The present paper is an attempt at the investigation of the nature of polarity contrast in natural languages. Truth conditions for natural language sentences are incomplete unless they include a proper definition of the conditions under which they are false. It is argued that the tertium non datur principle of classical bivalent logical systems is empirically invalid for natural languages: falsity cannot be equated with non-truth. Lacking a direct intuition about the conditions under which a sentence is false, we need (...) an independent foundation of the concept of falsity. The solution I offer is a definition of falsity in terms of the truth of a syntactic negation of the sentence. A definition of syntactic negation is proposed for English. (shrink)

A formal theory of oppositions and opposites is proposed on the basis of a non- Fregean semantics, where opposites are negation-forming operators that shed some new light on the connection between opposition and negation. The paper proceeds as follows. After recalling the historical background, oppositions and opposites are compared from a mathematical perspective: the first occurs as a relation, the second as a function. Then the main point of the paper appears with a calculus of oppositions, by means of a (...) non-Fregean semantics that redefines the logical values of various sorts of sentences. A num- ber of topics are then addressed in the light of this algebraic semantics, namely: how to construct value-functional operators for any logical opposition, beyond the classical case of contradiction; Blanché's "closure problem", i.e., how to find a complete structure connecting the sixteen binary sentences with one another. All of this is meant to devise an abstract theory of opposition: it encompasses the relation of consequence as subalternation, while relying upon the use of a primary "proto- negation" that turns any relatum into an opposite. This results in sentential negations that proceed as intensional operators, while negation is broadly viewed as a difference-forming operator without special constraints on it. (shrink)

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)

Duality in Logic and Language [draft--do not cite this article] Duality phenomena occur in nearly all mathematically formalized disciplines, such as algebra, geometry, logic and natural language semantics. However, many of these disciplines use the term ‘duality’ in vastly different senses, and while some of these senses are intimately connected to each other, others seem to be entirely … Continue reading Duality in Logic and Language →.

I provide arguments in favor of an implicature approach to Homogeneity where the basic meaning of the kids laughed is some of the kids laughed, and its strengthened meaning is all of the kids laughed. The arguments come from asymmetries between positive and negatives sentences containing definite plurals with respect to children’s behavior, the availability of Non-maximal readings, and the robustness of neither-true-nor-false judgments :205–248, 2015). I propose to avoid some problems of Magri’s analysis by modeling the Implicature account of (...) Homogeneity after the Implicature account of Free Choice, based on a hitherto unnoticed analogy between the two phenomena. The approach that emerges has the advantages of Magri’s implicature account of Homogeneity, while at the same time bears a close resemblance to recent approaches to Non-maximality :1–58. https://doi.org/10.3765/sp.5.3, 2012; Križ and Spector in Interpreting plural predication: homogeneity and non-maximality, Ms., Institut Jean Nicod, 2017), which enables restating their account of Non-maximality as following from the context-sensitivity of implicature calculation. (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)

The Generalized Quantifiers Theory, I will argue, in the second half of last Century has led to an important rapprochement, relevant both in logic and in linguistics, between logical quantification theories and the semantic analysis of quantification in natural languages. In this paper I concisely illustrate the formal aspects and the theoretical implications of this rapprochement.