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Logical nihilism is the view that there is no logic, or more precisely that no single, universal consequence relation governs natural language reasoning. Here, I present three different arguments for logical nihilism from philosophically palatable premises. The first argument comes from a combination of pluralism with the desideratum that logical consequence should be universal, properly understood. The second argument is a slippery slope argument against monists who support weak logical systems on account of their power to characterize a vast range of true theories. The third argument is a general strategy of generating counterexamples to any inference rule, including purportedly fundamental ones such as disjunction introduction. I close by discussing why a truth-conditional approach to the meaning of the logical connectives not only does not force us to reject such counterexamples but also reveals that right truth-conditions are far more general than the classical ones, at the price of nihilism.
Øystein Linnebo develops an abstractionist account of the natural numbers as ordinals. On this account, the natural numbers are abstracted from orderings of concrete numerals. But Linnebo also gestures towards an alternative version of his account, on which the restriction to concrete numerals is lifted. I develop something like this alternative account, show how it avoids the Burali-Forti paradox, and show how it guarantees that every number has a successor. Given these and other good features, I claim that Linnebo should prefer this alternative account to the one he develops.
Contextual analysis deals with systems of random variables. Each random variable within a system is labeled in two ways: by its content (that which the variable measures or responds to) and by its context (conditions under which it is recorded). Dependence of random variables on contexts is classified into (1) direct (causal) cross-influences and (2) purely contextual (non-causal) influences. The two can be conceptually separated from each other and measured in a principled way. The theory has numerous applications in quantum mechanics, and also in such areas as decision making and computer databases. A system of deterministic variables (as a special case of random variables) is always void of purely contextual influences. There are, however, situations when we know that a system is one of a set of deterministic systems, but we cannot know which one. In such situations we can assign epistemic (Bayesian) probabilities to possible deterministic systems, create thereby a system of epistemic random variables, and subject it to contextual analysis. In this way one can treat, in particular, such logical antinomies as the Liar paradox. The simplest systems of epistemic random variables describing the latter have no direct cross-influences and the maximal possible degree of purely contextual influences.
Kujala, J.V., Dzhafarov, E.N., & Larsson, J.-A. (2015). Necessary and sufficient conditions for extended noncontextuality in a broad class of quantum mechanical systems. Physical Review Letters 115, 150401 (available as arXiv:1407.2886.).
Dzhafarov, E.N., Cervantes, V.H., Kujla, J.V. (2017). Contextuality in canonical systems of random variables. Philosophical Transactions of the Royal Society A 375: 20160389 (available as arXiv:1703.01252).
Cervantes, V.H., & Dzhafarov, E.N. (2018). Snow Queen is evil and beautiful: Experimental evidence for probabilistic contextuality in human choices. Decision 5, 193-204 (available as arXiv:1711.00418).
The Curry-Howard correspondence between intuitionistic logic and the simply-typed lambda calculus forms an important bridge between logical and computational research. This talk develops a variant typed lambda calculus, called “core type theory”, that stands in a similar correspondence to Neil Tennant’s “core logic” (fka “intuitionistic relevant logic”), and shows some basic (and some surprising!) results about this calculus.
This talk will have two parts. First I will discuss the approach to semantics making no use of variable names, indices, or assignment functions that I have advocated in a series of papers (see especially Jacobson, 1999, Linguistics and Philosophy and 2000, Natural Language Semantics, also exposited in Jacobson 2014 textbook Compositional Semantics, OUP). There are a number of theoretical and empirical advantages to this approach, which will be just briefly reviewed. To mention the most obvious theoretical advantage: the standard use of variable names and indices in semantics requires meanings to be relativized to assignment functions (assignments of values to the variable names), adding a layer to the semantic machinery. This program eliminates this and treats all meanings as ‘healthy’ model theoretic objects (the meaning of a pronoun, for example, is simply the identity function on individuals, not a function from assignments to individuals). I will then show a new empirical payoff, which concerns competition effects found in ellipsis constructions. These competition effects have gone under the rubric of MaxElide in the linguistics literature. One example centers on the contrast in (1) (on the reading where each candidates hope is about their own success):
- a. Harris is hoping that South Carolina will seal the nomination for her, and Warren is too. (= ‘hoping that it will seal nomination for her (Warren)’)
b. ?*Harris is hoping that South Carolina will seal the nomination for her, and Warren is also hoping that it will. (= ‘seal the nomination for her (Warren)’)
The ‘standard’ wisdom is there is a constraint in the grammar that when material is ‘missing’ (or, ‘elided’) if a bigger constituent can be elided, the bigger ellipsis is required. Why the grammar should contain such a constraint is a total mystery; moreover I and others have argued elsewhere that grammatical competition constraints represent a real complication in the grammar. When there are competition effects they should be located in speakers and hearers (we know that speakers and hearers do compute alternatives – Gricean reasoning, for example, is based on that assumption). Under the variable free account, the missing material in (1a) is of a different type than that in (1b). In (1a), the listener need only supply the property ‘be an x such that x hopes that SC will seal nomination for x‘ which is the meaning of the VP in the first clause. In (1b) what must be supplied is the 2-place relation ‘seal the nomination for’ (note that this is in part because the pronoun her in the first clause is not a variable, and so [[seal the nomination for her]] is the function from an individual x to the property of sealing the nomination for x, which in turn is the two place relation named above. The competition effect is thus about types not size, and can be given a plausible explanation in terms of communicative pressures. Assuming that meanings of more complex types are more difficult to access than those of simpler types, there is a pressure for speakers to choose the simpler type ellipsis. The type competition story crucially relies on the claim that expressions containing pronouns unbound within them denote functions from individuals to something rather than functions from assignment functions.
I contend that semantic paradox shows we should regard the rules of inference for semantic notions as defeasible. Truth is a prominent semantic notion for which semantic paradox poses a problem, and so a first step in solving semantic paradox is handling a semantic notion of truth. This talk investigates how defeasible rules of inference for a semantic notion of truth can form the basis for a successful truth-conditional theory of meaning. I start by using Default Logic for representing the rules of inference for a truth predicate as defeasible, before adding preferences over rules and modifying the criteria for defeat.
The Liar paradox arises when we combine the assumption that a sentence can refer to itself with our naïve notion of truth and apply our unrevised logic. Most current approaches to the Liar paradox focus on revising our notion of truth and logic because nowadays almost everyone is convinced that there are self-referential sentences. I will argue against this conviction. My argument starts from observations about the metaphysics of expressions: A meaningful expression is based in a syntactic expression which in turn is based in a non-semiotic object, and these are pairwise distinct. As all objects of this three-fold ontology exist only relative to contexts, we can import ideas from tense logic about how existence and reference can interact in a contextualist metaphysics. Semantico-metaphysical reasoning then shows that in this dynamic setting, an object can be referred to only after it has started to exist. Hence the self-reference needed in the Liar paradox cannot occur, after all. As this solution is contextualist, it evades the expressibility problems of other proposals.
Intensional aspects from a second-order perspective
The talk will focus on harmony, the widely discussed condition that two collections of introduction and of elimination rules for a certain logical constant should satisfy in order to perfectly match each other. Attention will be restricted to rules for propositional connectives in single-conclusion natural deduction format.
I will first argue that a decent criterion of harmony should be (hyper-)intensional, in the sense that taken two equivalent collections of elimination rules (i.e. such that each rule of the first collection is derivable from the rules in the second collection, and vice versa), it may well be that only one of the two collections is in harmony with a given collection of introduction rules.
To precisely formulate an intensional criterion for harmony we propose to modify the (extensional) criterion for harmony recently proposed by Schroeder-Heister (for short SH-harmony) and based on the idea that to any collection of introduction and of elimination rules one can associate a formula of intuitionistic propositional logic with propositional quantification (IPC2). Two collections of introduction and elimination rules for a certain connective are in SH-harmony if the two formulas associated to them are interderivable in IPC2.
An intensional criterion is achieved by imposing on the two formulas a condition stricter than intederivability: isomorphim. We will present the notion of isomorphism (well-known from category theory and computational study of proof systems), stress its significance for proof-theoretic semantics, and finally apply it to clarify the relationship between different ways of generating collections of elimination rules from a given collection of introduction rules known in the literature.
Free choice sequences (also called ‘infinitely proceeding sequences’) are a concept from intuitionistic mathematics that are central to the development of the intuitionistic theory of the continuum. Free choice sequences have, however, been largely rejected or ignored both by classical and other constructive mathematicians. In this talk I argue that the objections to free choice sequences can be overcome by grounding the concept in our temporal intuition and formalizing the theory in modal logic. I will present a theory of free choice sequences as a modal extension of classical second-order arithmetic. The resulting theory is able to prove modal versions of the intuitionist’s axioms for so-called lawless sequences, suffices for the development of a theory of real number generators, and captures many results distinctive of intuitionistic analysis including the non-existence of functions on real numbers with definable discontinuities.