Putnam’s most famous contribution to mathematical logic was his role in investigating Hilbert’s Tenth Problem; Putnam is the ‘P’ in the MRDP Theorem. This volume, though, focusses mostly on Putnam’s work on the philosophy of logic and mathematics. It is a somewhat bumpy ride. Of the twelve papers, two scarcely mention Putnam. Three others focus primarily on Putnam’s ‘Mathematics without foundations’ (1967), but with no interplay between them. The remaining seven papers apparently tackle unrelated themes. Some of this disjointedness would (...) doubtless have been addressed, if Putnam had been able to compose his replies to these papers; sadly, he died before this was possible. In this review, I do my best to tease out some connections between the paper; and there are some really interesting connections to be made. (shrink)

This paper defends a reading of Wittgenstein’s philosophy of mathematics in the Lectures on the Foundation of Mathematics as a radical conventionalist one, whereby our agreement about the particular case is constitutive of our mathematical practice and ‘the logical necessity of any statement is a direct expression of a convention’ (Dummett 1959, p. 329). -/- On this view, mathematical truths are conceptual truths and our practices determine directly for each mathematical proposition individually whether it is true or false. Mathematical truths (...) are thus not consequences of a prior adoption of a convention or rules as orthodox conventionalism has it. -/- The goal of the paper is not merely exegetical, however, and argues that radical conventionalism can withstand some of the most difficult objections that have been brought forward against it, including those of Dummett himself, and thus that radical conventionalism has been prematurely excluded from consideration by philosophers of mathematics. (shrink)

There are several epistemic distinctions among truths that I have argued for in this paper. First, there are those truths which holdof every rationally accessible conceptual scheme (class A truths). Second, there are those truths which holdin every rationally accessible conceptual scheme (class B truths). And finally, there are those truths whose truthvalue status isindependent of the empirical sciences (class C truths). The last category broadly includes statementsabout systems and the statements they contain, as well as statements true by virtue (...) of the rules of language itself.At the risk of anachronism, I'll describe the positions of Carnap (1956); Quine (1980); Grice et al. (1956); the various Putnam's and myself in terms of the above distinctions: both Carnap and Quine (pretty much) think there are no class A or class B truths. Both Putnam (1975) and Putnam (1983c) think there are class A and class B truths, and that these classes overlap. I deny there are class B truths but affirm the existence of class A truths (although I haven't given explicit examples of the latter here). Finally, everyone here but Quine (1980) thinks there are class C truths (of one sort or another). Putnam (1975) attempts to show that certain class C truths are simultaneously class A and class B truths. Grice et al. (1956) take pains to distinguish the claim that there are class C truths from the claim that there are class A truths, and claim (against Quine, 1980) that no argument showing there are no class A truths shows there are no class C truths.On my interpretation of Quine (1980) he thinks that the nonexistence of class A truths shows there are no class C truths-given the extra bit of argument that a notion of ‘true by convention’ or ‘true by virtue of meaning’ without epistemic content, is a distinction without significance. But that issue, which is the one Grice et al. (1956) are concerned with, has not been the focus in this paper-and so in a sense I have shifted the terms of the original debate.Here I have been primarily concerned to distinguish epistemic notions and sort out how and in what ways they relate to each other. A primary tool in this exercise has been the explicit recognition that formal models of truth make universality an unlikely property of our conceptual schemes. If I have not convinced anyone that the epistemic notions sort out the way I think they do, I hope at least that some burden-shifting has occurred: that philosophers do not either take it for granted that certain notionsmust be expressible in any conceptual scheme or treat the fact that conceptual schemes must be (in some sense) limited as of little (philosophical) moment.On the other hand, if I am right about the epistemology, it follows that previous attempts to mark out the necessary structures in rationally accessible conceptual schemes via a priori truths is hopeless. What I think must replace their role, what I call globally incorrigible sets of sentences, is a topic for another time. (shrink)

The paper argues that the use of epistemic terms, prominently “… knows” and even “… knows a priori/a posteriori” is context-sensitive along several dimensions. Besides the best known dimension of quality of evidence (lower quality for less demanding context, and higher one for more demanding), there is a dimension of depth (shallow justification for superficial evaluation, and deeper justification for deeper probing evaluation contexts). This claim is illustrated by context-dependent ascription of apriority and aposteriority. The argument proposed here focuses upon (...) the status of propositions that are analytic in empirical concepts (like “Whales are animals”). It is a commonplace in epistemology that any analytic proposition (including e-analytic ones) is a priori. The paper claims that propositions analyzing empirical concepts are an interesting counterexample. It develops the following argument: Many such propositions have empirical counterparts that are expressed by the same form-of-words. (E.g., the form of words “Whales are mammals” can express both an e-analytic proposition and an empirical statement.) They normally derive from their empirical counterparts. Beliefs in such propositions, can be explicitly justified either a priori, by pointing out their conceptual, analytic status, or by reverting to their empirical counterparts. In contexts of very superficial evaluation, one may justify such an analytic belief in the first, conceptual way. In most contexts a belief in a proposition analyzing an empirical concept is being justified by appeal to its empirical counterparts. The empirical justification is normally taken as being ultimate. Empirical counterparts are derivationally deeper than the corresponding analytic propositions, and empirical justification is deeper than a priori one as well. Therefore, propositions analyzing empirical concepts are deeply a posteriori and superficially a priori. (shrink)

This paper addresses two issues in Richard Swinburne’s book Are We Bodies or Souls? I interpret Swinburne’s modal argument as an example of a priori synthetic knowledge. Swinburne’s thesis that every person possesses “thisness” is compared with Kant’s distinction between the empirical character and the intelligible character.

It is argued that mathematical statements are "a posteriori synthetic" statements of a very special sort, To be called "structure-Analytic" statements. They follow logically from the axioms defining the mathematical structure they are describing--Provided that these axioms are "consistent". Yet, Consistency of these axioms is an empirical claim: it may be "empirically verifiable" by existence of a finite model, Or may have the nature of an "empirically falsifiable hypothesis" that no contradiction can be derived from the axioms.

Epistemology has recently witnessed a number of efforts to rehabilitate rationalism, to defend the existence and importance of a priori knowledge or warrant construed as the product of rational insight or apprehension (Bealer 1987; Bigelow 1992; BonJour 1992, 1998; Burge 1998; Butchvarov 1970; Katz 1998; Plantinga 1993). This effort has sometimes been coupled with an attack on naturalistic epistemology, especially in BonJour 1994 and Katz 1998. Such coupling is not surprising, because naturalistic epistemology is often associated with thoroughgoing empiricism and (...) the rejection of the a priori. In this paper, however, I shall present a conception of naturalistic epistemology that is entirely compatible with a priori justification or knowledge. The resulting conception, I claim, gives us a better appreciation of the respective merits of the rational and the empirical, as well as a better understanding of how moderate epistemological naturalism comports, at least in principle, with moderate rationalism. This paper defends moderate rationalism; but it does not defend everything rationalists have often wanted, only what it is reasonable to grant them. (shrink)

Hilary Putnam first published the consistency objection against Ludwig Wittgenstein’s account of mathematics in 1979. In 1983, Putnam and Benacerraf raised this objection against all conventionalist accounts of mathematics. I discuss the 1979 version and the scenario argument, which supports the key premise of the objection. The wide applicability of this objection is not apparent; I thus raise it against an imaginary axiomatic theory T similar to Peano arithmetic in all relevant aspects. I argue that a conventionalist can explain the (...) consistency of T and suggest that an analogous explanation can be provided for the consistency of Peano arithmetic. (shrink)

Modality and Anti-Metaphysics critically examines the most prominent approaches to modality among analytic philosophers in the twentieth century, including essentialism. Defending both the project of metaphysics and the essentialist position that metaphysical modality is conceptually and ontologically primitive, Stephen McLeod argues that the logical positivists did not succeed in banishing metaphysical modality from their own theoretical apparatus and he offers an original defence of metaphysics against their advocacy of its elimination. -/- Seeking to assuage the sceptical worries which underlie modal (...) anti-realism, McLeod provides an original contribution to essentialist epistemology, engaging with current debates about modality and suggesting that standard essentialist approaches to some issues in the philosophies of logic and language require revision. -/- This book offers valuable insights to professional philosophers, postgraduates and advanced undergraduates interested in metaphysics, philosophy of logic or the history of twentieth-century analytic philosophy. (shrink)

I argue that Quine’s early critique of Carnap’s conventionalism is in serious tension with the holism of "Two Dogmas of Empiricism", since his critique of convention- alism makes a compelling case for a privileged status either for logic, or for some other principle by means of which to derive consequences. Based on this, I call for a modification of Quinean holism, on grounds internal to Quine’s views. The result motivates a rehabilitation of Carnap’s notion of framework principles, and a rethinking (...) of logic as a kind of relativised a priori. (shrink)