
Vol 9 (1) ◽  
pp. 18 ◽  
Author(s):  
Peter Simons
Keyword(s):  
Propositional Logic ◽  
Predicate Logic ◽  
Expressive Power ◽  
Predominant Form ◽  
Term Logic ◽  
Empty Term ◽  
Inference Tree

The predominant form of logic before Frege, the logic of terms has been largely neglected since. Terms may be singular, empty or plural in their denotation. This article, presupposing propositional logic, provides an axiomatization based on an identity predicate, a predicate of non-existence, a constant empty term, and term conjunction and negation. The idea of basing term logic on existence or non-existence, outlined by Brentano, is here carried through in modern guise. It is shown how categorical syllogistic reduces to just two forms of inference. Tree and diagram methods of testing validity are described. An obvious translation into monadic predicate logic shows the system is decidable, and additional expressive power brought by adding quantifiers enables numerical predicates to be defined. The system’s advantages for pedagogy are indicated.

scholarly journals Neutrosophic Modal Logic

2017 ◽  
Author(s):  
Florentin Smarandache
Keyword(s):  
Modal Logic ◽  
Propositional Logic ◽  
Predicate Logic ◽  
Neutrosophic Logic ◽  
Definition Of ◽  
First Time

I introduce now for the first time the neutrosophic modal logic. The Neutrosophic Modal Logic includes the neutrosophic operators that express the modalities. It is an extension of neutrosophic predicate logic, and of neutrosophic propositional logic. In order for the paper to be self-contained, I also recall the etymology and definition of neutrosophy and of neutrosophic logic. Several examples are presented as well.

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