Research

In his 1951 Gibbs Memorial Lecture, Kurt Gödel put forth his famous disjunction that either the power of the mind outstrips that of any machine or there are absolutely unsolvable problems. The view that there are no absolutely unsolvable problems is optimism, the view that there are such problems is pessimism. In his 1995—and, revised in 2013—Verificationism Then and Now, Per Martin-Löf presents an illustrative argument for a constructivist form of optimism. In response to that argument, Solomon Feferman points out that Martin-Löf’s reasoning relies upon constructive understandings of key philosophical notions. In the vein of Feferman’s analysis, one might be object to Martin-Löf’s argument for either its reliance upon constructivist (as opposed to classical) considerations, or for its appeal to non-

unproblematically mathematical premises. We argue that both of these responses fall short. On one hand, to be critical of Martin-Löf’s reasoning for its constructiveness is to reject what would otherwise be a scientific advance on the basis of the assumption of constructivism’s falsehood or implausibility, which is of course uncharitable at best. On the other hand, to object to the argument for its use of non-unproblematically mathematical premises is to assume that there is some philosophically neutral mathematics, which is im-

plausible. Martin-Löf’s argument relies upon his

third law, the claim that from the impossibility of a proof of a proposition we can construct a proof of its negation. We close with a discussion of some ways in which this claim can be criticized from the constructive point of view. Specifically, we contend that Martin-Löf’s third law is incompatible with what has been called “Poincaré’s Principle of Epistemic Conservation”, the thesis that genuine increase in mathematical knowledge requires subject-specific insight.

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In Semiotic Studies, 2020

In this paper, we investigate epistemic predicates in extensions of arithmetic. We use as our case study Kurt Gödel’s 1951 thesis that either the power of the human mind surpasses that of any finite machine or there are absolutely unsolvable problems. Because Gödel also claimed that his disjunction was a mathematically established fact, we must ask the following: what sort of syntactical object should formalize human reason? In this paper, we lay the foundations for a predicate treatment of this epistemic feature. We begin with a very general examination of the Gödel sentence in the arithmetical context. We then discuss two systems of modal predicates over arithmetic. The first, called coreflective arithmetic or 𝖢𝗈𝖯𝖠, extends 𝖯𝖠 with a coreflective modal predicate but does not contain a consistency statement. The second, called doxastic arithmetic or 𝖣𝖠⁠, has as its characteristic feature the consistency statement but does not contain coreflection or its instance, the 𝟦 axiom. We examine the logical properties of, motivations for and criticisms of both systems. We close with a brief comparison of the systems in the context of Gödel’s disjunction.

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In Journal of Logic and Computation, 2020

The logical analysis of epistemic paradoxes—including, for example, the Moore and Gödel-Buridan paradoxes—has traditionally been performed assuming the whole range of corresponding modal logic principles: {𝖣,𝖳,𝟦,𝟧}{D,T,4,5}. In this paper, it is discovered precisely which of those principles (including also the law of excluded middle, LEM) are responsible for the paradoxical behavior of the Moore, Gödel-Buridan, Dual Moore, and Commissive Moore sentences. Further, by reproducing these paradoxes intuitionistically we reject a conjecture that these paradoxes are caused by the LEM. An exploration of the Gödel-Buridan sentence prompts the inquiry into a system, Doxastic Arithmetic, 𝖣𝖠, designed to represent the arithmetical beliefs of an agent who accepts all specific arithmetical proofs and yet believes in the consistency of their own beliefs. For these reasons, 𝖣𝖠 may be regarded as an epistemic way to circumvent limitations of Gödel’s Second Incompleteness Theorem.

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In Logical Foundations of Computer Science, 2018

Given some doxastic logic, Hintikka paradoxical sentences are those sentences that are satisfiable but refutable if believed. The standard examples of doxastic paradoxes are the Moore and Buridan-Gödel sentences. Considering the standard logics for modeling doxastic agents, should we prefer logics in which the believed versions of these sentences are refutable? One might answer affirmatively assuming that to refute the believed sentence is to avoid paradox. One motivation for this approach is the thought that these sentences are intuitively unbelievable for rational agents. In this note, we examine the extent to which this intuition motivates the preference for logics in which the believed versions of these sentences are refutable. We argue that such a motivation is weak by providing a theory (namely, the recently introduced Doxastic Arithmetic (DA)) within which a rational agent believes a Buridan-Gödel sentence.

Link to Wittgenstein Symposium Program

Quine's translation argumnent figures centrally in his views on logic. The goal of this paper is to get clear on that argument. It can be interpreted as an argument to the effect that one should never translate somebody’s speech as going against a law of the translator’s logic. Key to this reading of the translation argument is the premise that one should never translate somebody's speech such that their speech is unintelligible. Ultimately, it is my aim to reject this reading. I argue that only a weaker conclusion—one that says “not most of the time” instead of the stronger “never”—should be attributed to Quine. Accordingly, I propose and defend a weaker version of the first premise that better coheres with the weaker conclusion of the translation argument. Instead of the claim that one should never translate somebody’s speech such that their speech is unintelligible I argue that we should only ascribe to Quine the claim that one should not most of the time translate somebody’s speech in a way that makes it unintelligible. I go on to sum up the results of my discussion and respond to a criticism of my reading.

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In Topoi, 2016