59. Conditionals

2011

Within linguistic semantics, it is near orthodoxy that the function of the word 'if'(in most cases) is to mark restrictions on quantification. Just as in the sentence 'Every man smokes', the common noun 'man'restricts the quantifier 'every', in the sentence 'Usually, if it's winter it's cold','it's winter'acts as a restrictor on the situational quantifier 'usually'. This view, originally due to Lewis (1975), has been greatly extended in work by Heim (1982) and, most notably, Kratzer (1981, 1986) into a rich theory of almost all uses of the word 'if'.

Journal of Philosophical Logic, 2015

2005

First, this paper proposes a new classification of ordinary language conditionals and discusses some of their properties. My approach is fundamentally based on the classical view that a conditional involves a sufficient condition (the antecedent) and a necessary condition (the consequent). Since the truth of the consequent is necessary for the truth of the antecedent, it is also an integral property of a typical conditional that the falsity of the consequent is sufficient for the falsity of the antecedent. Then the paper goes into the much discussed question of the truth conditions of a conditional. Some authors maintain that conditionals are not propositions and are not bearers of a truth-value.1 Contrary to these claims, I consider ordinary language conditionals as statements that are considered true or false by the speaker and the addressee and express propositions that may be true or false. Since it involves sufficiency and necessity, a conditional is seen here as a complex sort of modal statement. ‘If’ is considered as an ordinary language operator that establishes a triple modal relation between two propositions. The paradoxes of the material conditional are avoided, as well as some problems of the alternative theories of the conditional. On the other hand, as any theory, the present one has its own problems, which, for lack of space, I will not explore in this article.

2020

EPiC Series in Computing

We discuss the evaluation of conditionals. Under classical logic a conditional of the form "A implies B" is semantically equivalent to "not A or B". However, psychological experiments have repeatedly shown that this is not how humans understand and use conditionals.We introduce an innovative abstract reduction system under the three-valued Łukasiewicz logic and the weak completion semantics, that allows us to reason abductively and by revision with respect to conditionals, in three values. We discuss the strategy of minimal revision followed by abduction and discuss two notions of relevance.Psychological experiments will need to ascertain if these strategies and notions, or a variant of them, correspond to how humans reason with conditionals.

In this paper I will propose a refinement of the semantics of hypervaluations (Mura 2009), one in which a hypervaluation is built up on the basis of a set of valuations, instead of a single val-uation. I shall define validity with respect to all the subsets of valua-tions. Focusing our attention on the set of valid sentences, it may easily shown that the rule substitution is restored and we may use valid schemas to represent classes of valid sentences sharing the same logical form. However, the resulting semantical theory TH turns out to be throughout a modal three-valued theory (modal sym-bols being definable in terms of the non modal connectives) and a fragment of it may be considered as a three-valued version of S5 system. Moreover, TH may be embedded in S5, in the sense that for every formula ϕ of TH there is a corresponding formula ϕ' of S5 such that ϕ' is S5-valid iff ϕ is TH-valid. The fundamental property of this system is that it allows the definition of a purely semantical relation of logical consequence which is coextensive to Adams’ p-entailment with respect to simple conditional sentences, without be-ing defined in probabilistic terms. However, probability may be well be defined on the lattice of hypervaluated tri-events, and it may be proved that Adam’s p-entailment, once extended to all tri-events, coincides with our notion of logical consequence as defined in purely semantical terms.

Topoi, 2021

The variably strict analysis of conditionals does not only largely dominate the philosophical literature, since its invention by Stalnaker and Lewis, it also found its way into linguistics and psychology. Yet, the shortcomings of Lewis-Stalnaker's account initiated a plethora of modifications, such as non-vacuist conditionals, presuppositional indicatives, perfect conditionals, or other conditional constructions, for example: reason relations, difference-making conditionals, counterfactual dependency, or probabilistic relevance. Many of these new connectives can be treated as strengthened or weakened conditionals. They are definable conditionals. This article develops a technique to infer the logic for such definable conditionals from the known logic of the underlying defining conditional. The technique is applied to central examples. The results show that a large part of the zoo of conditionals arises from a basic conditional-a constant nucleus of the different contextual and conceptual variations of variably strict conditionals.

This is the first part of two in a study on the logic and semantics of the indicative conditional. This first part provides a global expressivist analysis of the indicative conditional along the lines of the Ramsey Test. The analysis is a form of 'global' expressivism in that it supplies acceptance and rejection conditions for all the sentence forming connectives of propositional logic (negation, disjunction, etc.) and so allows the conditional to embed in arbitrarily complex sentences (thus avoiding the Frege-Geach problem). The resulting mental usage semantics is provably not susceptible to triviality results and completely characterises a logic for the indicative conditional that respects many of its well known logical quirks (the failure of modus ponens, etc.). The second part provides a semantic analysis of the resulting structure and logic.

There is a profound, but frequently ignored relationship between logical consequence (formal implication) and material implication. The first repeats the patterns of the latter, but with a wider modal reach. It is argued that this kinship between formal and material implication simply means that they express the same kind of implication, but differ in scope. Formal implication is unrestricted material implication. This apparently innocuous observation has some significant corollaries: (1) conditionals are not connectives, but arguments; (2) the traditional examples of valid argumentative forms are metalogical principles that express the properties of logical consequence; (3) formal logic is not a useful guide to detect valid arguments in the real world; (4) it is incoherent to propose alternatives to the material implication while accepting the classical properties of formal implication; (5) some of the counter-examples to classical argumentative forms and known conditional puzzles are unsound.

Logic and Logical Philosophy, 2019

The paper proposes a first approach to systems whose language includes two primitives (> + and > −) as symbols for factual and counterfactual conditionals which are explicit, i.e. that are stated jointly with the truth or falsity of the antecedent clause. In systems based on this language, here called 2-conditional, the standard corner operator may be defined by (Def >) A > B := (A > + B) ∨ (A > − B), while in classical conditional systems one could introduce the two symbols for explicit conditionals by the definitions (Def > +) A > + B := A ∧ (A > B) and (Def > −) A > − B := ¬A ∧ (A > B). Two 2-conditional systems, V 2 and VW 2 , are axiomatized and proved to be definitionally equivalent to the monoconditional systems V and VW. A third system VWTr 2 is characterized by an axiom stating the transitivity of factual conditionals and is shown to be distinct from V 2 , from VW 2 and from the 2-conditional version of Lewis' well-known system VC, here named VC 2. The same may be said for a fourth system VW 2 ♦±, based on an axiom inderivable in VC 2 : ♦(A > + B) ⊃ ♦(¬A > − ¬B). VC 2 contains what is here called a "semi-collapse" of the operator > + and it is argued that it is inadequate as a logic for both factual and counterfactual conditionals. The last section shows that several different definitions of the corner operators in terms of > + and > − may be introduced as an alternative to (Def >).