The foundational problem of logic

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Abstract. The construction of a systematic philosophical foundation for logic is a notoriously difficult problem. In Part One I suggest that the problem is in large part methodological, having to do with the common philosophical conception of "providing a foundation". I offer an alternative to the common methodology which combines a strong foundational requirement (veridical justification) with the use of non-traditional, holistic tools to achieve this result. In Part Two I delineate an outline of a foundation for logic, employing the new methodology. The outline is based on an investigation of why logic requires a veridical justification, i.e., a justification which involves the world and not just the mind, and what features or aspect of the world logic is grounded in. Logic, the investigation suggests, is grounded in the formal aspect of reality, and the outline proposes an account of this aspect, the way it both constrains and enables logic (gives rise to logical truths and consequences), logic's role in our overall system of knowledge, the relation between logic and mathematics, the normativity of logic, the characteristic traits of logic, and error and revision in logic. Association for Symbolic LogicIt is an interesting fact that, with a small number of exceptions, a systematic philosophical foundation for logic, a foundation for logic rather than for mathematics or language, has rarely been attempted.1 In this essay I aim to understand why this is the case, utilize this understanding to develop an appropriate foundational methodology, and use this methodology to construct an outline of a philosophical foundation for logic. The notion of a philosophical foundation will be clear to some readers, but due to the diverse readership of this journal it would be useful to briefly spell out and motivate the kind of philosophical foundation I have in mind.Received October 8, 2010. Acknowledgments: An earlier version of this paper was presented at a meeting of the Southern California History and Philosophy of Logic and Mathematics Group. I would like to thank the participants for stimulating comments. I am also very thankful to anonymous reviewers of this journal. Special thanks are due to two editors of the journal for extremely helpful comments that led to a considerable improvement of the paper. Finally, I am greatly thankful to Peter Sher for important suggestions and support.*One recent exception is Maddy [2007, Part III], which differs from the present attempt in being thoroughly naturalistic. Another psychologically oriented attempt is Hanna [2006]. Due to limitations of space and in accordance with my constructive goal, I will limit comparisons and polemics to a minimum. 14...

Logos, logic and maximal infinity

“…8.Sher replaced permutation invariance with bijection invariance and added some further conditions. For a summary, see Sher (2013, 176).…”

Isomorphism Invariance and Overgeneration

“…The proponent of isomorphism invariance is in no way committed to, indeed rejects, the idea that 'a competent speaker should be able, on the basis of his or her semantic competence, to accept or reject the Continuum Hypothesis'. 25 Even in the first-order case, in which there is a sound and complete proof procedure, the order of discovery may very well run from mathematics to logic. For example, there is a statement φ N of first-order logic materially equivalent to the arithmetical statement 'N is prime'.…”

“…Then this route to knowledge of the primality of N is not purely logical, as it rests on extra-logical knowledge of the biconditional. Similarly, coming 25 Bonnay ([1], p. 65), with the original 'its' replaced by 'his or her'. 26 The statement φN is the negation of 1<b<N Mult (a, b, N ), a disjunction which consists of (N −2) 2 disjuncts .…”

The Roots of Logical Hylomorphism

“…Thus, various modifications of the substantial formality may be classified into two clusters, i.e. the formal as schematic (see [13], [30]) and the formal as model-theoretic invariance (see [6], [21], [22], [36]). In other words, the form of argument represents a scheme, in other words, a result of the substitution of all the non-logical terms with variables of the corresponding categories.…”