AN INTENSIONAL LEIBNIZ SEMANTICS FOR ARISTOTELIAN LOGIC

2010 ◽  
Vol 3 (2) ◽  
pp. 262-272 ◽  
Author(s):  
KLAUS GLASHOFF
Keyword(s):  
Natural Deduction ◽  
Predicate Logic ◽  
Formal System ◽  
Aristotelian Logic ◽  
Categorical Logic ◽  
Characteristic Numbers ◽  
Formal Syntax ◽  
Philosophical Standpoint ◽  
Classical Philosophy

Since Frege’s predicate logical transcription of Aristotelian categorical logic, the standard semantics of Aristotelian logic considers terms as standing for sets of individuals. From a philosophical standpoint, this extensional model poses problems: There exist serious doubts that Aristotle’s terms were meant to refer always to sets, that is, entities composed of individuals. Classical philosophy up to Leibniz and Kant had a different view on this question—they looked at terms as standing for concepts (“Begriffe”). In 1972, Corcoran presented a formal system for Aristotelian logic containing a calculus of natural deduction, while, with respect to semantics, he still made use of an extensional interpretation. In this paper we deal with a simple intensional semantics for Corcoran’s syntax—intensional in the sense that no individuals are needed for the construction of a complete Tarski model of Aristotelian syntax. Instead, we view concepts as containing or excluding other, “higher” concepts—corresponding to the idea which Leibniz used in the construction of his characteristic numbers. Thus, this paper is an addendum to Corcoran’s work, furnishing his formal syntax with an adequate semantics which is free from presuppositions which have entered into modern interpretations of Aristotle’s theory via predicate logic.

Logic in the twenties: the nature of the quantifier

1979 ◽  
Vol 44 (3) ◽  
pp. 351-368 ◽  
Author(s):  
Warren D. Goldfarb
Keyword(s):  
Predicate Logic ◽  
Formal System ◽  
Recent Book ◽  
Modern Logic ◽  
Nineteen Twenties ◽  
Quantification Theory ◽  
Michael Dummett ◽  
The Road ◽  
Semantic Treatment ◽  
Do So

We are often told, correctly, that modern logic originated with Frege. For Frege clearly depicted polyadic predication, negation, the conditional, and the quantifier as the bases of logic; moreover, he introduced the idea of a formal system, and argued that mathematical demonstrations, to be fully precise, must be carried out within a formal language by means of explicitly formulated syntactic rules.Consequently Frege has often been read as providing all the central notions that constitute our current understanding of quantification. For example, in his recent book on Frege [1973], Michael Dummett speaks of ”the semantics which [Frege] introduced for formulas of the language of predicate logic.” That is, “An interpretation of such a formula … is obtained by assigning entities of suitable kinds to the primitive nonlogical constants occurring in the formula … [T]his procedure is exactly the same as the modern semantic treatment of predicate logic” (pp. 89–90). Indeed, “Frege would therefore have had within his grasp the concepts necessary to frame the notion of the completeness of a formalization of logic as well as its soundness … but he did not do so” (p. 82).This common appraisal of Frege's work is, I think, quite misleading. Even given Frege's tremendous achievements, the road to an understanding of quantification theory was an arduous one. Obtaining such understanding and formulating those notions which are now common coin in the discussion of logical systems were the tasks of much of the work in logic during the nineteen-twenties.

Proof systems for lattice theory

2004 ◽  
Vol 14 (4) ◽  
pp. 507-526 ◽  
Author(s):  
SARA NEGRI ◽  
JAN VON PLATO
Keyword(s):  
Lattice Theory ◽  
Natural Deduction ◽  
Sequent Calculus ◽  
Predicate Logic ◽  
Relational Theory ◽  
Predicate Calculus ◽  
Alternative Formulation ◽  
Proof Systems ◽  
Decidable Classes ◽  
Classical Predicate

A formulation of lattice theory as a system of rules added to sequent calculus is given. The analysis of proofs for the contraction-free calculus of classical predicate logic known as G3c extends to derivations with the mathematical rules of lattice theory. It is shown that minimal derivations of quantifier-free sequents enjoy a subterm property: all terms in such derivations are terms in the endsequent.An alternative formulation of lattice theory as a system of rules in natural deduction style is given, both with explicit meet and join constructions and as a relational theory with existence axioms. A subterm property for the latter extends the standard decidable classes of quantificational formulas of pure predicate calculus to lattice theory.

Natural deduction

2021 ◽  
pp. 65-100
Author(s):  
Paolo Mancosu ◽  
Sergio Galvan ◽  
Richard Zach
Keyword(s):  
Classical Logic ◽  
Natural Deduction ◽  
Final Section ◽  
Predicate Logic ◽  
Original System ◽  
Axiomatic System ◽  
Proof System ◽  
Ordinary Life ◽  
Classical Predicate

Natural deduction is a philosophically as well as pedagogically important logical proof system. This chapter introduces Gerhard Gentzen’s original system of natural deduction for minimal, intuitionistic, and classical predicate logic. Natural deduction reflects the ways we reason under assumption in mathematics and ordinary life. Its rules display a pleasing symmetry, in that connectives and quantifiers are each governed by a pair of introduction and elimination rules. After providing several examples of how to find proofs in natural deduction, it is shown how deductions in such systems can be manipulated and measured according to various notions of complexity, such as size and height. The final section shows that the axiomatic system of classical logic presented in Chapter 2 and the system of natural deduction for classical logic introduced in this chapter are equivalent.

scholarly journals Maximum cuts in extended natural deduction

2010 ◽  
Vol 87 (101) ◽  
pp. 59-74
Author(s):  
Mirjana Borisavljevic
Keyword(s):  
Natural Deduction ◽  
Predicate Logic ◽  
Standard System

We consider a standard system of sequents and a system of extended natural deduction (which is a modification of natural deduction) for intuitionistic predicate logic and connect the special cuts, maximum cuts, from sequent derivations and maximum segments from derivations of extended natural deduction. We show that the image of a sequent derivation without maximum cuts is a derivation without maximum segments (i.e., a normal derivation) in extended natural deduction.

scholarly journals Growing into deduction

2020 ◽  
Vol 63 (1) ◽  
pp. 87-106
Author(s):  
Jovana Kostic ◽  
Katarina Maksimovic
Keyword(s):  
Mathematical Logic ◽  
20Th Century ◽  
Natural Deduction ◽  
Deductive Reasoning ◽  
Formal System ◽  
Formal Systems ◽  
Adults And Children ◽  
Logical Connectives

Psychologists have experimentally studied deductive reasoning since the beginning of the 20th century. However, as we will argue, there has not been much improvement in the field until relatively recently, due to how the experiments were designed. We deem the design of the majority of conducted experiments inadequate for two reasons. The first one is that psychologists have, for the most part, ignored the development of mathematical logic and based their research on syllogistic inferences. The second reason is the influence of the view, which is dogmatically still prevalent in semantics and logic in general, that the categorical notions, such as the notion of truth, are more important than the hypothetical notions, such as the notion of deduction. The influence of this dogma has been twofold. In studies concerning logical connectives in adults and children, much more emphasis has been put on the semantical aspects of the connectives - the truth functions, than on the deductive inferences. And secondly, even in the studies that investigated deductive inferences by using formal systems, the dogma still influenced the choice of the formal system. Researchers, in general, preferred the axiomatic formal systems over the systems of natural deduction, even though the systems of the second kind are much more suitable for studying deduction.

The Grammatical Background of Kant's General Logic

2008 ◽  
Vol 13 (1) ◽  
pp. 116-140
Author(s):  
Kurt Mosser
Keyword(s):  
Natural Deduction ◽  
Necessary Conditions ◽  
Predicate Logic ◽  
Specific Model ◽  
Symbolic Logic ◽  
Pure Reason ◽  
Critique Of Pure Reason ◽  
First Order ◽  
General Logic ◽  
First Order Predicate Logic

In theCritique of Pure Reason, Kant conceives of general logic as a set of universal and necessary rules for the possibility of thought, or as a set of minimal necessary conditions for ascribing rationality to an agent (exemplified by the principle of non-contradiction). Such a conception, of course, contrasts with contemporary notions of formal, mathematical or symbolic logic. Yet, in so far as Kant seeks to identify those conditions that must hold for the possibility of thought in general, such conditions must holda fortiorifor any specific model of thought, including axiomatic treatments of logic and standard natural deduction models of first-order predicate logic. Kant's general logic seeks to isolate those conditions by thinking through – or better, reflecting on – those conditions that themselves make thought possible.

THE QUANTIFIED ARGUMENT CALCULUS

2014 ◽  
Vol 7 (1) ◽  
pp. 120-146 ◽  
Author(s):  
HANOCH BEN-YAMI
Keyword(s):  
Natural Language ◽  
Predicate Logic ◽  
Formal System ◽  
Proof System ◽  
Value Assignment ◽  
Truth Value ◽  
Necessary Existence ◽  
Modal Predicate Logic ◽  
Converse Relation ◽  
Main Ideas

AbstractI develop a formal logic in which quantified arguments occur in argument positions of predicates. This logic also incorporates negative predication, anaphora and converse relation terms, namely, additional syntactic features of natural language. In these and additional respects, it represents the logic of natural language more adequately than does any version of Frege’s Predicate Calculus. I first introduce the system’s main ideas and familiarize it by means of translations of natural language sentences. I then develop a formal system built on these principles, the Quantified Argument Calculus or Quarc. I provide a truth-value assignment semantics and a proof system for the Quarc. I next demonstrate the system’s power by a variety of proofs; I prove its soundness; and I comment on its completeness. I then extend the system to modal logic, again providing a proof system and a truth-value assignment semantics. I proceed to show how the Quarc versions of the Barcan formulas, of their converses and of necessary existence come out straightforwardly invalid, which I argue is an advantage of the modal Quarc over modal Predicate Logic as a system intended to capture the logic of natural language.

scholarly journals Minimum segments in sequent derivations

2003 ◽  
Vol 74 (88) ◽  
pp. 5-18
Author(s):  
Mirjana Borisavljevic
Keyword(s):  
Natural Deduction ◽  
Predicate Logic ◽  
Special Kind

In a system of sequents for intuitionistic predicate logic derivations without a special kind of cuts (maximum cuts) will be considered. The following be shown: in a derivation without maximum cuts there are paths of the same form as paths in a normal derivation of natural deduction, i.e., these paths have the E-part, the I-part, and one minimum part which corresponds to a minimum segment in a normal derivation.

scholarly journals Normal form theorem for systems of sequents

2007 ◽  
pp. 37-53
Author(s):  
Mirjana Borisavljevic
Keyword(s):  
Normal Form ◽  
Natural Deduction ◽  
Predicate Logic ◽  
Special Kind ◽  
Form Theorem ◽  
Normal Form Theorem

In a system of sequents for intuitionistic predicate logic a theorem, which corresponds to Prawitz?s Normal Form Theorem for natural deduction, are proved. In sequent derivations a special kind of cuts, maximum cuts, are defined. Maximum cuts from sequent derivations are connected with maximum segments from natural deduction derivations, i.e., sequent derivations without maximum cuts correspond to normal derivations in natural deduction. By that connection the theorem for the system of sequents (which correspond to Normal Form Theorem for natural deduction) will have the following form for each sequent derivation whose end sequent is ??A there is a sequent derivation without maximum cuts whose end sequent is ??A.

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