Abstract
Is a substantive standard of truth for theories of the world by and for humans possible? What kind of standard would that be? How intricate would it be? How unified would it be? How would it work in “problematic” fields of truth like mathematics? The paper offers an answer to these questions in the form of a “composite” correspondence theory of truth. By allowing variations in the way truths in different branches of knowledge correspond to reality the theory succeeds in rendering correspondence universal, and by investigating, rather than taking as given, the structure of the correspondence relation in various fields of knowledge, it makes a substantive account of correspondence possible. In particular, the paper delineates a “composite” type of correspondence applicable to mathematics, traces its roots in views of other philosophers, and shows how it solves well-known problems in the philosophy of mathematics, due to Benacerraf and others.
Earlier versions of this paper was presented at the Truth at Work Conference in Paris, 2011 and at the philosophy colloquium at UC Santa Barbara the same year. I would like to thank the audiences at both events for very constructive comments. This paper continues my earlier work on truth, knowledge, and logic, e.g., Sher (2004, 2010, 2011).
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Notes
- 1.
The compromise concerning unity results from their willingness to assign altogether different standards of truth—e.g., correspondence vs. coherence standards—to different disciplines. The compromise concerning the connection between truth and reality arises from their willingness to assign a non-correspondence standard of truth to allegedly “problematic” disciplines. (It’s important to note, however, that neither Wright nor Lynch is in principle averse to assigning a correspondence standard to any discipline).
- 2.
Indeed, given that we have more options in treating the simple cases than the complex cases, combinatorically it makes sense to start with the latter.
- 3.
It should be noted, though, that Horgan approaches truth from a somewhat different direction than we do. In particular, Horgan is interested in solving the problem of vagueness while we are interested in solving the epistemic problem of truth. These problems are not completely disconnected, but the difference is in certain ways insignificant.
- 4.
A universe is a non-empty set of individuals. The structures <U, P> and <U’,P’> are isomorphic iff there is a 1-1 and onto function f from U to U’ such that P’ is the image of P in U’ under f.
- 5.
This problem is raised by nominalists of various stripes (e.g., Goodman and Quine 1947), mathematical finitists, supporters of V = L, and others who feel uncomfortable with the huge ontology of contemporary (classical) set-theory. Here I focus on the issue of size.
- 6.
- 7.
The issue of algebraic vs. non-algebraic mathematical theories is discussed in, e.g., Shapiro (1997).
- 8.
This account (and explanation) applies smoothly to geometry, but Aristotle uses a somewhat different (and arguably weaker) account for arithmetic. For us, post-Fregean philosophers, however, it’s natural to extend Aristotle’s account of geometry to arithmetic, by viewing numbers as representing cardinality properties of physical objects.
- 9.
For a discussion of this point, though not as it relates to Aristotle, see Sher (2010).
- 10.
- 11.
This will be briefly discussed in the next section. See references there.
- 12.
- 13.
An earlier version, directed as modal fictionalism, is due to Rosen (1990).
- 14.
Our speech is engaged when we speak from within a given game, disengaged when we speak from outside it.
- 15.
- 16.
The logical truth “Pa ∨ ~ Pa” is an image of the mathematical truth that a is in the union of P and its complement (in the given universe), the logical inference “Pa∨Qa, ~ Pa; therefore Qa” is the image of the mathematical inference that if a is in the union of P and Q, yet is not in P, then it is in Q, and so on.
- 17.
This correspondence would be properly composite (i.e., will involve more than one step) if logical laws are construed as directly concerning linguistic entities and indirectly the world.
- 18.
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Sher, G. (2015). Truth as Composite Correspondence. In: Achourioti, T., Galinon, H., Martínez Fernández, J., Fujimoto, K. (eds) Unifying the Philosophy of Truth. Logic, Epistemology, and the Unity of Science, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9673-6_9
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