Zur Einheit der modalen Syllogistik des Aristoteles

2008 ◽  
Vol 13 ◽  
pp. 54-86
Author(s):  
Klaus J. Schmidt
Keyword(s):  
Modal Logic ◽  
Simple Structure ◽  
Predicate Logic ◽  
Modern Logic ◽  
Basic Formula ◽  
Complexity Results ◽  
Modal Predicate Logic

On the unity of modal syllogistics in Aristotle. The goal of this paper is an interpretation of Aristotle’s modal syllogistics closely oriented on the text using the resources of modern modal predicate logic. Modern predicate logic was successfully able to interpret Aristotle’s assertoric syllogistics uniformly, that is, with one formula for universal premises. A corresponding uniform interpretation of modal syllogistics by means of modal predicate logic is not possible. This thesis does not imply that a uniform view is abandoned. However, it replaces the simple unity of the assertoric by the complex unity of the modal. The complexity results from the fact that though one formula for universal premises is used as the basis, it must be moderated if the text requires.Aristotle introduces his modal syllogistics by expanding his assertoric syllogistics with an axiom that links two apodictic premises to yield a single apodictic sentence. He thus defines a regular modern modal logic. By means of the regular modal logic that is thus defined, he is able to reduce the purely apodictic syllogistics to assertoric syllogistics. However, he goes beyond this simple structure when he looks at complicated inferences.In order to be able to link not only premises of the same modality, but also premises with different modalities, he introduces a second axiom, the T-axiom, which infers from necessity to reality or – equivalently – from reality to possibility. Together, the two axioms, the axiom of regularity and the T-axiom, define a regular T-logic. It plays an important role in modern logic. In order to be able to account for modal syllogistics adequately as a whole, another modern axiom is also required, the so-called B-axiom. It is very difficult to decide whether Aristotle had the B-axiom. Each of the two last named axioms is sufficient to achieve the required contextual moderation of the basic formula for universal propositions.

Modal logic

2018 ◽  
Author(s):  
Steven T. Kuhn
Keyword(s):  
Modal Logic ◽  
Possible Worlds ◽  
Predicate Logic ◽  
Modus Ponens ◽  
Possible World ◽  
Metaphysical Necessity ◽  
Truth Values ◽  
Classical Counterpart ◽  
Modern Development ◽  
Modal Predicate Logic

Modal logic, narrowly conceived, is the study of principles of reasoning involving necessity and possibility. More broadly, it encompasses a number of structurally similar inferential systems. In this sense, deontic logic (which concerns obligation, permission and related notions) and epistemic logic (which concerns knowledge and related notions) are branches of modal logic. Still more broadly, modal logic is the study of the class of all possible formal systems of this nature. It is customary to take the language of modal logic to be that obtained by adding one-place operators ‘□’ for necessity and ‘◇’ for possibility to the language of classical propositional or predicate logic. Necessity and possibility are interdefinable in the presence of negation: □A↔¬◊¬A and  ◊A↔¬□¬A hold. A modal logic is a set of formulas of this language that contains these biconditionals and meets three additional conditions: it contains all instances of theorems of classical logic; it is closed under modus ponens (that is, if it contains A and A→B it also contains B); and it is closed under substitution (that is, if it contains A then it contains any substitution instance of A; any result of uniformly substituting formulas for sentence letters in A). To obtain a logic that adequately characterizes metaphysical necessity and possibility requires certain additional axiom and rule schemas: K □(A→B)→(□A→□B) T □A→A 5 ◊A→□◊A Necessitation A/□A. By adding these and one of the □–◇ biconditionals to a standard axiomatization of classical propositional logic one obtains an axiomatization of the most important modal logic, S5, so named because it is the logic generated by the fifth of the systems in Lewis and Langford’s Symbolic Logic (1932). S5 can be characterized more directly by possible-worlds models. Each such model specifies a set of possible worlds and assigns truth-values to atomic sentences relative to these worlds. Truth-values of classical compounds at a world w depend in the usual way on truth-values of their components. □A is true at w if A is true at all worlds of the model; ◇A, if A is true at some world of the model. S5 comprises the formulas true at all worlds in all such models. Many modal logics weaker than S5 can be characterized by models which specify, besides a set of possible worlds, a relation of ‘accessibility’ or relative possibility on this set. □A is true at a world w if A is true at all worlds accessible from w, that is, at all worlds that would be possible if w were actual. Of the schemas listed above, only K is true in all these models, but each of the others is true when accessibility meets an appropriate constraint. The addition of modal operators to predicate logic poses additional conceptual and mathematical difficulties. On one conception a model for quantified modal logic specifies, besides a set of worlds, the set Dw of individuals that exist in w, for each world w. For example, ∃x□A is true at w if there is some element of Dw that satisfies A in every possible world. If A is satisfied only by existent individuals in any given world ∃x□A thus implies that there are necessary individuals; individuals that exist in every accessible possible world. If A is satisfied by non-existents there can be models and assignments that satisfy A, but not ∃xA. Consequently, on this conception modal predicate logic is not an extension of its classical counterpart. The modern development of modal logic has been criticized on several grounds, and some philosophers have expressed scepticism about the intelligibility of the notion of necessity that it is supposed to describe.

scholarly journals Constant Domain Quantified Modal Logics Without Boolean Negation

2005 ◽  
Vol 3 ◽  
Author(s):  
Greg Restall
Keyword(s):  
Modal Logic ◽  
Predicate Logic ◽  
Difficult Problem ◽  
Modal Logics ◽  
Modal Operator ◽  
Completeness Proof ◽  
Constant Domain ◽  
Relevant Logics ◽  
Modal Predicate Logic

his paper provides a sound and complete axiomatisation for constant domain modal logics without Boolean negation. This is a simpler case of the difficult problem of providing a sound and complete axiomatisation for constant-domain quantified relevant logics, which can be seen as a kind of modal logic with a two-place modal operator, the relevant conditional. The completeness proof is adapted from a proof for classical modal predicate logic (I follow James Garson’s 1984 presentation of the completeness proof quite closely), but with an important twist, to do with the absence of Boolean negation.

Syllogism and quantification

1962 ◽  
Vol 27 (1) ◽  
pp. 58-72 ◽  
Author(s):  
Timothy Smiley
Keyword(s):  
Predicate Logic ◽  
Modern Logic ◽  
First Order ◽  
First Order Predicate Logic

Anyone who reads Aristotle, knowing something about modern logic and nothing about its history, must ask himself why the syllogistic cannot be translated as it stands into the logic of quantification. It is now more than twenty years since the invention of the requisite framework, the logic of many-sorted quantification.In the familiar first-order predicate logic generality is expressed by means of variables and quantifiers, and each interpretation of the system is based upon the choice of some class over which the variables may range, the only restriction placed on this ‘domain of individuals’ being that it should not be empty.

A proof-theoretic study of the correspondence of classical logic and modal logic

2003 ◽  
Vol 68 (4) ◽  
pp. 1403-1414 ◽  
Author(s):  
H. Kushida ◽  
M. Okada
Keyword(s):  
Modal Logic ◽  
Classical Logic ◽  
Predicate Logic ◽  
Transformation Method ◽  
Modal Operator ◽  
Theoretic Method ◽  
Barcan Formula ◽  
Theoretic Proof ◽  
Classical Predicate

AbstractIt is well known that the modal logic S5 can be embedded in the classical predicate logic by interpreting the modal operator in terms of a quantifier. Wajsberg [10] proved this fact in a syntactic way. Mints [7] extended this result to the quantified version of S5; using a purely proof-theoretic method he showed that the quantified S5 corresponds to the classical predicate logic with one-sorted variable. In this paper we extend Mints' result to the basic modal logic S4; we investigate the correspondence between the quantified versions of S4 (with and without the Barcan formula) and the classical predicate logic (with one-sorted variable). We present a purely proof-theoretic proof-transformation method, reducing an LK-proof of an interpreted formula to a modal proof.

The complexity of the modal predicate logic of “true in every transitive model of ZF”

1997 ◽  
Vol 62 (4) ◽  
pp. 1371-1378
Author(s):  
Vann McGee
Keyword(s):  
Lower Bound ◽  
Set Theory ◽  
Predicate Logic ◽  
Alternative Conception ◽  
Predicate Calculus ◽  
Modal Formula ◽  
Sentential Calculus ◽  
Transitive Model ◽  
Modal Predicate Logic

Robert Solovay [8] investigated the version of the modal sentential calculus one gets by taking “□ϕ” to mean “ϕ is true in every transitive model of Zermelo-Fraenkel set theory (ZF).” Defining an interpretation to be a function * taking formulas of the modal sentential calculus to sentences of the language of set theory that commutes with the Boolean connectives and sets (□ϕ)* equal to the statement that ϕ* is true in every transitive model of ZF, and stipulating that a modal formula ϕ is valid if and only if, for every interpretation *, ϕ* is true in every transitive model of ZF, Solovay obtained a complete and decidable set of axioms.In this paper, we stifle the hope that we might continue Solovay's program by getting an analogous set of axioms for the modal predicate calculus. The set of valid formulas of the modal predicate calculus is not axiomatizable; indeed, it is complete .We also look at a variant notion of validity according to which a formula ϕ counts as valid if and only if, for every interpretation *, ϕ* is true. For this alternative conception of validity, we shall obtain a lower bound of complexity: every set which is in the set of sentences of the language of set theory true in the constructible universe will be 1-reducible to the set of valid modal formulas.

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.

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