A deductive system for Lewis counterfactuals is presented, based directly on the influential generalisation of relational semantics through ternary similarity relations introduced by Lewis. This deductive system builds on a method of enriching the syntax of sequent calculus by labels for possible worlds. The resulting labelled sequent calculus is shown to be equivalent to the axiomatic system VC of Lewis. It is further shown to have the structural properties that are needed for an analytic proof system that supports root-first proof search. Completeness of the calculus is proved in a direct way, such that for any given sequent either a formal derivation or a countermodel is provided; it is also shown how finite countermodels for unprovable sequents can be extracted from failed proof search, by which the completeness proof turns into a proof of decidability. §1. Introduction. Counterfactual conditionals have been of interest in Philosophy ever since the classic accounts of Chisholm (1946) and Goodman (1947), which were motivated by the large role that counterfactual reasoning appears to play in ordinary life and scientific inquiry. Counterfactuals still play a crucial role in current theories of causation, natural laws, in epistemology, and in metaphysics. In English, counterfactuals are usually expressed by subjunctive conditionals; to use the famous example by , followed by his proposed two-place connective for the counterfactual conditional:If kangaroos had no tails, they would topple over. Kangaroos have no tail Kangaroos topple overCounterfactuals share some properties with other kinds of conditionals, most notably the material conditional and the strict conditional introduced by C. I. Lewis. For instance, they all validate modus ponens and modus tollens. Yet the logic of counterfactuals can't be simply analyzed in terms of either. Transitivity (Hypothetical Syllogism) is validated by material and strict conditionals, but fails for counterfactuals:If Hoover had been a Communist, he would have been a traitor. If Hoover had been born in Russia, he would have been a Communist.∴ If Hoover had been born in Russia, he would have been a traitor.

We discuss a well-known puzzle about the lexicalization of logical operators in natural language, in particular connectives and quantifiers. Of the many logically possible operators, only few appear in the lexicon of natural languages: the connectives in English, for example, are conjunction and, disjunction or, and negated disjunction nor; the lexical quantifiers are all, some and no. The logically possible (negated conjunction) and (negated universal) are not expressed by lexical entries in English, nor in any natural language. Moreover, the lexicalized operators are all upward or downward monotone, an observation known as the Monotonicity Universal. We propose a logical explanation of lexical gaps and of the Monotonicity Universal, based on the dynamic behaviour of connectives and quantifiers. We define update potentials for logical operators as procedures to modify the context, under the assumption that an update by $$ \phi $$ ϕ depends on the logical form of $$ \phi $$ ϕ and on the speech act performed: assertion or rejection. We conjecture that the adequacy of update potentials determines the limits of lexicalizability for logical operators in natural language. Finally, we show that on this framework the Monotonicity Universal follows from the logical properties of the updates that correspond to each operator.

Given current data, only a few binary Boolean operators are expressed in lexically simple fashion in the world's languages: and, or, nor. These do not occur in every combination, for example, nor is not observed by itself. To explain these cross‐linguistic patterns, we propose an encoding of Boolean operators as update procedures to accept or reject information in a context. We define a measure of conceptual simplicity for such updates, on which attested operators are conceptually simpler than the remaining Booleans. Moreover, we show that language evolution selects for the attested lexical inventories by minimizing the complexity of using a lexical inventory compositionally to convey precise information.

Lewis's counterfactual logics are a class of conditional logics that are defined as extensions of classical propositional logic with a twoplace modal operator expressing conditionality. Labelled proof systems are proposed here that capture in a modular way Burgess's preferential conditional logic PCL, Lewis's counterfactual logic V, and their extensions. The calculi are based on preferential models, a uniform semantics for conditional logics introduced by Lewis. The calculi are analytic, and their completeness is proved by means of countermodel construction. Due to termination in root-first proof search, the calculi also provide a decision procedure for the logics.

According to the Maxim of Quality, rational agents tend to speak honestly. Due to the influence of Grice, a connection between linguistic rationality and honesty is often taken for granted. However, the connection is not obvious: structural rationality in language use does not require honesty, any more than it requires dishonesty. In particular, Quality does not follow from the Cooperative Principle and structural rationality. But then what is honest rational speech? I propose to move the discussion in the context of Stalnaker's theory of assertion. From this perspective, although there is no most rational way to behave, Quality follows from the structure of Stalnakerian conversations if interlocutors are sensitive to credibility. In this case, honesty is built on the expectation of reciprocity, and cooperativity is an outcome and not a precondition of rational communication. A benefit of my discussion is that the account of linguistic rationality falls under the more general view of interactive rationality familiar from theoretical economics.