Browse Papers

The purpose of this note is to present a strong form of the liar paradox. It is strong because the logical resources needed to generate the paradox are weak, in each of two senses. First, few expressive resources required: conjunction, negation, and identity. In particular, this form of the liar does not need to make any use of the conditional. Second, few inferential resources are required. These are: (i) conjunction introduction; (ii) substitution of identicals; and (iii) the inference: From ¬(p ∧ p), infer ¬ p. It is, interestingly enough, also essential to the argument that the ‘strong’ form of the diagonal lemma be used: the one that delivers a term λ such that we can prove: λ = ¬ T(⌈λ⌉); rather than just a sentence Λ for which we can prove: Λ ≡ ¬T(⌈Λ⌉).

The truth-theoretic principles used to generate the paradox are these: ¬(S ∧ T(⌈¬S⌉); and ¬(¬S ∧ ¬T(⌈¬S⌉). These are classically equivalent to the two directions of the T-scheme, but they are intuitively weaker.

The lesson I would like to draw is: There can be no consistent solution to the Liar paradox that does not involve abandoning truth-theoretic principles that should be every bit as dear to our hearts as the T-scheme. So we shall have to learn to live with the Liar, one way or another.

Our understanding of subjunctive conditionals has been greatly enhanced through the use of possible world semantics and, more precisely, by the idea that they involve variably strict quantification over possible worlds. I propose to extend this treatment to ceteris paribus conditionals – that is, conditionals that incorporate a ceteris paribus or ‘other things being equal’ clause. Although such conditionals are commonly invoked in scientific theorising, they traditionally arouse suspicion and apprehensiveness amongst philosophers. By treating ceteris paribus conditionals as a species of variably strict conditional I hope to shed new light upon their content and their logic.

Oystein Linnebo has shown that the existence of successors cannot be proven in predicative Frege arithmetic, that is, predicative second-order logic plus "Hume's Principle" and Frege's definitions of zero, predecessor, and natural number. It is shown in the present paper that the existence of successors can be proven if the logic is strengthened to ramified predicative second-order logic. It then follows from work by John Burgess and Allen Hazen that Robinson arithmetic, Q, can be interpreted in ramified Frege arithmetic.

While half a century ago answering the question, ‘What is logic, really?’, would have been easy, it is somewhat problematic today when mathematical logic has proved inadequate for the use for which it was originally intended: to give mathematics the most secure foundation. It has also proved inadequate for being a tool for the new sciences, from computer science and artificial intelligence to life sciences, not to speak of social sciences. This makes imperative to reconsider the nature of logic and its role in science and in human life generally. This task involves reconsidering the relations between logic and method and between logic and nature. These questions are discussed in this paper which is intended to be the Introduction to a book in progress.

It is often said that diagonalization allows one to construct sentences that are self-referential. This paper investigates the sense in which that is true. I argue first that, in the standard language of arithmetic, in which we have only the symbols 0, S, +, and ×, truly self-referential sentences cannot be constructed. This is shown by considering sentences like "The right-hand side of this biconditional is false iff its left-hand side is true". This sentence is intuitively inconsistent, but the sentence constructed by using diagonalization in the usual way is true and, in fact, provable in Q. This problem can be resolved by expanding the language to include function-symbols for all primitive recursive functions. It can also be resolved by proving a stronger form of the diagonal lemma that I call the "structural" diagonal lemma. At the end of the paper, it is argued, however, that there are some contexts in which the latter method is insufficient. In particular, the theory of truth containing the following two claims:

• ~A is true iff A is not true
• "t is true" is true iff the sentence t denotes is true

is intuitively inconsistent, but it is consistent with PA (unless we use a very non-standard Gödel numbering).

Addresses the question of why we find Fitch's knowability 'paradox' argument surprising.