Alexander W. Kocurek

Hyperlogic is a hyperintensional system designed to regiment metalogical claims (e.g., "Intuitionistic logic is correct" or "The law of excluded middle holds") into the object language, including within embedded environments such as attitude reports and counterfactuals. This paper is the first of a two-part series exploring the logic of hyperlogic. This part presents a minimal logic of hyperlogic and proves its completeness. It consists of two interdefined axiomatic systems: one for classical consequence (truth preservation under a classical interpretation of the connectives) and one for "universal" consequence (truth preservation under any interpretation). The sequel to this paper explores stronger logics that are sound and complete over various restricted classes of models as well as languages with hyperintensional operators.

There are many different ways to talk about the world. Some ways of talking are more expressive than others—that is, they enable us to say more things about the world. But what exactly does this mean? When is one language able to express more about the world than another? In my dissertation, I systematically investigate different ways of answering this question and develop a formal theory of expressive power, translation, and notational variance. In doing so, I show how these investigations help to clarify the role that expressive power plays within debates in metaphysics, logic, and the philosophy of language.

Sentences about logic are often used to show that certain embedding expressions are hyperintensional. Yet it is not clear how to regiment “logic talk” in the object language so that it can be compositionally embedded under such expressions. In this paper, I develop a formal system called hyperlogic that is designed to do just that. I provide a hyperintensional semantics for hyperlogic that doesn’t appeal to logically impossible worlds, as traditionally understood, but instead uses a shiftable parameter that determines the interpretation of the logical connectives. I argue this semantics compares favorably to the more common impossible worlds semantics, which faces difficulties interpreting propositionally quantified logic talk.

Counterfactual skepticism holds that most ordinary counterfactuals are false. The main argument for this view appeals to a ‘chance undermines would’ principle: if ψ would have some chance of not obtaining had φ obtained, then φ []–> ψ is false. This principle seems to follow from two fairly weak principles, viz., that ‘chance ensures could’ and that φ []–> ψ and φ <>–> ψ clash. Despite their initial plausibility, I show that these principles are independently problematic: given some modest closure principles, they entail absurdities. Moreover, on the most promising strategy for saving these principles, they do not, in the relevant sense, entail the chance-undermines-would principle. Instead, they entail a principle that only supports counterfactual indeterminism, the view that most ordinary counterfactuals are chancy, i.e., not settled true. I demonstrate this by developing an indeterminist semantics that vindicates the clash and chance-ensures-could principles but not the chance-undermines-would principle. This view, I argue, offers a better account of our credal and linguistic judgments than counterfactual skepticism.

Arguments have always played a central role within logic and philosophy. But little attention has been paid to arguments as a distinctive kind of discourse, with its own semantics and pragmatics. The goal of this essay is to study the mechanisms by means of which we make arguments in discourse, starting from the semantics of argument connectives such as `therefore'. While some proposals have been made in the literature, they fail to account for the distinctive anaphoric behavior of `therefore', as well as uses of argument connectives in complex arguments, suppositional arguments, arguments with non-declarative conclusions, as well as arguments with parenthetical remarks. A comprehensive account of arguments requires imposing a distinctive tree-like structure on contexts. We show how to extend our account to accommodate modal subordination and and different flavors of `therefore'.

We offer a novel account of metalinguistic comparatives, such as 'Al is more wise than clever'. On our view, metalinguistic comparatives express comparative commitments to conventions. Thus, 'Al is more wise than clever' expresses that the speaker has a stronger commitment to a convention on which Al is wise than to a convention on which she is clever. This view avoids problems facing previous approaches to metalinguistic comparatives. It also fits within a broader framework—independently motivated by metalinguistic negotiations and convention-shiftingexpressions— that gives linguistic conventions a role in the semantics.

We develop and defend a new approach to counterlogicals. Non-vacuous counterlogicals, we argue, fall within a broader class of counterfactuals known as counterconventionals. Existing semantics for counterconventionals, 459–482 ) and, 1–27 ) allow counterfactuals to shift the interpretation of predicates and relations. We extend these theories to counterlogicals by allowing counterfactuals to shift the interpretation of logical vocabulary. This yields an elegant semantics for counterlogicals that avoids problems with the usual impossible worlds semantics. We conclude by showing how this approach can be extended to counterpossibles more generally.

In this paper, we axiomatize the deontic logic in Fusco, which uses a Stalnaker-inspired account of diagonal acceptance and a two-dimensional account of disjunction to treat Ross’s Paradox and the Puzzle of Free Choice Permission. On this account, disjunction-involving validities are a priori rather than necessary. We show how to axiomatize two-dimensional disjunction so that the introduction/elimination rules for boolean disjunction can be viewed as one-dimensional projections of more general two-dimensional rules. These completeness results help make explicit the restrictions Fusco’s account must place on free-choice inferences. They are also of independent interest, as they raise difficult questions about how to “lift” a Kripke frame for a one-dimensional modal logic into two dimensions.

In this paper, we axiomatize the deontic logic in Fusco 2015, which uses a Stalnaker-inspired account of diagonal acceptance and a two-dimensional account of disjunction to treat Ross’s Paradox and the Puzzle of Free Choice Permission. On this account, disjunction-involving validities are a priori rather than necessary. We show how to axiomatize two-dimensional disjunction so that the introduction/elimination rules for boolean disjunction can be viewed as one-dimensional projections of more general two-dimensional rules. These completeness results help make explicit the restrictions Fusco’s account must place on free-choice inferences. They are also of independent interest, as they raise difficult questions about how to ‘lift’ a Kripke frame for a one- dimensional modal logic into two dimensions.

While standard first-order modal logic is quite powerful, it cannot express even very simple sentences like “I could have been taller than I actually am” or “Everyone could have been smarter than they actually are”. These are examples of cross-world predication, whereby objects in one world are related to objects in another world. Extending first-order modal logic to allow for cross-world predication in a motivated way has proven to be notoriously difficult. In this paper, I argue that the standard accounts of cross-world predication all leave something to be desired. I then propose an account of cross-world predication based on quantified hybrid logic and show how it overcomes the limitations of these previous accounts. I will conclude by discussing various philosophical consequences and applications of such an account.

Many authors have noted that there are types of English modal sentences cannot be formalized in the language of basic first-order modal logic. Some widely discussed examples include “There could have been things other than there actually are” and “Everyone who is actually rich could have been poor.” In response to this lack of expressive power, many authors have discussed extensions of first-order modal logic with two-dimensional operators. But claims about the relative expressive power of these extensions are often justified only by example rather than by rigorous proof. In this paper, we provide proofs of many of these claims and present a more complete picture of the expressive landscape for such languages.

In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these accounts use, however, is too strong, as the standard translation from modal logic to first-order logic is not compositional in this sense. In light of this, we will explore a weaker version of this notion that we will call schematicity and show that there is no schematic translation either from first-order logic to propositional logic or from intuitionistic logic to classical logic.

Conventional wisdom has it that truth is always evaluated using our actual linguistic conventions, even when considering counterfactual scenarios in which different conventions are adopted. This principle has been invoked in a number of philosophical arguments, including Kripke’s defense of the necessity of identity and Lewy’s objection to modal conventionalism. But it is false. It fails in the presence of what Einheuser (2006) calls c-monsters, or convention-shifting expressions (on analogy with Kaplan’s monsters, or context-shifting expressions). We show that c-monsters naturally arise in contexts, such as metalinguistic negotiations, where speakers entertain alternative conventions. We develop an expressivist theory—inspired by Barker (2002) and MacFarlane (2016) on vague predications and Einheuser (2006) on counterconventionals—to model these shifts in convention. Using this framework, we reassess the philosophical arguments that invoked the conventional wisdom.

Sentences about logic are often used to show that certain embedding expressions, including attitude verbs, conditionals, and epistemic modals, are hyperintensional. Yet it not clear how to regiment “logic talk” in the object language so that it can be compositionally embedded under such expressions. This paper does two things. First, it argues against a standard account of logic talk, viz., the impossible worlds semantics. It is shown that this semantics does not easily extend to a language with propositional quantifiers, which are necessary for regimenting some logic talk. Second, it develops an alternative framework based on logical expressivism, which explains logic talk using shifting conventions. When combined with the standard S5π+ semantics for propositional quantifiers, this framework results in a well-behaved system that does not face the problems of the impossible worlds semantics. It can also be naturally extended with hybrid operators to regiment a broader range of logic talk, e.g., claims about what laws hold according to other logics. The resulting system, called hyperlogic, is therefore a better framework for modeling logic talk than previous accounts.

It is widely held that counterfactuals, unlike attitude ascriptions, preserve the referential transparency of their constituents, i.e., that counterfactuals validate the substitution of identicals when their constituents do. The only putative counterexamples in the literature come from counterpossibles, i.e., counterfactuals with impossible antecedents. Advocates of counterpossibilism, i.e., the view that counterpossibles are not all vacuous, argue that counterpossibles can generate referential opacity. But in order to explain why most substitution inferences into counterfactuals seem valid, counterpossibilists also often maintain that counterfactuals with possible antecedents are transparency‐preserving. I argue that if counterpossibles can generate opacity, then so can ordinary counterfactuals with possible antecedents. Utilizing an analogy between counterfactuals and attitude ascriptions, I provide a counterpossibilist‐friendly explanation for the apparent validity of substitution inferences into counterfactuals. I conclude by suggesting that the debate over counterpossibles is closely tied to questions concerning the extent to which counterfactuals are more like attitude ascriptions and epistemic operators than previously recognized.

A counteridentical is a counterfactual with an identity statement in the antecedent. While counteridenticals generally seem non-trivial, most semantic theories for counterfactuals, when combined with the necessity of identity and distinctness, attribute vacuous truth conditions to such counterfactuals. In light of this, one could try to save the orthodox theories either by appealing to pragmatics or by denying that the antecedents of alleged counteridenticals really contain identity claims. Or one could reject the orthodox theory of counterfactuals in favor of a hyperintensional semantics that accommodates non-trivial counterpossibles. In this paper, I argue that none of these approaches can account for all the peculiar features of counteridenticals. Instead, I propose a modified version of Lewis’s counterpart theory, which rejects the necessity of identity, and show that it can explain all the peculiar features of counteridenticals in a satisfactory way. I conclude by defending the plausibility of contingent identity from objections.