Conditionals and modals in contexts

1. For millennia philosophers have been grappling with the problem of the logic of conditionals-the problem of determining which principles of inference hold for conditional statements. Surprisingly enough (considering the nature of philosophers) there has been almost universal agreement on at least a small number of these principles, including, for example, modus ponens, modus tollens, hypothetical syllogism and contraposition. Most of the dispute about the logic of conditionals has concerned whether or not certain additional principles, such as those which generate the so-called paradoxes of material implication, hold for conditionals. Recently, however, several philosophers have presented putative counterexamples to hypothetical syllogism and to contraposition, and have argued that these patterns of inference are not valid for conditionals. I will try to show that these philosophers are mistaken because their examples involve certain hypothetical statements which are not genuine conditionals at all. Since this strategy presupposes that hypotheticals (i.e., 'if '-statements) can be legitimately divided into at least two categories (which I shall call conditionals on the one hand and theticals on the other), I must at least address the problem of providing a comprehensive account of hypotheticals.1 I shall first try to show that 'if '-sentences are used in ordinary discourse to make several distinguishable kinds of statements, one of these, the genuine conditional, involving an assertion that the state of affairs described by the apodosis is conditional upon, or a consequence of, the state of affairs described by the protasis. This kind of statement, in the making of which the speaker posits a consequential connection between the protasis-state and the apodosisstate, does seem to obey the principles of hypothetical syllogism and contraposition. Secondly, I shall try to show that the putative counterexamples to these principles involve theticals-hypotheticals which are not assertions that the apodosis-state is a consequence of the protasis-state. Finally, I shall attempt to show that the relevant theticals not only fail to obey the rules of hypothetical syllogism and contraposition, but fail to obey other basic principles, such as modus ponens and modus tollens. If my arguments are sound, it will follow that a theorist who tries to devise a logic of hypotheticals-

The past decade has seen a protracted debate over the semantics of epistemic modals. According to contextualists, epistemic modals quantify over the possibilities compatible with some contextually determined group’s information. Relativists often object that contextualism fails to do justice to the way we assess utterances containing epistemic modals for truth or falsity. However, recent empirical work seems to cast doubt on the relativist’s claim, suggesting that ordinary speakers’ judgments about epistemic modals are more closely in line with contextualism than relativism (Knobe & Yalcin 2014; Khoo 2015). This paper furthers the debate by reporting new empirical research revealing a previously overlooked dimension of speakers’ truth-value judgments concerning epistemic modals. Our results show that these judgments vary systematically with the question under discussion in the conversational context in which the utterance is being assessed. We argue that this ‘QUD effect’ is difficult to explain if contextualism is true, but is readily explained by a suitably flexible form of relativism. EARLY ACCESS

The debate over Hypothetical Syllogism is locked in stalemate. Although putative natural language counterexamples to Hypothetical Syllogism abound, many philosophers defend Hypothetical Syllogism, arguing that the alleged counterexamples involve an illicit shift in context. The proper lesson to draw from the putative counterexamples, they argue, is that natural language conditionals are context-sensitive conditionals which obey Hypothetical Syllogism. In order to make progress on the issue, I consider and improve upon Morreau’s proof of the invalidity of Hypothetical Syllogism. The improved proof relies upon the semantic claim that conditionals with antecedents irrelevant to the obtaining of an already true consequent are themselves true. Moreover, this semantic insight allows us to provide compelling counterexamples to Hypothetical Syllogism that are resistant to the usual contextualist response.

A variety of problems emerged investigating electronic circuits, computer devices and cellular automata motivated a number of attempts to create a differential and integral calculus for Boolean functions. In the present article, we extend this kind of calculus in order to include the semantic of classical logical operations. We show that this extension to logics is strongly helped if we submerge the elementary logical calculus in a matrix-vector formalism that naturally includes a kind of fuzzy-logic. In this way, guided by the laws of matrix algebra, we can construct compact representations for the derivatives and the integrals of logical functions. Inside this semantic-algebraic calculus, we obtain expressions for the derivatives of some of the basic logical operations and show the general way to obtain the derivatives of any well-formed formula of propositional calculus. We show that some of the basic tautologies (Excluded middle, Modus ponens, Hypothetical syllogism) are members of a kind of hierarchical system linked by the differentiation algorithm. In addition using the logical derivatives we show that relatively complex formulas can collapse in simple expressions that reveal clearly their hidden logical meaning. The search for the antiderivatives produces naturally an integral calculus. Within this logical formalism an indefinite integral can always be found for any logical expression. Moreover, particular integrals can be constructed based on detachment properties that lead to logical expressions of growing complexity. We show that these particular integrals have some similarities with the 'generalizing deduction' procedures investigated by Lukasiewicz.

The implication operator (→) plays a significant role in the classical two-valued logic. Firstly, from the classical implication one can obtain all other basic logical connectives of the binary logic, viz., the binary operators - and (\(\land\)), or (\(\lor\)) - and the unary negation operator (¬). Secondly, the implication operator holds the center stage in the inference mechanisms of any logic, like modus ponens, modus tollens, hypothetical syllogism in classical logic.

On functional equations and inequalities related to some reasoning schemes that involve fuzzy implications

We introduce a new formal logic, called equality propositional logic. It has two basic connectives, $boldsymbol{wedge}$ (conjunction) and $equiv$ (equivalence). Moreover, the $Rightarrow$ (implication) connective can be derived as $ARightarrow B:=(Aboldsymbol{wedge}B)equiv A$. We formulate the equality propositional logic and demonstrate that the resulting logic has reasonable properties such as Modus Ponens(MP) rule, Hypothetical Syllogism(HS) rule and completeness, etc. Especially, we provide two ways to prove the completeness of this logic system. We also introduce two extensions of equality propositional logic. The first one is involutive equality propositional logic, which is equality propositional logic with double negation. The second one adds prelinearity which is rich enough to enjoy the strong completeness property. Finally, we introduce additional connective $Delta$(delta) in equality propositional logic and demonstrate that the resulting logic holds soundness and completeness.

This paper explores the relationship between dynamic and truth conditional semantics for epistemic modals. It provides a generalization of a standard dynamic update semantics for modals. This new semantics derives a Kripke semantics for modals and a standard dynamic semantics for modals as special cases. The semantics allows for new characterizations of a variety of principles in modal logic, including the inconsistency of ‘p and might not p’. Finally, the semantics provides a construction procedure for transforming any truth conditional semantics for modals into a dynamic semantics for modals with similar properties.

This paper addresses the role of modal expressions as argumentative indicators and the importance of considering the semantics of modality in the reconstruction of argumentation. Modal expressions include modal verbs, such as the English may, can, must, should, will and many other lexical and grammatical devices. For instance, in English we find modal adjectives (possible, necessary, probable, likely, certain), adverbs (maybe, necessarily, certainly, probably, likely, perhaps) and nouns (possibility, necessity, but also less obvious ones like risk). The present investigation builds on previous linguistic work on the semantics of modality (see Rocci, 1997, 2000a, 2000b, 2005a, 2005b, 2006, 2007) which showed that modals can function as indicators of relations between discourse units in a text or dialogue, and that epistemic modals, in particular, can serve as signals of inferential, and hence argumentative, relations. Here we consider what this analysis entails for the task of reconstructing the structure of argumentation. To do so we will first go back to Toulmin’s views on the relevance of modality for understanding the structure of arguments and, more precisely, to the views he states in the very first chapter of The Uses of Argument concerning modals and fields of arguments. We will argue that Toulmin is indeed after something which is crucial for the semantics of the modals and relevant for the reconstruction of arguments, but, at the same time, his analysis is problematic on important respects. In the second part of the article we will present a view of modality, originating in linguistic research on formal semantics, which we believe can capture Toulmin’s insights but is not plagued by the same problems and thus can serve as the basis for a treatment of modals in their role of indicators of argumentation. Chapter 15 Modalities as Indicators in Argumentative Reconstruction

The predicates typical of scientific discourse shown, proved, established , etc., which are used to present the results of the study are discussed in the article. The relevance of the study is due to the scientific significance of addressing the problem of discursive research of epistemic indicators of reliability in the context of the principles of the language postclassical logical analysis. The purpose of the study is to justify the “modality” of these constructions and describe the epistemic modal meaning that they bring to the proposition, as specific propositional attitudes. The methodology used in the article is based on some basic principles of modal (in particular, epistemic) logic and linguistic semantics of J. Lyons and E. V. Paducheva. The novelty of the study is in the fact that the pragmatic functions of the considered predicates are described. It is shown that, in addition to a simple statement, their value contains an epistemic modal component, which is based on the value of “provability” described by modal logic. It is established that this modal meaning correlates with the category of objective language modality. Particular attention is paid to logical and linguistic grounds on which these predicates are classified as modal. The question is raised about the functioning of these predicates in contexts complicated by indicators of subjective epistemic modality with a value of problematic certainty. It is concluded that the predicates shown, proved, established, and similar ones indicate the “objective” (verifiable) nature of the information presented and the author’s conviction of the reliability of the described data, and constructions with the indicated predicates also perform a rhetorical function and enhance the “persuasiveness” of the text.

The inferences of contraposition (A ⇒ C ∴ ¬C ⇒ ¬A), the hypothetical syllogism (A ⇒ B, B ⇒ C ∴ A ⇒ C), and others are widely seen as unacceptable for counterfactual conditionals. Adams convincingly argued, however, that these inferences are unacceptable for indicative conditionals as well. He argued that an indicative conditional of form A ⇒ C has assertability conditions instead of truth conditions, and that their assertability ‘goes with’ the conditional probability p(C|A). To account for inferences, Adams developed the notion of probabilistic entailment as an extension of classical entailment. This combined approach (correctly) predicts that contraposition and the hypothetical syllogism are invalid inferences. Perhaps less well-known, however, is that the approach also predicts that the unconditional counterparts of these inferences, e.g., modus tollens (A ⇒ C, ¬C ∴ ¬A), and iterated modus ponens (A ⇒ B, B ⇒ C, A ∴ C) are predicted to be valid. We will argue both by example and by calling to the results from a behavioral experiment (N = 159) that these latter predictions are incorrect if the unconditional premises in these inferences are seen as new information. Then we will discuss Adams’ (1998) dynamic probabilistic entailment relation, and argue that it is problematic. Finally, it will be shown how his dynamic entailment relation can be improved such that the incongruence predicted by Adams’ original system concerning conditionals and their unconditional counterparts are overcome. Finally, it will be argued that the idea behind this new notion of entailment is of more general relevance.

This paper considers some questions related to the determination of epistemic modal domains. Specifically, given situations in which groups of agents have epistemic states that bear on a modal domain, how is the domain best restricted? This is a metasemantic project, in which I combine a standard semantics for epistemic modals, as developed by Kratzer, with a standard story about conversational dynamics, as developed by Stalnaker. I show how a standard framework for epistemic logic can model their interaction. I contend that if groups have epistemic states that bear on the modal domain, as the data suggests, then careful attention must be paid to the method of combination selected for the epistemic states of the individuals within the group. Specifically, there should be some explanation of the flow of information in a group-deliberative setting. Through this study we find a novel explanation of modal disagreement and uptake.

This chapter defends a semantics for epistemic modals and probability operators. This semantics is probabilistic—that is, sentences containing these expressions have sets of probability spaces as their semantic values relative to a context. Existing non-truth-conditional semantic theories of epistemic modals face serious problems when it comes to interpreting nested modal constructions such as ‘it must be possible that Jones smokes’. The semantics in this chapter solves these problems, accounting for several significant features of nested epistemic vocabulary. The chapter ends by defending a probabilistic semantics for simple sentences that do not contain any epistemic vocabulary, and by using this semantics to illuminate the relationship between credence and full belief.

Communication facilitates coordination, but coordination might fail if there's too much uncertainty. I discuss a scenario in which vagueness-driven uncertainty undermines the possibility of publicly sharing a belief. I then show that asserting an epistemic modal sentence, 'Might p', can reveal the speaker's uncertainty, and that this may improve the chances of coordination despite the lack of a common epistemic ground. This provides a game-theoretic rationale for epistemic modality. The account draws on a standard relational semantics for epistemic modality, Stalnaker's theory of assertion as informative update, and a Bayesian framework for reasoning under uncertainty.

The paper explores the syntax and semantics of epistemic modality in Thai grammar. Four syntactic categories are found to express epistemic modality—preverbal auxiliaries, initial particles, adverbs and final particles. Epistemic modality can also be further classified; five categories are found to be distinctive in the system of Thai grammar—assumptive, speculative, deductive, dubitative and certainty.