Deflationism, Arithmetic, and the Argument from Conservativeness

Research output: Journal PublicationsJournal Article (refereed)

3 Citations (Scopus)

Abstract

Many philosophers believe that a deflationist theory of truth must conservatively extend any base theory to which it is added (roughly: talking about truth shouldn't allow us to establish any new claims about subject-matters not involving truth). But when applied to arithmetic, it's argued, the imposition of a conservativeness requirement leads to a serious objection to deflationism: for the Gödel sentence for Peano Arithmetic (PA) is not a theorem of PA, but becomes one when PA is extended by adding plausible principles governing truth. This paper argues that no such objection succeeds. The issue turns on how we understand the notion of logical consequence implicit in any conservativeness requirement, and whether we possess a categorical conception of the natural numbers (i.e. whether we can rule out so-called "non-standard models"). I offer a disjunctive response: if we possess a categorical conception of arithmetic, then deflationists have principled reason to accept a rich notion of logical consequence according to which the Gödel sentence follows from PA. But if we do not, then the reasons for requiring the derivation of the Gödel sentence lapse, and deflationists are free to accept a conservativeness requirement stated proof-theoretically. Either way, deflationism is in the clear.
Original languageEnglish
Pages (from-to)429-463
Number of pages35
JournalMind
Volume126
Issue number502
Early online date19 Jan 2017
DOIs
Publication statusPublished - Apr 2017
Externally publishedYes

Fingerprint

Deflationism
Deflationist
Logical Consequence
Categorical
Conception
Imposition
Subject Matter
Theory of Truth
Philosopher
Natural numbers
Lapse

Cite this

@article{14e4226a0607413f8654549de13f379d,
title = "Deflationism, Arithmetic, and the Argument from Conservativeness",
abstract = "Many philosophers believe that a deflationist theory of truth must conservatively extend any base theory to which it is added (roughly: talking about truth shouldn't allow us to establish any new claims about subject-matters not involving truth). But when applied to arithmetic, it's argued, the imposition of a conservativeness requirement leads to a serious objection to deflationism: for the G{\"o}del sentence for Peano Arithmetic (PA) is not a theorem of PA, but becomes one when PA is extended by adding plausible principles governing truth. This paper argues that no such objection succeeds. The issue turns on how we understand the notion of logical consequence implicit in any conservativeness requirement, and whether we possess a categorical conception of the natural numbers (i.e. whether we can rule out so-called {"}non-standard models{"}). I offer a disjunctive response: if we possess a categorical conception of arithmetic, then deflationists have principled reason to accept a rich notion of logical consequence according to which the G{\"o}del sentence follows from PA. But if we do not, then the reasons for requiring the derivation of the G{\"o}del sentence lapse, and deflationists are free to accept a conservativeness requirement stated proof-theoretically. Either way, deflationism is in the clear.",
author = "Daniel WAXMAN",
year = "2017",
month = "4",
doi = "10.1093/mind/fzv182",
language = "English",
volume = "126",
pages = "429--463",
journal = "Mind",
issn = "0026-4423",
publisher = "Oxford University Press",
number = "502",

}

Deflationism, Arithmetic, and the Argument from Conservativeness. / WAXMAN, Daniel.

In: Mind, Vol. 126, No. 502, 04.2017, p. 429-463.

Research output: Journal PublicationsJournal Article (refereed)

TY - JOUR

T1 - Deflationism, Arithmetic, and the Argument from Conservativeness

AU - WAXMAN, Daniel

PY - 2017/4

Y1 - 2017/4

N2 - Many philosophers believe that a deflationist theory of truth must conservatively extend any base theory to which it is added (roughly: talking about truth shouldn't allow us to establish any new claims about subject-matters not involving truth). But when applied to arithmetic, it's argued, the imposition of a conservativeness requirement leads to a serious objection to deflationism: for the Gödel sentence for Peano Arithmetic (PA) is not a theorem of PA, but becomes one when PA is extended by adding plausible principles governing truth. This paper argues that no such objection succeeds. The issue turns on how we understand the notion of logical consequence implicit in any conservativeness requirement, and whether we possess a categorical conception of the natural numbers (i.e. whether we can rule out so-called "non-standard models"). I offer a disjunctive response: if we possess a categorical conception of arithmetic, then deflationists have principled reason to accept a rich notion of logical consequence according to which the Gödel sentence follows from PA. But if we do not, then the reasons for requiring the derivation of the Gödel sentence lapse, and deflationists are free to accept a conservativeness requirement stated proof-theoretically. Either way, deflationism is in the clear.

AB - Many philosophers believe that a deflationist theory of truth must conservatively extend any base theory to which it is added (roughly: talking about truth shouldn't allow us to establish any new claims about subject-matters not involving truth). But when applied to arithmetic, it's argued, the imposition of a conservativeness requirement leads to a serious objection to deflationism: for the Gödel sentence for Peano Arithmetic (PA) is not a theorem of PA, but becomes one when PA is extended by adding plausible principles governing truth. This paper argues that no such objection succeeds. The issue turns on how we understand the notion of logical consequence implicit in any conservativeness requirement, and whether we possess a categorical conception of the natural numbers (i.e. whether we can rule out so-called "non-standard models"). I offer a disjunctive response: if we possess a categorical conception of arithmetic, then deflationists have principled reason to accept a rich notion of logical consequence according to which the Gödel sentence follows from PA. But if we do not, then the reasons for requiring the derivation of the Gödel sentence lapse, and deflationists are free to accept a conservativeness requirement stated proof-theoretically. Either way, deflationism is in the clear.

UR - https://www.scopus.com/inward/record.uri?eid=2-s2.0-85027303335&doi=10.1093%2fmind%2ffzv182&partnerID=40&md5=522f6401d8b25397d83833d769df466c

U2 - 10.1093/mind/fzv182

DO - 10.1093/mind/fzv182

M3 - Journal Article (refereed)

VL - 126

SP - 429

EP - 463

JO - Mind

JF - Mind

SN - 0026-4423

IS - 502

ER -