Online research in philosophy

Here is an account of logical consequence inspired by Bolzano and Tarski. Logical validity is a property of arguments. An argument is a pair of a set of interpreted sentences (the premises) and an interpreted sentence (the conclusion). Whether an argument is logically valid depends only on its logical form. The logical form of an argument is fixed by the syntax of its constituent sentences, the meanings of their logical constituents and the syntactic differences between their non-logical constituents, treated as variables. A constituent of a sentence is logical just if it is formal in meaning, in the sense roughly that its application is invariant under permutations of individuals.1 Thus ‘=’ is a logical constant because no permutation maps two individuals to one or one to two; ‘∈’ is not a logical constant because some permutations interchange the null set and its singleton. Truth functions, the usual quantifiers and bound variables also count as logical constants. An argument is logically valid if and only if the conclusion is true under every assignment of semantic values to variables (including all non-logical expressions) under which all its premises are true. A sentence is logically true if and only if the argument with no premises of which it is the conclusion is logically valid, that is, if and only if the sentence is true under every assignment of semantic values to variables. An interpretation assigns values to all variables.

He argues that the intuitively provable arithmetic sentences constitute a recursively enumerable set, which has a Gödel sentence which is itself intuitively provable. The incompleteness theorem does not apply, since the set of provable arithmetic sentences is not consistent. The purpose of this article is to sharpen Priest's argument, avoiding reference to informal notions, consensus, or Church's thesis. We add Priest's dialetheic semantics to ordinary Peano arithmetic PA, to produce a recursively axiomatized formal system PA that contains its own truth predicate. Whether one is a dialetheist or not, PA is a legitimate, rigorously defined formal system, and one can explore its proof-theoretic properties. The system is inconsistent (but presumably non-trivial), and it proves its own Gödel sentence as well as its own soundness. Although this much is perhaps welcome to the dialetheist, it has some untoward consequences. There are purely arithmetic (indeed, 0) sentences that are both provable and refutable in PA. So if the dialetheist maintains that PA is sound, then he must hold that there are true contradictions in the most elementary language of arithmetic. Moreover, the thorough dialetheist must hold that there is a number g which both is and is not the code of a derivation of the indicated Gödel sentence of PA. For the thorough dialetheist, it follows ordinary PA and even Robinson arithmetic are themselves inconsistent theories. I argue that this is a bitter pill for the dialetheist to swallow.

Neil Tennant and Joseph Salerno have recently attempted to rigorously formalize Michael Dummett's argument for logical revision. Surprisingly, both conclude that Dummett commits elementary logical errors, and hence fails to offer an argument that is even prima facie valid. After explicating the arguments Salerno and Tennant attribute to Dummett, I show how broader attention to Dummett's writings on the theory of meaning allows one to discern, and formalize, a valid argument for logical revision. Then, after correctly providing a rigorous statement of the argument, I am able to delineate four possible anti-Dummettian responses. Following recent work by Stewart Shapiro and Crispin Wright, I conclude that progress in the anti-realist's dialectic requires greater clarity about the key modal notions used in Dummett's proof.

In a well-known passage in the last chapter of Frege: Philosophy of Mathematics Michael Dummett suggests that Frege’s major “mistake”—the key to the collapse of the project of Grundgesetze—consisted in “his supposing there to be a totality containing the extension of every concept defined over it; more generally [the mistake] lay in his not having the glimmering of a suspicion of the existence of indefinitely extensible concepts” (Dummett [1991, 317]). Now, claims of the form, Frege fell into paradox because……. are notoriously difficult to assess even when what replaces the dots is relatively straightforward. Offerings have included, for instance, that — (A) Unrestricted quantification: Frege fell into paradox because he allowed himself to quantify over a single, all-inclusive domain of objects (Russell, Dummett).

Of all the cases made against classical logic, Michael Dummett's is the most deeply considered. Issuing from a systematic and original conception of the discipline, it constitutes one of the most distinctive achievements of twentieth century British philosophy. Although Dummett builds on the work of Brouwer and Heyting, he provides the case against classical logic with a new, explicit and general foundation in the philosophy of language. Dummett's central arguments, widely celebrated if not widely endorsed, concern the implications of the relation between meaning and use for both the inference rules that govern logical connectives and the relation between truth and its recognition. It is less often noted that Dummett has a further argument against classical logic, one based on the semantic and set-theoretic paradoxes. That is the topic of this paper.

Neil Tennant and Joseph Salerno have recently attempted to rigorously formalize Michael Dummett's argument for logical revision. Surprisingly, both conclude that Dummett commits elementary logical errors, and hence fails to offer an argument that is even prima facie valid. After explicating the arguments Salerno and Tennant attribute to Dummett, I show how broader attention to Dummett's writings on the theory of meaning allows one to discern, and formalize, a valid argument for logical revision. Then, after correctly providing a rigorous statement of the argument, I am able to delineate four possible anti-Dummettian responses. Following recent work by Stewart Shapiro and Crispin Wright, I conclude that progress in the anti-realist's dialectic requires greater clarity about the key modal notions used in Dummett's proof.

The purpose of this paper is to assess the prospects for a neo-logicist development of set theory based on a restriction of Frege's Basic Law V, which we call (RV): PQ[Ext(P) = Ext(Q) [(BAD(P) & BAD(Q)) x(Px Qx)]] BAD is taken as a primitive property of properties. We explore the features it must have for (RV) to sanction the various strong axioms of Zermelo–Fraenkel set theory. The primary interpretation is where ‘BAD’ is Dummett's ‘indefinitely extensible’. 1 Background: what and why? 2 Framework 3 GOOD candidates, indefinite extensibility 4 The framework of (RV) alone, or almost alone 5 The axioms 6 Brief closing.

Various authors of logic texts are cited who either suggest or explicitly state that the Gödel incompleteness result shows that some unprovable sentence of arithmetic is true. Against this, the paper argues that the matter is one of philosophical controversy, that it is not a mathematical or logical issue.

A central theme in the foundational debates in the early Twentieth century in response to the paradoxes was to invoke the notion of the indefinite extensibility of certain concepts e,g. definability (the Richard paradox) and class (the Zermelo-Russell contradiction). Dummett has recently revived the notion, as the real lesson of the paradoxes and the source of Frege's error in basic law five of the Grundgesetze. The paper traces the historical and conceptual evolution of the concept and critices Dummett's argument that the proper lesson of the paradoxes is that set theory is a theory of indefinitely extensible domains.

Some earlier remarks Michael Dummett made on Gödel’s theorem have recently inspired attempts to formulate an alternative to the standard demonstration of the truth of the Gödel sentence. The idea underlying the non-standard approach is to treat the Gödel sentence as an ordinary arithmetical one. But the Gödel sentence is of a very specific nature. Consequently, the non-standard arguments are conceptually mistaken. In this paper, both the faulty arguments themselves and the general reasons underlying their failure are analysed. The analysis reveals the true nature of the epistemological relation between the Gödel sentence and its numerical instances.