Abstract
In this paper, I introduce and defend a notion of analyticity for formal languages. I first uncover a crucial flaw in Timothy Williamson’s famous argument template against analyticity, when it is applied to sentences of formal mathematical languages. Williamson’s argument targets the popular idea that a necessary condition for analyticity is that whoever understands an analytic sentence assents to it. Williamson argues that for any given candidate analytic sentence, there can be people who understand that sentence and yet who fail to assent to it. I argue that, on the most natural understanding of the notion of assent when it is applied to sentences of formal mathematical languages, Williamson’s argument fails. Formal analyticity is the notion of analyticity that is based on this natural understanding of assent. I go on to develop the notion of formal analyticity and defend the claim that there are formally analytic sentences and rules of inference. I conclude by showing the potential payoffs of recognizing formal analyticity.
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Notes
A sentence stating a rule of inference for a formal language is a sentence of the metalanguage. For instance, the sentence ‘B may be inferred from \(\lnot A \vee B\) and A’, which states the rule of inference of disjunctive syllogism, is part of the metalanguage (here English), and ‘A’ and ‘B’ are syntactic variables in the metalanguage that range over formulas of the formal language in question. Strictly speaking, ‘\(\lnot A\)’ and ‘\(A \vee B\)’ should be written respectively as ‘\(\lnot\)’ \({} ^\frown A\) and \(A^\frown\)‘\(\vee\)’\(^\frown B,\) but I will use the former shorthands throughout.
See also: “It must be emphasized that the concept of analyticity has an exact definition only in the case of a language system, namely a system of semantical rules, not in the case of an ordinary language, because in the latter the words have no clearly defined meaning” (Carnap 1952b, 427). A. J. Ayer notes in a similar vein, as he explains how to understand ‘truth in virtue of meaning’: “There is ground for saying that the philosopher is always concerned with an artificial language. For the conventions which we follow in our actual usage of words are not altogether systematic and precise” (Ayer 1936, 70, fn. 1).
Williamson takes this to be a definition of the “epistemological conception of analyticity” (Williamson 2007, 73). Notice that Williamson doesn’t need to accept any definition of ‘analytic’; premise 1 (and whatever definition of ‘analytic’ underlies it) is understood to be proposed by Williamson’s opponents.
Later Williamson goes on to argue against “watered down” understanding–assent links with ‘assent’ replaced throughout with ‘disposition to assent’ (Williamson 2007, 100–109). He states that “[h]aving a disposition to assent does not entail assenting” (Williamson 2007, 100). There are thus two notions of “disposition to assent” at play for Williamson; here I will remain neutral between the two, as this won’t make a difference to my arguments.
We may want to leave it open whether all sentences in \(\Pi _w\) are analytic (i.e. such that understanding them requires assenting to them). Indeed, it may be that for some word(s), w, understanding w requires assenting to some sentence, s, which doesn’t contain w, and such that for any word, \(w'\), in s, \(s \notin \Pi _{w'}\). In this case (also assuming that understanding the syntax of s doesn’t require assenting to s), understanding s wouldn’t require assenting to s, even though \(s \in \Pi _w\).
See for instance Williamson (2007, 123f., 125).
One could also propose a weaker definition 1e\('-\): A sentence, s, of a formal system, \(\Sigma\), is analytic in \(\Sigma\) just in case, necessarily, whoever understands s believes that s is part of a formal system that correctly represents Reality. My arguments against 1e\('\) below also work against 1e\('-\), so I won’t consider 1e\('-\) separately.
A famous example of a “formalist” mathematician is Paul Cohen. See for instance Cohen (1971).
Of note is that these definitions are very similar to the ones proposed by Carnap (1937, 1939). For Carnap, analytic sentences of a mathematical formal system are (roughly) all those that are true according to the formal semantics of that formal system (Carnap 1937, 111). Analyticity for Carnap is thus a purely formal feature of a sentence of a formal system; it doesn’t make reference to the notion of linguistic understanding because Carnap found this notion is too imprecise (Carnap 1939, 12f.). However, Carnap agrees with the idea motivating formal analyticity that understanding formal languages has to do with knowing their derivation rules: “Since to know the truth conditions of a sentence is to know what is asserted by it, the given semantical rules determine for every sentence of [the formal system] […] what it asserts—in usual terms, its “meaning”—or, in other words, how it is to be translated into English. […] Therefore, we shall say that we understand a language system, or a sign, or an expression, or a sentence in a language system, if we know the semantical rules of the system” (Carnap 1939, 10f.).
In general, a formal derivation in \(\Sigma\) is a sequence of formulas such that each formula in the sequence is either an axiom, or else results by a rule of inference from formulas that precede it in the sequence. So axioms of a formal system are trivially derivable in that formal system. I make this assumption here concerning Syntactic Formal Analyticity because I want to make room for a (possibly) weaker reading.
I.e. \(\forall x, y (x = y \leftrightarrow \forall z (z \in x \leftrightarrow z \in y))\).
Throughout, I will use ‘knowing what are the basic axioms of ZFC’ to mean knowing of the basic axioms of ZFC that they are axioms of ZFC, without necessarily knowing or believing that these axioms are basic.
In order to count as an expert, one has to be able to use the language in a way that exhibits understanding. And thus, if one of the basic uses of the language is to do formal derivations, this ability also needs to involve understanding, as opposed to mere manipulation of symbols.
Other contemporary thinkers also endorse Understanding–Ability, albeit in different contexts. For instance, John Bengson and Mark A. Moffett (2007) argue that some concepts are “ability-based concepts,” i.e. concepts understanding of which entails having certain abilities [see also Bengson (2016), Pavese (2015), and Wikforss (2017)].
Williamson also clearly assumes something like Understanding–Ability throughout his arguments against analyticity. For example, in making the case that his hypothetical example, Peter, understands the sentence ‘Vixen are vixen’ but doesn’t assent to it, Williamson states that Peter is able to have normal conversations about vixens with members of his linguistic community, that he is able to adjust his conversations so that his odd beliefs about vixens don’t get him in trouble, etc. (Williamson 2007, 88–91).
See also Putnam (1970).
See also Williamson (2007, 97).
Perhaps one can also have justification or “entitlement” to believe axioms to be true even without knowing the correct metasemantic theory, just in virtue of understanding the expressions, and hence knowing that certain sentences have a privileged status within the formal systems. This is connected to Paul Boghossian’s project: In a number of papers, Boghossian aimed to show that, if certain rules of inference have content-determining statuses, and if understanding the logical constants entails assenting to these rules of inference, then we can be justified or at least “blameless” in inferring according to these rules of inference (Boghossian 1996, 2003a, b). Of course, the same strategy can’t be used here because understanding expressions of a formal language, I argued, only requires “internal” assent, not something like “inferring according to the rules in ordinary reasoning.”
Sheldon Smith makes a point similar to this in arguing against Rey’s claim that Leibniz deferred to an optimal theory for the concept of the derivative (Smith 2015, 15).
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Acknowledgements
I am grateful to Olivia Bailey, Selim Berker, Matti Eklund, Ned Hall, Harold Hodes, Jens Kipper, Peter Koellner, Doug Kremm, Elizabeth Miller, Bernhard Nickel, Shantia Rahimian, Mark Richard, Jacob Rosen, Susanna Siegel, audiences at the 15th Congress on Logic, Methodology and Philosophy of Science in Helsinki, and at the M & E Workshop at Harvard for very helpful discussions and comments on various drafts of this paper.
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Soysal, Z. Formal analyticity. Philos Stud 175, 2791–2811 (2018). https://doi.org/10.1007/s11098-017-0982-6
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DOI: https://doi.org/10.1007/s11098-017-0982-6