Showing papers by "Neil Tennant published in 2014"

Logic, Mathematics, and the A Priori, Part II: Core Logic as Analytic, and as the Basis for Natural Logicism

Abstract: This is Part II of a two-part study of the foundations of mathematics through the lenses of (i) apriority and analyticity, and (ii) the resources supplied by Core Logic. [In Part I we explained what is meant by apriority, as the notion applies to knowledge and possibly also to truths in general. We distinguished grounds for

Aristotle's Syllogistic and Core Logic

Abstract: I use the Corcoran–Smiley interpretation of Aristotle's syllogistic as my starting point for an examination of the syllogistic from the vantage point of modern proof theory. I aim to show that fresh logical insights are afforded by a proof-theoretically more systematic account of all four figures. First I regiment the syllogisms in the Gentzen–Prawitz system of natural deduction, using the universal and existential quantifiers of standard first-order logic, and the usual formalizations of Aristotle's sentence-forms. I explain how the syllogistic is a fragment of my (constructive and relevant) system of Core Logic. Then I introduce my main innovation: the use of binary quantifiers, governed by introduction and elimination rules. The syllogisms in all four figures are re-proved in the binary system, and are thereby revealed as all on a par with each other. I conclude with some comments and results about grammatical generativity, ecthesis, perfect validity, skeletal validity and Aristotle's chain principle.

The logical structure of evolutionary explanation and prediction: Darwinism’s fundamental schema

Abstract: We present a logically detailed case-study of Darwinian evolutionary explanation. Special features of Darwin’s explanatory schema made it an unusual theoretical breakthrough, from the point of view of the philosophy of science. The schema employs no theoretical terms, and puts forward no theoretical hypotheses. Instead, it uses three observational generalizations—Variability, Heritability and Differential Reproduction—along with an innocuous assumption of Causal Efficacy, to derive Adaptive Evolution as a necessary consequence. Adaptive Evolution in turn, with one assumption of scale (‘Deep Time’), implies the observational generalization of Adaptation. It is a fascinating methodological task to regiment the premises and make the reasoning both rigorous and clear. Doing so reveals how surprisingly small an amount of mathematics is needed in order to carry out the argument. The investigation also reveals the crucial role played by heritability, and how heritability itself admits of Darwinian explanation.

Logic, Mathematics, and the A Priori, Part I: A Problem for Realism

Abstract: This is Part I of a two-part study of the foundations of mathematics through the lenses of (i) apriority and analyticity, and (ii) the resources supplied by Core Logic. Here we explain what is meant by apriority, as the notion applies to knowledge and possibly also to truths in general. We distinguish grounds for knowledge from grounds of truth, in light of our recent work on truthmakers. We then examine the role of apriority in the realism/anti-realism debate. We raise a hitherto unnoticed problem for any Orthodox Realist who attempts to explain the a priori.