Partial Semantics for Iterated if-Clauses

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.

Journal of Computer and System Sciences, 2010

In classical logics, the meaning of a formula is invariant with respect to the renaming of bound variables. This property, normally taken for granted, has been shown not to hold in the case of Independence Friendly (IF) logics. In this paper we argue that this is not an inherent characteristic of these logics but a defect in the way in which the compositional semantics given by Hodges for the regular fragment was generalized to arbitrary formulas. We fix this by proposing an alternative formalization, based on a variation of the classical notion of valuation. Basic metatheoretical results are proven. We present these results for Hodges' slash logic (from which these can be easily transferred to other IF-like logics) and we also consider the flattening operator, for which we give novel game-theoretical semantics.

Paraconsistency: Logic and Applications, 2012

Although propositional logic is about the analysis of all logical connectives, we must undoubtedly recognise a primus inter pares in this class: the conditional connective "if.. . then". Since the ancient times reams of paper have been depleted, and rivers of ink have been spilt, in order to discuss the logical properties of conditionals-even crows on the roofs once did so, according to an oft-quoted passage by Callimachus. Here I'll beg those birds to move over and let me join them in croaking about which conditionals are sound and which are not. Given the massive proportions of such a debate, it is to some extent surprising that there is comparably little agreement among the specialists on how to classify conditional sentences in natural languages like English. For the purpose of the present discussion, let us focus on what is in my opinion the most accurate taxonomy of conditionals sentences from a logical viewpoint. This taxonomy, or something closely resembling it, is to be found in several places in the literature (e.g. Routley et al. 1982; Mares 2004); conditionals are ranked in decreasing order according to the logical cogency of the connection between their antecedents and their consequents. • At the top of the ladder we find entailments, where the degree of logical cogency is maximal: necessarily, if the antecedent holds true, then so does the consequent. For example,

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.

Erkenntnis, 2007

Bradley has argued that a truth-conditional semantics for conditionals is incompatible with an allegedly very weak and intuitively compelling constraint on the interpretation of conditionals. I argue that the example Bradley offers to motivate this constraint can be explained along pragmatic lines that are compatible with the correctness of at least one popular truth-conditional semantics for conditionals.

In this paper we present a new approach to evaluate indicative conditionals with respect to some background information specified by a logic program. Because the weak completion of a logic program admits a least model under the three-valued Lukasiewicz semantics and this semantics has been successfully applied to other human reasoning tasks, conditionals are evaluated under these least L-models. If such a model maps the condition of a conditional to unknown, then abduction and revision are applied in order to satisfy the condition. Different strategies in applying abduction and revision might lead to different evaluations of a given conditional. Based on these findings we outline an experiment to better understand how humans handle those cases.

LACL 2016, LNCS 10054, M. Amblard et al. (Eds.), pp.291-307

This paper presents the first compositional semantics for if then conditionals. The semantics of each element are first examined separately. The meaning of if is modeled according to a possible worlds semantics. The particle then is analyzed as an anaphoric word that places its focused element inside the context settled by a previous element. Their meanings are subsequently combined in order to provide a formal semantics of if A then C conditionals, which differs from the simple if A, C form. This semantics has the particularity of validating contraposition for the first type but invalidating it for the second type. Finally, a detailed examination of the sentences presented in the literature opposing this schema of reasoning shows that these counterexamples do not generally concern if then conditionals but, rather, even if conditionals and that contraposition is therefore a valid means of reasoning with regard to if then conditionals in natural language, as this system predicts.

In this article I define a strong conditional for classical sentential logic, and then extend it to three non-classical sentential logics. It is stronger than the material conditional and is not subject to the standard paradoxes of material implication, nor is it subject to some of the standard paradoxes of C. I. Lewis’s strict implication. My conditional has some counterintuitive consequences of its own, but I think its pros outweigh its cons. In any case, one can always augment one’s language with more than one conditional, and it may be that no single conditional will satisfy all of our intuitions about how a conditional should behave. Finally, I will make no claim that the strong conditional is a good model for any particular use of the indicative conditional in English or other natural languages, though it would certainly be a nice bonus if some modified version of it could serve as one. I shall begin by exploring some of the disadvantages of the material conditional, the strict conditional, and some relevant conditionals. I proceed to define a strong conditional for classical sentential logic. I go on to adapt this account to Graham Priest’s Logic of Paradox, to S. C. Kleene’s logic K3, and then to J. Łukasiewicz’s logic Ł, a standard version of fuzzy logic.

Synthese, 2020

The aim of this paper is to give a general solution to the paradoxes of the material conditional, including the paradoxes generated by embedded conditionals. The solution consists in a pragmatic reinterpretation of the formal languages of classical logic LK and relevant logic LR (which rejects the validity of weakening and splits the connectives into two) as presented in Paoli [2002],[2007]. In particular I argue that the material conditional in the classical logic LK captures the truth conditions of "if...then", but ignores certain pragmatic enrichments that are associated to it, while relevant logic LR can give a systematic diagnostic of the cases in which a conditional is pragmatically enriched and those cases in which it is not. This diagnostic shows the reason why the paradoxes seem unacceptable and why they are, nevertheless, truth-preserving. This reinterpretation will also cover the solution that Paoli [2005] gives to McGee's paradoxes of the Modus Ponens.

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.