Abstract
Starting from certain metalogical results (the completeness theorem, the soundness theorem, and Lindenbaum-Scott theorem), I argue that first-order logical truths of classical logic are a priori and necessary. Afterwards, I formulate two arguments for the idea that first-order logical truths are also analytic, namely, I first argue that there is a conceptual connection between aprioricity, necessity, and analyticity, such that aprioricity together with necessity entails analyticity; then, I argue that the structure of natural deduction systems for FOL displays the analyticity of its truths. Consequently, each philosophical approach to these truths should account for this evidence, i.e., that first-order logical truths are a priori, necessary, and analytic, and it is my contention that the semantic account is a better candidate.
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Notes
The current debate on logical anti-exceptionalism, i.e., the idea that logic does not have a special status in our overall system of knowledge and, thus, logical truths are on the same par with the scientific truths, brings into focus the importance of the problem of explaining the nature of logical truths and could be also seen as a confirmation of Carnap’s dictum quoted above. See (Hjortland 2017) for a discussion of logical anti-exceptionalism.
The term “logic” and its derivatives will occur often in my article, so the reader may expect some remarks on the notion of logicality. However, since I am explicitly concerned in this article with classical FOL and we know from practice which are the logical terms of this system (we can simply enumerate them), for the present purposes the problem of logicality is assumed as being settled and, thus, the term “logic” is taken rather as primitive. I do not think that this treatment raises difficulties since, as [(Tarski 1986): 145] emphasizes, even a criterion of logicality has to be in agreement in the end, for adequacy to the intuitive concepts, with “at least with one usage which is actually encountered in practice”.
This definition goes back to [(Kant 1781/1998): 137]’s Critique of Pure Reason: “we will understand by a priori cognitions not those that occur independently of this or that experience, but rather those that occur absolutely independently of all experience. Opposed to them are empirical cognitions, or those that are possible only a posteriori, i.e., through experience.”, and closely resembles [(Kripke 1980/2001): 34]’s formulation of Kant’s idea: “the traditional characterization from Kant goes something like: a priori truths are those which can be known independently of any experience”.
“Now of these relations of statements to statements, one of conspicuous importance is the relation of logical implication: the relation of any statement to any that follows logically from it. […] Yet implication is not really an added factor; for, to say that one statement logically implies a second is the same as saying that a third statement of the system, an ‘if-then’ compound formed from the other two is logically true, or ‘valid’.”
It is natural to treat here the relation of logical consequence syntactically, since the logic of the web will have an associated formal system in which the deductions will be carried out –in the process of revision, in particular.
The equivalence would fail in this case because there would be no conditional sentence to correspond to that infinitary deduction –the implicit assumption being here that any statement from the web has a finite length. In particular, the commitment to this equivalence forbids us to accept deductions governed by the ω-rule, i.e., the rule which allows to infer (∀n)φ(n) from the infinite number of premises φ(0), φ(1), φ(2),….
Certainly, as one reviewer kindly pointed out, if we take the transfer principle as primitive, then the deduction theorem is deducible from it in conjunction with the soundness and weak completeness theorems: assuming ˹φ1,…,φn ⊢ ψ˺, it follows by the transfer principle that ˹⊨ (φ1&…&φn) → ψ˺, and from that by weak completeness that ˹⊢ (φ1&…&φn) → ψ˺. Conversely, if ˹⊢ (φ1&…&φn) → ψ˺, then it follows by soundness that ˹⊨ (φ1&…&φn) → ψ˺ and from there by the transfer principle that ˹φ1,…,φn ⊢ ψ˺. However, I think that we should not take the transfer principle as primitive. If we explicitly distinguish between (L1) the logic of the web and (L2) the formal logical system meant to formalize this logic of the web, then we can easily recognize that the transfer principle belongs prima facie to L1. The identification of the metatheoretical presuppositions of the transfer principle is the problem of finding out which are the correspondent metatheoretical properties of that formal logical system, meant to be an adequate formalization of L1, in which the transfer principle would be valid. In this sense, the transfer principle presupposes the deduction theorem, i.e., if the deduction theorem does not hold in L2, then the transfer principle also does not hold in L2 and, thus, that formal system cannot be an adequate formalization of the logic of the web (i.e., of L1).
The completeness theorem is used in fact by Quine as one of the arguments for the idea that first-order logic is the only logic. Of course, there are different notions of completeness that logicians work with (see for instance [(Manzano and Alonso 2013): 51–53]), but the relevant one for the present discussion is the notion of weak completeness of a calculus, i.e., each logical truth has a proof in that calculus (if ⊨ Φ then ⊢ Φ). In addition, since the deduction theorem holds in FOL (i.e., ˹φ1,…,φn ⊢ ψ˺ is equivalent to ˹⊢ (φ1&…&φn) → ψ)˺ the notion of strong completeness can be accounted via weak completeness.
The conclusion is free of any assumption in the sense that it does not rely on any other formula of the object language. The conclusion is still dependent, for instance, on the structural rules that define the concept of logical consequence in classical logic. These structural assumptions, however, seem to be unproblematic since they hold even in the formal logical system that was taken by some logicians to display the empirical character of logic, i.e., quantum logic (see for instance (Bacciagaluppi 2009) for a recent discussion of (Putnam 1975)).
In general, in scientific practice, and it is hard to see how it can be otherwise, the evidence for a logico-mathematical statement is taken to be (given by) a proof for that statement.
Some remarks about the status of these rules are formulated in section IV.
Although each logical truth is based on its proof, its justification also depends on the metatheorems because the proof is basically for the logical theorem, and in order to make the correspondence between the logical theorems and the logical truths we need to involve the soundness and completeness theorems.
Certainly, I do not want to suggest that completeness is a necessary condition for aprioricity. For instance, Peano Arithmetic may be a priori even though it is incomplete.
For a critical discussion and analysis of this received view, which is in fact one of the tenets of logical empiricism, see for instance (Sloman 1965).
This theorem is a result proved by (Scott 1974), who saw it as generalization of Lindenbaum’s Theorem (see (Scott 1974: 416). This is why some authors mention it as Lindenbaum-Scott Theorem (see (Koslow 1992) and (Rumfitt 2015)) and some others as Scott-Lindenbaum Theorem (see Payette and Schotch 2014).
[(Russell 1919): 153–154], for instance, argued that logical consequence has no modal ingredient and that its essential feature is only truth-preservingness.
For this version of the theorem and its relation with the version mentioned above see also the theorems 8.13 and 8.15 from (Koslow 1992). (Scott 1974) formulated the theorem in a multiple-conclusion framework, but its restriction to single conclusion does not raise difficulties. For a discussion of the relation between implication and valuations see also [(Scott 1971): 795–798].
The relation between the possibilities generated by an implicative relation and the models of the system of logic that defines that implicative relation requires further investigations. It seems natural to take every model of FOL as representing a possibility from Π⇒ and to take every possibility from Π⇒ as being represented by a model of FOL, but, as [(Shapiro 2018): 5] points out, if we move to second order calculus (where we have both standard and Henkin models), this correspondence is no longer so straightforward.
As (Quine 1953) argued, the combination between modal operators and quantifiers raises problems when we are allowed to attach the necessity sign to open sentences and then to quantify into these modal contexts. However, this problem does not affect the present discussion of the necessity of FOLTs because they are closed statements and, thus, we can treat their modalized form as having, in Quine’s terms, only two grades of modal involvement (as statement predicate or statement operator). We do not need to consider formulas of this sort ˹(∃x)□Fx˺ since they are part of first order modal logic, but not of FOL. See also [(Rumfitt 2015): 76] for a brief discussion of this aspect.
See [(Rumfitt 2015): 52–55] for a discussion of this idea.
More precisely, according to this line of thought, logico-mathematical statements are a priori because they are necessary.See (McEvoy 2013) for a discussion of this idea.
The logical empiricist view on logic is usually described by saying that logical truths are true by linguistic convention. However, Rudolf Carnap disapproved the use of “linguistic conventions” as applying to his account to logical truths. The choice of the meanings of the logical terms may be a matter of convention, but once these meanings are fixed, there is not conventional at all which statements are logically true: “once the meanings of the individual words in a sentence … are given (which may be regarded as a matter of convention), then it is no longer a matter of convention or of arbitrary choice whether or not to regard the sentence as true; the truth of such sentence is determined by the logical relations holding between given meanings” [(Carnap 1963): 915–916]. The distinction is important and, as [(Hacking 1979): 317] emphasized, “to reject truth by convention is not to reject truth in virtue of meaning”.
[(Boghossian 1996): 363] distinguishes between two notions of analyticity: an epistemological and a metaphysical one. In the latter sense, a statement is analytic if it owes its truth value completely to its meaning, and not at all to ‘the facts’. A statement S is analytic in the former sense if mere grasp of S’s meaning by a person P sufficed for P’s being justified in holding S true. Boghossian’s concern is to show that the analytic theory of a priori could be maintained if analyticity is understood in the epistemological sense, and that the metaphysical notion of analyticity, which is supposed to ground the linguistic theory of necessity, could not be defended. However, since I do not want to ground neither aprioricity, nor necessity in analyticity, I do not find this distinction helpful. It would be so if we want to pass directly from analyticity to aprioricity, or to necessity.
The philosophers who endorse the view that we can know empirical facts independently of experience usually postulate a special faculty of our mind, called intuition (or rational insight). Prominent scientists have recognized the importance of intuition (anschaulicher Einsicht) for the construction of mathematical and physical theories (see [(Hilbert 1930/2005): 1161–63; (Einstein 1934): 163–165]), but the principles gained by this intuitive insight, as [(Einstein 1934): 165] explicitly states it, have no a priori justification. Their justification is obtained by confronting the whole theory with the observational data. Kant’s assertion of the existence of synthetic a priori knowledge rests in the end on the assumption that we impose on experience, in sense perception, the relational structure of space and time. However, the development of modern geometry and physics undermined Kant’s assumption in the specific form that he gave it (see also [(Hilbert 1930/2005): 1162–63]). More generally, as [(Hintikka 1999): 145] emphasizes, the assertion of synthetic a priori knowledge rests on the failure to disentangle between heuristics and justification. (The reader may find in Hintikka’s article an excellent analysis and criticism of the concept of ‘intuitive knowledge’.)
Due to Carnap’s proofs from his 1943 book, Formalization of Logic, a problem with this approach is that if we start only from the formal system, the formal rules and axioms are not categorical in their standard formulation, i.e., they do not uniquely determine the meanings of the logical terms. Nevertheless, this problem could be easily solved if we adopt a multiple conclusion calculus and transfinite rules for the quantifiers, as (Carnap 1943) did, or we simply impose some semantic restrictions on interpretations, as (Bonnay and Westerståhl 2016) proposed. In the present context of the analysis of FOLTs, the use of both model and proof-theoretic instruments is allowed since they do not engage empirical considerations.
The elimination rules are not reduced to the introduction ones in the common sense of reduction, i.e., that of being formally derived, but rather by showing how “they can be justified in some semantic way” [(Prawitz 2006):510].
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Acknowledgements
I would like to thank Mircea Dumitru, Mircea Flonta, Matti Eklund, Ilie Pârvu, Gabriel Sandu, and Iulian Toader for their feedback. Special thanks to a reviewer for this journal whose substantial comments helped me improve this paper.
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Brîncuş, C.C. Philosophical Accounts of First-Order Logical Truths. Acta Anal 34, 369–383 (2019). https://doi.org/10.1007/s12136-019-00381-5
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DOI: https://doi.org/10.1007/s12136-019-00381-5