Abstract
The paper argues that the theory of Implicit Definition cannot give an account of knowledge of logical principles. According to this theory, the meanings of certain expressions are determined such that they make certain principles containing them true; this is supposed to explain our knowledge of the principles as derived from our knowledge of what the expressions mean. The paper argues that this explanation succeeds only if Implicit Definition can account for our understanding of the logical constants, and that fully understanding a logical constant in turn requires the ability to apply it correctly in particular cases. It is shown, however, that Implicit Definition cannot account for this ability, even if it draws on introduction rules for the logical constants. In particular, Implicit Definition cannot account for our ability to apply negation in particular cases. Owing to constraints relating to the unique characterisation of logical constants, invoking the notion of rejection does not remedy the situation. Given its failure to explain knowledge of logic, the prospects of Implicit Definition to explain other kinds of a priori knowledge are even worse.
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Notes
See Prawitz (2006) for a recent discussion.
In a series of papers following his 1996 and 1997 (Boghossian 2001, 2003a, b), Boghossian develops a meaning-based account of the a priori justification of inferences. While his 2001 still employs the idea that meaning-constituting inferences are truth-preserving by virtue of a kind of implicit definition, viz. “implicit stipulation” (2001, p. 33), 2003a and 2003b differ in spirit by dispensing with this requirement. It is not entirely clear whether Boghossian’s later meaning-based account of the justification of individual a priori inferences supersedes his earlier account of a priori beliefs, since, even if a priori inferences are involved in the production of a priori beliefs, appropriate, and appropriately justified, premises are needed too. In fact, in 2003b (p. 34) he states that the account of a priori inferences can be applied to what he calls the “Implicit Definition Template”, suggesting that the two accounts complement each other, and in the introduction to a collection of papers of his that contains all of those cited above, he professes agnosticism about what a successful meaning-based account of the a priori will ultimately look like (2008, p. 5). For critical discussion of Boghossian’s different accounts and similar approaches by other authors, see Horwich (2005) chapter 4, and Horwich (2010) chapter 10. Williamson (2003) is a direct response to Boghossian (2003a). Irrespective of the dynamics of Boghossian's views, his original account of Implicit Definition keeps drawing philosophical attention (Ebert 2005, Jenkins 2008 chapter 2, and García-Carpintero and Pérez Otero 2009 are recent examples) and merits to be discussed in its own right.
Compare Boghossian (1997): “Implicit definition: It is by arbitrarily stipulating that certain sentences of logic are to be true, or that certain inferences are to be valid, that we attach a meaning to the logical constants. More specifically, a particular constant means that logical object, if any, which would make valid a specified set of sentences and/or inferences involving it” (p. 348).
Unless the constituent is in quotation marks.
The example is also discussed in Hale and Wright (2000, p. 294).
This ability might still be constitutive of understanding in the sense of being a prior, that is, more fundamental, necessary condition for it; however, my argument does not require this stronger assumption.
It might also generally be more demanding to come to know that such-and-such a sentence is true than to form a justified belief that such-and-such a sentence is true. The reason why application requires the ability for the former is also that it facilitates the presentation of later arguments. Like the requirement of meta-linguistic knowledge, this more demanding requirement is harmless given the ceteris paribus condition, which rules out cases of justified belief without knowledge, such as Gettier cases and, plausibly, skeptical scenarios. This is not to say that such cases are irrelevant for the epistemology of logic; on Gettier cases, see Besson (2009). Peacocke (1999, chapter 2) endorses necessary conditions for the possession of concepts that are formulated in terms of knowledge.
It might be objected that understanding + application does not follow from understanding and application for the following reason. ‘Ceteris paribus’ functions like a modal operator: ‘Ceteris paribus, A’ means that, in all normal circumstances, A is the case. However, what the normal circumstances are varies with the sentence following the ceteris paribus clause and the context of the utterance. Thus, it might be that ‘Ceteris paribus, all Fs are G’ and ‘Ceteris paribus, all Gs are H’ are both true while ‘Ceteris paribus, all Fs are H’ is false owing to different circumstances being relevant for the different sentences. I agree with everything in this objection, except that I deny that the general failure of transitivity of ceteris paribus conditionals (or universally quantified conditionals) affects our case. For there is no reason to suppose that understanding, application and understanding + application give rise to different circumstances for the respective ceteris paribus operators. In all cases, the normal circumstances seem to involve a competent speaker of the non-logical fragment of English (perhaps one with slightly idealised mental powers) who is not distracted or deluded, etc.
I am closely following Gentzen’s (1969) formalism here, but mutatis mutandis the following arguments would also apply to other systems.
Alternatively, we could conceive of ⊥ as a constantly false sentence. A reading of ⊥ as an absurdity is discussed at the end of this section.
Dummett (1993, p. 258) anticipates this result when he remarks that “it would be difficult to provide for the derivation of ‘¬A’ with A atomic by means of a purely logical rule”. For a similar formal result see Milne (1994), who also draws pessimistic conclusions about the prospect of defining negation by its introduction rule.
Note that this result is very general: it does not require that the logical rules be introduction rules or that they involve only a single logical constant each.
One might be attracted to a more restrictive definition of unique characterisation according to which any pair of sentences such that one sentence is the result of replacing O1 with O2 in the other—irrespective of whether or not O1/O2 is the principal operator—are logically equivalent given that O1 and O2 satisfy P 1, …, P n . However, satisfaction of this more restrictive definition will follow from satisfaction of the definition endorsed here given that the logic in question has the so-called congruentiality property, according to which the logical equivalence of sentences B 1 and B 2 implies the logical equivalence of χ(B 1) and χ(B 2) for all contexts χ. Classical and intuitionistic propositional logic are congruential; see, for instance, Humberstone (2010).
See Williamson (1988, pp. 111–112), who also draws attention to the consequences for the dispute about Double Negation Elimination that is discussed below.
For raa and efq are valid in intuitionistic logic while dne is not; see Dummett (1977, p. 26).
The characterisations of ‘−’ and ‘+’ in this paragraph follow Rumfitt (2000, pp. 800–803).
Calling a formula beginning with ‘+’ or ‘−’ ‘true’ might be inadmissible, since “‘+A’ is true” would translate into the ungrammatical “‘Is it the case that A? Yes’ is true” according to the definition of ‘+’ (similarly for ‘–’). Instead, we might call a formula of the form ‘+A’ or ‘−A’ correct if and only if ‘Yes’ or ‘No’ would be, respectively, the correct answer to the question ‘Is it the case that A?’. Consequently, the validity of rules such as +-¬-I, +-¬-E, −-¬-I, and −-¬-E should be understood as correctness-preservation as opposed to truth-preservation.
Proof. To show that −-¬-I cannot be derived from +-¬-I and +-¬-E, evaluate +A as true relative to an assignment of truth-values to atomic sentences if and only if A is true or atomic (where A is evaluated in the usual way if complex), and evaluate –A as true if and only if A (evaluated as before) is false. Then +-¬-I and +-¬-E are truth-preserving relative to all assignments, but −-¬-I is not (let A be false and atomic). Similarly, to show that −-¬-E cannot be derived from +-¬-I and +-¬-E, evaluate +A as true relative to an assignment if and only if A is true and non-atomic, and evaluate –A as true if and only if A is false. Then +-¬-I and +-¬-E are truth-preserving relative to all assignments, but −-¬-E is not (let A be true and atomic).
At least if we assume further that acceptance (‘+’) is primitive as well.
The Poincaré-Hilbert approach to geometry might be regarded as a possible niche for Implicit Definition. For a recent discussion, see Ben-Menahem (2006).
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Acknowledgments
Thanks to Corine Besson, Franz Huber, Wolfgang Künne, Erik Stei, Timothy Williamson, and an anonymous referee of Philosophical Studies for very helpful comments and suggestions.
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Kroedel, T. Implicit definition and the application of logic. Philos Stud 158, 131–148 (2012). https://doi.org/10.1007/s11098-010-9675-0
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DOI: https://doi.org/10.1007/s11098-010-9675-0