Truth, indefinite extensibility, and fitch's paradox

In Joe Salerno (ed.), New Essays on the Knowability Paradox. Oxford University Press (2009)
Abstract
A number of authors have noted that the key steps in Fitch’s argument are not intuitionistically valid, and some have proposed this as a reason for an anti-realist to accept intuitionistic logic (e.g. Williamson 1982, 1988). This line of reasoning rests upon two assumptions. The first is that the premises of Fitch’s argument make sense from an anti-realist point of view – and in particular, that an anti-realist can and should maintain the principle that all truths are knowable. The second is that we have some independent reason for thinking that classical logic is not appropriate in this area. This paper explores these two assumptions in the context of Michael Dummett’s version of anti-realism, with particular reference to the argument from indefinite extensibility developed at various points in Dummett’s writings (e.g. Dummett 1991 Ch. 24). Dummett argues that certain concepts, the indefinitely extensible concepts, are such that we cannot form a clear and determinate conception of all the objects that fall under them. The most familiar examples of indefinitely extensible concepts are mathematical. Dummett discusses the concepts ordinal number, real number, and natural number, which are indefinitely extensible because any conception that one might form of their complete extension can be extended to a more inclusive conception (as, for example, in Cantor’s proof of the non-denumerability of the set of real numbers). This paper argues that the concept of a truth is indefinitely extensible. This gives a Dummettian anti-realist an independent motivation for rejecting the classical understanding of the quantifiers in this area. At the same time, however, it places in doubt the admissibility of the knowability principle, which seems to involve quantification over the “totality” of truths. As Dummett is at pains to point out (1991: 316), some sentences that purport to quantify over the extension of an indefinitely extensible concept plainly have a truth-value (we can truly say, for example, that every ordinal number has a successor, even though when we say that we are not quantifying over the set of all ordinals). But is the knowability principle one of these sentences?
Keywords fitch's paradox  indefinite extensibility  set  ordinals
Categories (categorize this paper)
Options
 Save to my reading list
Follow the author(s)
My bibliography
Export citation
Find it on Scholar
Edit this record
Mark as duplicate
Revision history Request removal from index
 
Download options
PhilPapers Archive


Upload a copy of this paper     Check publisher's policy on self-archival     Papers currently archived: 11,826
External links
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
Through your library
References found in this work BETA

No references found.

Citations of this work BETA
Similar books and articles
Analytics

Monthly downloads

Sorry, there are not enough data points to plot this chart.

Added to index

2009-01-28

Total downloads

0

Recent downloads (6 months)

0

How can I increase my downloads?

My notes
Sign in to use this feature


Discussion
Start a new thread
Order:
There  are no threads in this forum
Nothing in this forum yet.