Frege’s Unification

What makes certain definitions fruitful? And how can definitions play an explanatory role? The purpose of this paper is to examine these questions via an investigation of Frege’s treatment of definitions. Specifically, I pursue this issue via an examination of Frege’s views about the scientific unification of logic and arithmetic. In my view, what interpreters have failed to appreciate is that logicism is a project of unification, not reduction. For Frege, unification involves two separate steps: (1) an account of the content expressed by arithmetical claims and (2) the justification of that content. The distinction between these steps allows us to see that there are two notions of definition at play in Frege’s logicist work, viz., one concerned with conceptual analysis, the other concerned with the construction of gap-free proof. I then use this discussion to explain how Frege employs his definitions to defend an epistemological thesis about arithmetic, and to clarify Grundlagen’s fruitfulness condition of definitions, and thereby address two interpretive puzzles from the recent literature.     

Frege’s Result: Frege’s Theorem and Related Matters

One of the remarkable results of Frege’s Logicism is Frege’s Theorem, which holds that one can derive the main truths of Peano arithmetic from Hume’s Principle (HP) without using Frege’s Basic Law V. This result was rediscovered by the Neo-Fregeans and their allies. However, when applied in developing a more advanced theory of mathematics, their fundamental principles—the abstraction principles—incur some problems, e.g., that of inflation. This paper finds alternative paths for such inquiry in extensionalism and object theory.     

Analyzing Sterba’s argument

International Journal for Philosophy of Religion -     

Frege’s philosophy of geometry

Synthese - In this paper, I critically discuss Frege’s philosophy of geometry with special emphasis on his position in The Foundations of Arithmetic of 1884. In Sect. 2, I argue that...     

Understanding Frege’s notion of presupposition

Why did Frege offer only proper names as examples of presupposition triggers? Some scholars claim that Frege simply did not care about the full range of presuppositional phenomena. This paper argues, in contrast, that he had good reasons for employing an extremely narrow notion of ‘Voraussetzung’. On Frege’s view, many devices that are now construed as presupposition triggers either express several thoughts at once or merely ‘illuminate’ a thought in a particular way. Fregean presuppositions, in contrast, are essentially tied to names.

     

Diagrammatic reasoning in Frege’s Begriffsschrift

In Part III of his 1879 logic Frege proves a theorem in the theory of sequences on the basis of four definitions. He claims in Grundlagen that this proof, despite being strictly deductive, constitutes a real extension of our knowledge, that it is ampliative rather than merely explicative. Frege furthermore connects this idea of ampliative deductive proof to what he thinks of as a fruitful definition, one that draws new lines. My aim is to show that we can make good sense of these claims if we read Frege??s notation diagrammatically, in particular, if we take that notation to have been designed to enable one to exhibit the (inferentially articulated) contents of concepts in a way that allows one to reason deductively on the basis of those contents.     

Strawson’s modest transcendental argument

Although Peter Strawson’s ‘Freedom and Resentment’ was published over fifty years ago and has been widely discussed, its main argument is still notoriously difficult to pin down. The most common – but in my view, mistaken – interpretation of Strawson’s argument takes him to be providing a ‘relentlessly’ naturalistic framework for our responsibility practices. To rectify this mistake, I offer an alternative interpretation of Strawson’s argument. As I see it, rather than offering a relentlessly naturalistic framework for moral responsibility, Strawson actually develops a transcendental argument, which grounds our moral responsibility practices in the practical perspective of social agents. However, the aims of this essay are not purely interpretative. Strawson’s essay continues to have important implications for a number of issues that arise in the contemporary debates that concern free will and moral responsibility. In particular, it puts significant pressure on moral responsibility sceptics like Derk Pereboom [Living Without Free Will. Cambridge: Cambridge University Press, 2001] who think that the truth of moral responsibility scepticism has no worrisome implications for our lives with others.     

O’Connor’s argument for indeterminism

Timothy O’Connor has recently defended a version of libertarianism that has significant advantages over similar accounts. One of these is an argument that secures indeterminism on the basis of an argument that shows how causal determinism threatens agency in virtue of the nature of the causal relation involved in free acts. In this paper, I argue that while it does turn out that free acts are not causally determined on O’Connor’s view, this fact is merely stipulative and the argument that he presents for this conclusion begs the question.     

Formality of logic and Frege’s Begriffsschrift

This paper challenges a standard interpretation according to which Frege’s conception of logic (early and late) is at odds with the contemporary one, because on the latter’s view logic is formal, while on Frege’s view it is not, given that logic’s subject matter is reality’s most general features. I argue that Frege – in Begriffsschrift – retained the idea that logic is formal; Frege sees logic as providing the ‘logical cement’ that ties up together the contentful concepts of specific sciences, not the most general truths. Finally, I discuss how Frege conceives of the application of Begriffsschrift, and of its status as a ‘lingua characteristica’.     

Hale’s argument from transitive counting

Synthese - A core commitment of Bob Hale and Crispin Wright’s neologicism is their invocation of Frege’s Constraint—roughly, the requirement that the core empirical applications...     

Does Semantic Relationism Solve Frege’s Puzzle?

In a series of recent works, Kit Fine (The Journal of Philosophy, 100(12), 605–631, 2003, 2007) has sketched a novel solution to Frege’s puzzle. Radically departing from previous solutions, Fine argues that Frege’s puzzle forces us to reject compositionality. In this paper we first provide an explicit formalization of the relational semantics for first-order logic suggested, but only briefly sketched, by Fine. We then show why the relational semantics alone is technically inadequate, forcing Fine to enrich the syntax with a coordination schema. Given this enrichment, we argue, that that the semantics is compositional. We then examine the deep consequences of this result for Fine’s proposed solution to Frege’s puzzle. We argue that Fine has mis-diagnosed his own solution–his attempted solution does not deny compositionality. The correct characterization of Fine’s solution fits him more comfortably among familiar solutions to the puzzle.     

Sharon Street’s unsuccessful argument against theism

International Journal for Philosophy of Religion - Sharon Street has argued that we should reject theism because we can accept it only at the cost of having good reason to doubt the reliability of...     

A repair of Frege’s theory of thoughts

Frege’s writings contain arguments for the thesis (i) that a thought expressed by a sentence S is a structured object whose composition pictures the composition of S, and for the thesis (ii) that a thought is an unstructured object. I will argue that Frege’s reasons for both (i) and (ii) are strong. Frege’s explanation of the difference in sense between logically equivalent sentences rests on assumption (i), while Frege’s claim that the same thought can be decomposed differently makes (ii) plausible. Thoughts are supposed to do work that requires that they be structured and work that requires that they be unstructured. But this cannot be! While the standard response to this problem is to reject either (i) or (ii), I propose a charitable repair in the spirit of Frege’s theory that accepts both. The key idea can be found in Frege’s Basic Laws of Arithmetic(BL, GGA). Frege argues that the thought expressed by a sentence is determined by the truth-conditions that can be derived from the semantic axioms for the sentence constituents. The fact that the same axiomatic truth-condition can be derived in different ways from different semantic axioms suggests a Fregean solution of the dilemma: A thought is a type that is instantiated by all sequences of senses (decomposed thoughts) that have the same axiomatic truth-conditions. This allows for multiple decomposability of the same thought (for different decomposed thoughts can have the same axiomatic truth-conditions) and for a notion of containment (the decomposed thought contains those senses whose semantic axioms are needed in the derivation of the truth-conditions). My proposal combines the virtues of (i) and (ii) without inheriting their vices.     

Hume’s better argument for motivational skepticism

On a standard interpretation, Hume argued that reason is not practical, because its operations are limited to “demonstration” and “probability.” But recent critics claim that by limiting reason’s operations to only these two, his argument begs the question. Despite this, a better argument for motivational skepticism can be found in Hume’s text, one that emphasizes reason’s inability to generate motive force against contrary desires or passions. Nothing can oppose an impulse but a contrary impulse, Hume believed, and reason cannot generate an impulse. This better argument is here developed and defended. Two lines of objection to it can be anticipated: (1) that reason actually can generate impulsive force, based on contents of its normative judgments and (2) that reason neither can nor needs to generate an impulse, since the actions of rational agents are not determined by forceful impulses of desire, as Hume supposed. These objections are answered by pointing out their unsatisfying consequences.