Operators in the paradox of the knower

In his rich "The Truth Predicate vs. the Truth Connective. On taking connectives seriously."

2014

We simplify and slightly modify the theory of types that Church provided with semantic primitive predicates. Two goals are pursued. The first goal is to present a simple application of Church's approach to paradoxes and to point out some aspects of this ap- proach. The second, perhaps more interesting, goal is to show that when type distinctions are removed some basic Churchian principles need to be restricted and different restrictions correspond to Tarski's and Kripke's different approaches to truth. Finally, we briefly hint at how to move in the direction of Field's recent approach to truth by giving up some specific essential points of the Churchian frame- work.

Logic, Rationality, and Interaction

The semantic paradoxes and the paradoxes of vagueness ('soritical paradoxes') display remarkable family resemblances. In particular, the same nonclassical logics have been (independently) applied to both kinds of paradoxes. These facts have been taken by some authors to suggest that truth and vagueness require a uni ed logical framework (see e.g. [5,3]). Some authors go further, and argue that truth is itself a vague or indeterminate concept (see e.g. [7,4]). Importantly, however, there currently is no identi cation of what the common features of semantic and soritical paradoxes exactly consist in. This is what we aim to do in this work: we analyze semantic and soritical paradoxes, and develop our analysis into a theory of paradoxicality. The uni cation of the paradoxes of truth and vagueness we propose here has a wide scope, but for the sake of concreteness we focus on four three-valued logics.

Notre Dame Journal of Formal Logic

The Logic of Paradox, LP, is a first-order, three-valued logic that has been advocated by Graham Priest as an appropriate way to represent the possibility of acceptable contradictory statements. Second-order LP is that logic augmented with quantification over predicates. As with classical second-order logic, there are different ways to give the semantic interpretation of sentences of the logic. The different ways give rise to different logical advantages and disadvantages, and we canvass several of these, concluding that it will be extremely difficult to appeal to second-order LP for the purposes that its proponents advocate, until some deep, intricate, and hitherto unarticulated metaphysical advances are made. 1 Background on the "Logic of Paradox" Over the past three or four decades, but importantly in his [10], Graham Priest has investigated a variety of paradoxical topics-the semantic paradoxes are the ones that come first to a logician's mind, but he has also studied puzzles arising from vagueness, and motion, and Buddhist philosophy and, in his recent book [14], metaphysical perplexities arising out the notion of parthood and in relation to the question of the unity of the proposition-all from a dialetheist perspective: one that considers it possible that there are true contradictions, i.e., that some propositions are both true and false or, equivalently, that some true propositions have true negations. As a basic logical framework for his investigations he has adopted the system he calls LP (for "Logic of Paradox"), which is perhaps the simplest modification of classical logic to allow non-trivial contradictions. This is a 3-valued logic, with values True, False, and Both. It has the usual propositional connectives ¬ (negation), ∧ (conjunction), ∨ (disjunction), and universal (∀) and existential (∃) quantifiers. The truth-functions ⊃ and ≡ are usually treated as defined connectives: (ϕ ⊃ ψ) = d f (¬ϕ ∨ ψ)

Journal of Philosophical Logic, 1998

Logics in which a relation R is semantically incomplete in a particular universe E, i.e. the union of the extension of R with its anti-extension does not exhaust the whole universe E, have been studied quite extensively in the last years. (Cf. van Benthem (1985), , for partial predicate logic; Muskens (1996), for the applications of partial predicates to formal semantics, and Doherty (1996) for applications to modal logic.) This is not so with semantically incomplete generalized quantifiers which constitute the subject of the present paper. The only systematic study of these quantifiers from a purely logical point of view, is, to the best of my knowledge, that by van Eijck (1995). We shall take here a different approach than that of van Eijck and mention some of the abstract properties of the resulting logic. Finally we shall prove that the two approaches are interdefinable.

A number of authors have noted that the key steps in Fitch’s argument are not intuitionistically valid, and some have proposed this as a reason for an anti-realist to accept intuitionistic logic (e.g. Williamson 1982, 1988). This line of reasoning rests upon two assumptions. The first is that the premises of Fitch’s argument make sense from an anti-realist point of view – and in particular, that an anti-realist can and should maintain the principle that all truths are knowable. The second is that we have some independent reason for thinking that classical logic is not appropriate in this area. This paper explores these two assumptions in the context of Michael Dummett’s version of anti-realism, with particular reference to the argument from indefinite extensibility developed at various points in Dummett’s writings (e.g. Dummett 1991 Ch. 24). Dummett argues that certain concepts, the indefinitely extensible concepts, are such that we cannot form a clear and determinate conception of all the objects that fall under them. The most familiar examples of indefinitely extensible concepts are mathematical. Dummett discusses the concepts ordinal number, real number, and natural number, which are indefinitely extensible because any conception that one might form of their complete extension can be extended to a more inclusive conception (as, for example, in Cantor’s proof of the non-denumerability of the set of real numbers). This paper argues that the concept of a truth is indefinitely extensible. This gives a Dummettian anti-realist an independent motivation for rejecting the classical understanding of the quantifiers in this area. At the same time, however, it places in doubt the admissibility of the knowability principle, which seems to involve quantification over the “totality” of truths. As Dummett is at pains to point out (1991: 316), some sentences that purport to quantify over the extension of an indefinitely extensible concept plainly have a truth-value (we can truly say, for example, that every ordinal number has a successor, even though when we say that we are not quantifying over the set of all ordinals). But is the knowability principle one of these sentences?

Paradoxes and Theorems, 2023

This book is a critique of self-reference in ordinary language and mathematics. From this critical point of view, the most well-known semantic paradoxes are analyzed, such as the liar's paradox, Russel's barber's paradox, the paradox of the set of all sets that do not belong to themselves, Richard's paradox, Berry's paradox, etc. It also includes a critical analysis of the interpretation in natural language of Gödel's sentence that is the protagonist of his famous incompleteness theorem, because this sentence has the same formal structure as the paradoxes analyzed in the book.

This article first describes a phenomenological approach used to develop a simple logical system complying with the limits and logical constraints of natural language. Then, the resulting 3-valued, modal logic, which, nevertheless, is extensional and encompasses classical binary logic, is successfully applied to the analysis of different well-known paradoxes, including the Liar Paradox and the Surprise Examination Paradox.

This paper will presumably appear in the Routledge Handbook of Propositions, ed. Chris Tillman. It sets out various resolutions of various propositional paradoxes short of the ramified theory of types.

We present a graph-theoretic analysis of the semantic paradoxes for the language of first-order Peano arithmetic augmented with a primitive truth predicate.