Abstract
Vann McGee has recently argued that Belnap’s criteria constrain the formal rules of classical natural deduction to uniquely determine the semantic values of the propositional logical connectives and quantifiers if the rules are taken to be open-ended, i.e., if they are truth-preserving within any mathematically possible extension of the original language. The main assumption of his argument is that for any class of models there is a mathematically possible language in which there is a sentence true in just those models. I show that this assumption does not hold for the class of models of classical propositional logic. In particular, I show that the existence of non-normal models for negation undermines McGee’s argument.
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Notes
McGee (2015) proposes an understanding of the semantic role of sentences in terms of possible worlds; namely, a sentence is taken to express a proposition and the latter is understood, following R. Stalnaker’s account, as a set of possible worlds. He then formulates propositional rules for the propositional connectives, i.e., the counterparts of the sentential ones. For simplicity, we shall not consider this propositional approach here, but what we say below is applicable, mutatis mutandis, to it.
In a non-normal interpretation, disjunction violates the fourth row from the normal truth table (NTT), i.e., it is true although both of its disjuncts are false (the first three rows are secured by the vI rule). This happens because vE rule does not fix the fourth row of the NTT. Nevertheless, if negation is normal, then disjunction is also normal, otherwise the Disjunctive Syllogism Rule (AvB, ~ A ˫ B) would become unsound (i.e., if “A” and “B” are false and negation is normal (thus, “~A” is true), then “AvB” cannot be true). However, since negation and disjunction form a functionally complete set of connectives, then all the other connectives will be normal.
The notion of categoricity used in this paper differs from the standard notion of categoricity defined in modern model theory, where a theory T is categorical in a cardinal κ (or κ-categorical) if and only if it has exactly one model of cardinality κ up to isomorphism. The present notion of categoricity simply points out to the fact that the formal rules of deduction are compatible with truth-tables (the normal and the non-normal ones) that are not isomorphic. A precise definition of isomorphic truth-tables was given by Kalicki (1950: p. 175) by adapting Tarski (1938: p. 106)’s definition of isomorphic matrices. The main idea of the definition is that, in at least one row, for the same input, the normal truth-table gives a designated value, while the non-normal truth-table gives an undesignated one –or the other way around. A general definition of categoricity in this sense could be given following Scott’s (1971: pp. 795–798) terminology: a formal system of logic is categorical if and only if the only valuation that is consistent with the syntactical relation of logical consequence in that system is the standard one. A valuation ν is consistent with a consequence relation ⊢ if and only if, whenever Γ ⊢ σ, if ν(φ) = 1 for all φ ∈ Γ, then ν(σ) = 1. For a discussion of this notion of categoricity see Hjortland (2014: pp. 447–51) and Bonnay and Westerståhl (2016: pp. 726–27). It is worth mentioning, however, that the property of categoricity, in this sense, is relative to the format of the proof system. For instance, if we strengthen the proof-theoretic framework of propositional logic, e.g. by allowing multiple-conclusions [like Carnap (1943) and Shoesmith and Smiley (1978) suggested], or by resorting to a bilateral formalisation of logic [see e.g. Smiley (1996) and Rumfitt (2000)], categoricity can be regained.
One moral that we can drawn from Carnap’s discovery of the non-normal interpretations is that syntactical uniqueness is not a sufficient condition for semantic uniqueness, i.e., for determining a unique meaning for the syntactical connectives.
The argument starts with the assumption that θ is a sentence that is true in just those models in which neither ϕ nor ~ ϕ is true. Then, by valid reasoning we find out, at line (9), that θ is inconsistent, i.e., it has no model (10). Since (1) and (10) constitute an inconsistent pair of sentences -(1) says that θ has at least one model and (10) says that θ has no models-, an absurdity follows at line (11). What led us, however, from the very beginning to this inconsistency was the assumption (1), which does not simply say that θ is true in general, but that it is true just in a certain class of models. Therefore, the negation of (1) has to be inferred by reductio and not simply the negation of θ. Actually, as Carnap showed, if θ is true in those models in which neither ϕ nor ~ ϕ is true, then θ is true in all the models of PC.
As a reviewer kindly suggested, someone may say that since assumption (1) leads to a contradiction, then one may classically draw any conclusion whatsoever from this contradiction. Hence, the derivation of (12) from it is not a non-sequitur. Indeed, from a strictly formal point of view, it is not a mere non-sequitur. However, when we arrive at a contradiction, it is more reasonable to see what false ideas led us to that contradiction, and not to start deriving any conclusion from it. Since McGee’s argument is in the meta-theory, and uses both proof-theoretic and model-theoretic resources, and it is a logical fact (more precisely, a model-theoretic fact) that there are models in which a sentence and its negation are both false, the truth value of the starting assumption should be first investigated.
This proof could be found in a different terminology in Carnap (1943: pp. 91–92).
As a reviewer suggested to me, the main point made in this paper can be formulated by saying that McGee's argument already presupposes the notion of an admissible model, i.e., a model that respects the meanings of the logical constants. For if we let McGee's claim be about any model, we get that the claim must also hold for non-standard interpretations of propositional logic, e.g. interpretations that make both φ and not-φ true. But there is no sentence that is (actually) true in exactly those models.
Certainly, Carnap refers to the standard formulations, i.e, those with a single conclusion. If we use multiple conclusions, categoricity can be regained. (see Hjortland 2014).
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Acknowledgements
I would like to thank my audiences in Hejnice, Klagenfurt, Novara, and Tübingen, where parts of my work on this paper have been presented, in particular Andrzej Indrzejczak, David Miller, Jaroslav Peregrin, Andrea Sereni and Peter Schroeder-Heister, for helpful discussions. Special thanks to Gabriel Sandu, Iulian Toader, and the reviewers for their useful feedback.
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Brîncuș, C.C. Are the open-ended rules for negation categorical?. Synthese 198, 7249–7256 (2021). https://doi.org/10.1007/s11229-019-02519-9
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DOI: https://doi.org/10.1007/s11229-019-02519-9