One of the manuscripts of Buridan’s Summulae contains three figures, each in the form of an octagon. At each node of each octagon there are nine propositions. Buridan uses the figures to illustrate his doctrine of the syllogism, revising Aristotle's theory of the modal syllogism and adding theories of syllogisms with propositions containing oblique terms (such as ‘man’s donkey’) and with ‘propositions of non-normal construction’ (where the predicate precedes the copula). O-propositions of non-normal construction (i.e., ‘Some S (some) P is (...) not’) allow Buridan to extend and systematize the theory of the assertoric (i.e., non-modal) syllogism. Buridan points to a revealing analogy between the three octagons. To understand their importance we need to rehearse the medieval theories of signification, supposition, truth and consequence. (shrink)

In this paper I propose a set-theoretical interpretation of the logical square of opposition, in the perspective opened by generalized quantifier theory. Generalized quantifiers allow us to account for the semantics of quantificational Noun Phrases, and of other natural language expressions, in a coherent and uniform way. I suggest that in the analysis of the meaning of Noun Phrases and Determiners the square of opposition may help representing some semantic features responsible to different logical properties of these expressions. I will (...) conclude with some consideration on scope interactions between quantifiers. (shrink)

In this paper, I raise the following problem: How does Avicenna define modalities? What oppositional relations are there between modal propositions, whether quantified or not? After giving Avicenna's definitions of possibility, necessity and impossibility, I analyze the modal oppositions as they are stated by him. This leads to the following results: The relations between the singular modal propositions may be represented by means of a hexagon. Those between the quantified propositions may be represented by means of two hexagons that one (...) could relate to each other.This is so because the exact negation of the bilateral possible, i.e. ‘necessary or impossible’ is given and applied to the quantified possible propositions.Avicenna distinguishes between the scopes of modality which can be either external or internal . His formulations are external unlike al-F̄ar̄ab̄;’s ones. However his treatment of modal oppositions remains incomplete because not all the relations between the modal propositi.. (shrink)

In this paper I examine Avicenna’s conception of the consequence relation. I will consider in particular his categorical and hypothetical logics. I will first analyse his definition of the implication and will show that this relation is not a consequence relation in his frame. Unlike the medieval logicians, he does not distinguish explicitly between material and formal consequences. The arguments discussed in al-Qiyās, where the conclusion is true only in some matters, and would seem close to a material consequence for (...) that reason, are rejected explicitly as not syllogistic. He also rejects the ‘enthymemes’ unless they are complemented by their missing premise and the superfluous premises which, according to him, should promptly be ruled out. It seems then that the consequence relation in his theory is formal. It can be characterized as being ‘productivity in all matters’ or ‘necessary truth preserving’. It is illustrated by some single premise arguments, and above all by all kinds of syllogisms which, in his theory, are more numerous and various than in Aristotle’s one. The syllogism may contain two or more premises, including disjunctive ones. When it is hypothetical, it may lead to several conclusions. The premises may be in conflict, but then, the conclusion is false. He thus rejects the principle according to which ‘anything follows from a contradiction’. But, unlike what some scholars say, he does not admit any connexive principle. In the compound syllogisms, the conclusion follows by steps, each step taking two premises at once. (shrink)

In this article, the author studies some central concepts in Avicenna's and sī's modal logics as presented in Avicenna's Al-Ish r t wa'l Tan īh t ( Pointers and Reminders ) and in sī's commentary. In this work, Avicenna introduces some remarkable distinctions in order to interpret Aristotle's modal syllogistic in the Prior Analytics . The author outlines a new interpretation of absolute sentences as temporally indefinite sentences and argues on the basis of this that Avicenna seems to subscribe to (...) the Principle of Plenitude. He also shows that he has no valid proof of the modal conversion rules and that he uses some rather ad hoc distinctions to show that Aristotle's modal syllogistic is correct. The author also notes some interesting differences between Avicenna's and sī's approaches to modal logic. (shrink)

Although he does not have an explicit theory of supposition as is found in the works of Latin medieval philosophers, Avicenna has two doctrines giving something equivalent: the threefold distinction of quiddity, corresponding to a division of simple, personal and material supposition, and his analyses of truth conditions for categorical propositions, where sentential context determines in part the reference of their terms. While he does address which individuals are being referred to by the universal terms used there, Avicenna concentrates more (...) on the varied temporal durations of the predications being made. In Western terms, he has incorporated ampliation and restriction into the theory of supposition itself. (shrink)

What is the function of logic in al-Kindī's corpus? What kind of relation does it have with mathematics? This article tackles these questions by examining al-Kindī's theory of categories as it was presented in his epistle On the Number of Aristotle's Books, from which we can learn about his special attitude towards Aristotle theory of categories and his interpretation, as well. Al-Kindī treats the Categories as a logical book, but in a manner different from that of the classical Aristotelian tradition. (...) He ascribes a special status to the categories Quantity and Quality, whereas the rest of the categories are thought to be no more than different combinations of these two categories with the category Substance. The discussion will pay special attention to the function of the categories of Quantity and Quality as mediators between logic and mathematics. (shrink)

The reception history of Aristotle's Prior Analytics in the Islamic world began even before its ninth-century translation into Arabic. Three generations earlier, Arabic authors already absorbed echoes of the varied and extensive logical teaching tradition of Greek- and Syriac-speaking religious communities in the new Islamic state. Once translated into Arabic, the Prior Analytics inspired a rich tradition of logical studies, culminating in the creation of an independent Islamic logical tradition by Ibn Sina (d. 1037), Ibn Rušd (d. 1098) and others. (...) This article traces the translation and commentary tradition of the Prior Analytics in Syriac and Arabic in the sixth to ninth centuries and sketches its appropriation, revision and, ultimately, transformation by Islamic philosophers between the ninth and eleventh centuries. (shrink)

I try to explain the difference between three kinds of negation: external negation, negation of the predicate and privation. Further I use polygons of opposition as heuristic devices to show that a logic which contains all three mentioned kinds of negation must be a fragment of a Łukasiewicz-four-valued predicate logic. I show, further, that, this analysis can be elaborated so as to comprise additional kinds of privation. This would increase the truth-values in question and bring fragments of (more generally speaking) (...) Łukasiewicz-n-valued predicate logics into the scene. (shrink)

The square of opposition (as part of a lattice) is used as a natural way to represent different and opposite ways of who makes decisions, and in what way, in/for a group or a society. Majority logic is characterized by multiple logical squares (one for each possible majority), with the “discursive dilemma” as a consequence. Three-valued logics of majority decisions with discursive dilemma undecided, of veto, consensus, and sequential voting are analyzed from the semantic point of view. For instance, the (...) paraconsistent and paracomplete logics M3, M3veto and C3 are described. The distinction of designated and non-designated values is not used, and instead, the consequence relation is defined as a preservation of the minimum truth-value of the implying set of sentences. The consequence and opposition relations of the logics described are compared, and an ordering of the logics with respect to their opposition relations is established. (shrink)

This paper analyzes a classification of different types of demonstration introduced by Alfarabi in his Kitāb al-Burhān. Alfarabi identifies eight combinations of demonstrative syllogisms, grouped in function of the different types of per se relations expressed by their premises and conclusions, where terms are definitionally connected with one another. The list contains a total of thirty-nine moods illustrated by a rich array of examples drawn from various scientific disciplines, including arithmetic, geometry, and natural philosophy. The combinations and moods are discussed (...) extensively by Averroes in the section of his Epitome of the Organon devoted to the Posterior Analytics and in his Quaesita on logic. Alfarabi’s classification also possibly inspired a simplified taxonomical effort in Avicenna’s Kitāb al-Burhān. (shrink)

Aquinas’ famous comments in his early Scriptum on the Sentences (In I Sent., d. 38, q. 1, a. 3) regarding the intellect’s apprehension of essence and esse have traditionally been interpreted as grounding Aquinas’ doctrine on the judgment of esse. For Aquinas, it appears, what the intellect apprehends in a simple concept is essence. Since esse, for him, is not an essence, it cannot, on the received view, be the object of conceptualization. Therefore, esse is grasped by the intellect only (...) in judgment. The claim that no genuine concept of esse is possible, however, is inconsistent with Aquinas’ theory of signification. A term’s signification is constituted, at least in part, in its “signing relation” with some “concept” in the mind. If, as on the traditional reading, there is no concept of esse, the term ‘esse’ is left without signification. To respond that the term ‘esse’ signs, not a concept, but the judgment in which esse is apprehended is in direct conflict with Aquinas’ claim elsewhere that no term, including ‘ens’ and ‘esse’, signifies a judgment. I propose an alternative interpretation to In I Sent., d. 38, q. 1, a. 3, one that allows for the possibility of a conceptual apprehension of esse. The alternative reading explains Aquinas’ remarks there in light of the theoretical context of the discussion, in particular the theory of propositions and their objects. I show how and why Aquinas develops a distinctive theory of the object of the proposition, that it is concerned with “the esse of a thing” as a complex. Although this complex includes the simple act of being, as for the traditional position (contrary to its major critics), it cannot be reduced to anything simple. Despite the Sentences’ affirmation of a real distinction between a thing’s esse and essence, to which correspond two different operations of the intellect, it does not follow that the human intellect cannot conceive esse, just as it conceives essences, in a simple conception. (shrink)

This dissertation offers a new and comprehensive analysis of Abū Naṣr al-Fārābī's cosmology by focusing on various important issues that have been largely neglected by the modern scholarship. It provides an examination of the physical, metaphysical, and astronomical aspects of al-Fārābī's cosmology by adopting a multidisciplinary approach that takes into account the history of philosophy and the history of astronomy. Accordingly, my dissertation explores how al-Fārābī attempted to reconcile features of Ptolemaic astronomy with Aristotelian and Neoplatonic theories, an endeavor which (...) resulted in the formulation of innovative cosmological ideas. Chapters I and II provide background information on al-Fārābī's activity as a commentator and his relation to the Greek commentatorial tradition and assess the relevant primary sources. In addition, they examine al-Fārābī's approach to cosmology and his scientific method in terms of both the Ptolemaic astronomical tradition and the Aristotelian corpus. Chapter III addresses problems related to the structure of al-Fārābī's cosmology, especially the origin of his ennadic cosmological model and his spherology. Particular attention is devoted to the question of how Aristotle's unmoved movers and Ptolemy's astronomical theories are reconciled. Chapters IV, V, and VI analyze the place of celestial matter, intellect, and motion respectively in al-Fārābī's cosmology and attempt to redefine the second teacher's relation to the various trends of Greek philosophy. The study stresses al-Fārābī's connection with the Neoplatonic and Peripatetic currents, particularly with thinkers such as Alexander of Aphrod. (shrink)