Some Notes on Truths and Comprehension

In this paper we study several translations that map models and formulae of the language of second-order arithmetic to models and formulae of the language of truth. These translations are useful because they allow us to exploit results from the extensive literature on arithmetic to study the notion of truth. Our purpose is to present these connections in a systematic way, generalize some well-known results in this area, and to provide a number of new results. Sections 3 and 4 contain some recursion- and proof-theoretic results about Kripke-style fixed-point theories of truth. Section 5 shows how to derive full second-order arithmetic from principles of truth. Section 6 investigates the proof-theoretic strength of disquotation without an arithmetical base theory

Comprehension, Demonstration, and Accuracy in Aristotle

according to aristotle's posterior analytics, scientific expertise is composed of two different cognitive dispositions. Some propositions in the domain can be scientifically explained, which means that they are known by "demonstration", a deductive argument in which the premises are explanatory of the conclusion. Thus, the kind of cognition that apprehends those propositions is called "demonstrative knowledge".1 However, not all propositions in a scientific domain are demonstrable. Demonstrations are ultimately based on indemonstrable principles, whose knowledge is called "comprehension".2 If the knowledge of all scientific propositions were..

Faith and Reason

This contribution discusses Leibniz’s conception of faith and its relation to reason. It shows that, for Leibniz, faith embraces both cognitive and non-cognitive dimensions: although it must be grounded in reason, it is not merely reasonable belief. Moreover, for Leibniz, a truth of faith (like any truth) can never be contrary to reason but can be above the limits of comprehension of human reason. The latter is the epistemic status of the Christian mysteries. This view raises the problem of how it can be determined whether a doctrine above the full grasp of human reason does or does not imply contradiction. The notion of ‘presumption’ and the ‘strategy of defence’ are presented and discussed as Leibniz’s way to tackle this issue. Finally, the article explores the ‘motives of credibility’ which, according to Leibniz, can and should be produced to uphold the credibility of a putative divine revelation, including his account of miracles

Computer Science and Metaphysics: A Cross-Fertilization

Computational philosophy is the use of mechanized computational techniques to unearth philosophical insights that are either difficult or impossible to find using traditional philosophical methods. Computational metaphysics is computational philosophy with a focus on metaphysics. In this paper, we (a) develop results in modal metaphysics whose discovery was computer assisted, and (b) conclude that these results work not only to the obvious benefit of philosophy but also, less obviously, to the benefit of computer science, since the new computational techniques that led to these results may be more broadly applicable within computer science. The paper includes a description of our background methodology and how it evolved, and a discussion of our new results.Comment: 39 pages, 3 figure

Logicism, Ontology, and the Epistemology of Second-Order Logic

In two recent papers, Bob Hale has attempted to free second-order logic of the 'staggering existential assumptions' with which Quine famously attempted to saddle it. I argue, first, that the ontological issue is at best secondary: the crucial issue about second-order logic, at least for a neo-logicist, is epistemological. I then argue that neither Crispin Wright's attempt to characterize a `neutralist' conception of quantification that is wholly independent of existential commitment, nor Hale's attempt to characterize the second-order domain in terms of definability, can serve a neo-logicist's purposes. The problem, in both cases, is similar: neither Wright nor Hale is sufficiently sensitive to the demands that impredicativity imposes. Finally, I defend my own earlier attempt to finesse this issue, in "A Logic for Frege's Theorem", from Hale's criticisms

Abstracta and Possibilia: Modal Foundations of Mathematical Platonism

This paper aims to provide modal foundations for mathematical platonism. I examine Hale and Wright's (2009) objections to the merits and need, in the defense of mathematical platonism and its epistemology, of the thesis of Necessitism. In response to Hale and Wright's objections to the role of epistemic and metaphysical modalities in providing justification for both the truth of abstraction principles and the success of mathematical predicate reference, I examine the Necessitist commitments of the abundant conception of properties endorsed by Hale and Wright and examined in Hale (2013); and demonstrate how a two-dimensional approach to the epistemology of mathematics is consistent with Hale and Wright's notion of there being non-evidential epistemic entitlement rationally to trust that abstraction principles are true. A choice point that I flag is that between availing of intensional or hyperintensional semantics. The hyperintensional semantic approach that I advance is a topic-sensitive epistemic two-dimensional truthmaker semantics. Epistemic and metaphysical states and possibilities may thus be shown to play a constitutive role in vindicating the reality of mathematical objects and truth, and in providing a conceivability-based route to the truth of abstraction principles as well as other axioms and propositions in mathematics