Abstract
In this paper, we give a groundwork for the foundations of the semantic concept of logical consequence. We first give an opinionated survey of recent discussions on the model-theoretic concept, in particular Etchemendy’s criticisms and responses, alluding to Kreisel’s squeezing argument. We then present a view that in a sense the semantic concept of logical consequence irreducibly depends on the meaning of logical expressions but in another sense the extensional adequacy of the semantic account of first-order logical consequence is also of fundamental importance. We further point out a connection with proof-theoretic semantics.
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Notes
- 1.
In the following we often switch the terms i) “l.c.” and ii) “logical truth or validity” as a special case. The difference never affects our philosophical points.
- 2.
This is obviously condensed. For details, see, e.g., [6].
- 3.
- 4.
Glanzberg [11] discusses the issue of what sort of “truth conditions” we consider in different schools of the semantics of natural language. According to him, Davidsonians consider the absolute notion of truth condition, Montagovians initially considered the one based on “truth in a model” but these days they also use the absolute notion.
- 5.
The problem has been aware of, e.g., “Mathematics as a whole – this is the lesson of the set theoretic antinomies – is not a structure itself, i.e., an object of mathematical investigation, nor is it isomorphic to one ( [1], p. 7).”
- 6.
Indeed, Etchemendy states, “it is hard to understand why in the semantics for first-order languages we vary the domain of quantification (p. 290, [8]).”
- 7.
We do not go into historical issues here.
- 8.
The second weakness can be fixed by avoiding equality and by using any relation symbol s.t. (*) that R is transitive and irreflexive implies that there exists x for all y s.t. \(\lnot R (x,y)\). The negation of (*) has only infinite models. But we keep using equality.
- 9.
This is a kind of relativization, but we universally quantify over relativizations.
- 10.
He adds that the issue of the choice of logical constants is red herring, for the dependence on extralogical facts can arise even when all expressions are “logical constants,” e.g., \(\forall x \forall y \forall P (Px \rightarrow Py)\). This is true in a world with essentially one object.
- 11.
In [8], Etchemendy emphasizes that Kripke semantics is a good case of representational semantics. However, he elsewhere suggests that there is a severe limitation in representational semantics. “\(2 + 2 \ne 4\) (p. 62, [7])” is easy to make sense in interpretational semantics but makes no sense in representational one (since mathematical truth is a necessary truth). This can be a reason why interpretational semantics is also needed in addition to the representational view. See Sect. 3, Sect. 4.
- 12.
Schurz [23] also claims that the intensions of logical terms are determined by the recursive truth definition.
- 13.
He does not explain the phrase “the ordinary concept of logical truth,” either.
- 14.
- 15.
We omit the details of the complicated background of this notion of Val.
- 16.
- 17.
Shapiro takes modality involved in l.c. to be a “logical modality” given with respect to the isomorphism property (a necessary condition for a logical term). But Hanson’s modality is not particularly “logical.”
- 18.
Due to the limited space, we often state our points without detailed arguments.
- 19.
One might immediately object that \(=\) is not a logical symbol. However, such a change’s raising significant difference would already make dubious the robustness of the view. Also, from an intensional viewpoint, it is out of the question whether we can extensionally accommodate this pre-theoretic concept by a subsitutional account.
- 20.
L.c. can be a special case of “analytic” consequence. A system of transformation rules which transforms an atomic formula to another can be taken to give an analytic consequence. Formal systems of logic are often conservative extensions thereof (cf. [21]).
- 21.
We refrain from entirely agreeing with Etchemendy and Shapiro about the details.
- 22.
Even this may not be entirely unproblematic. There is an issue called “non-categoricity” in propositional logic first pointed out by Carnap [4].
- 23.
- 24.
Etchemendy addresses the issue (p. 275, footnote 6, [8]), saying that the finitist overgenerates. But there may be a further problem: the use of the axiom of infinity in proving completeness makes Etchemendy’s squeezing argument circular.
- 25.
This roughly means that logical truth corresponds to a lack of counterexample.
- 26.
- 27.
This issue is not simple, since those who argue that the infinity of a domain can be equipped on purely logical ground consider only first-order logic. To invalidate a formula in first-order language, we only need a countable model (see \(V^\omega \)). But things are more complicated in second-order logic, since falsifying a sentence in the language of second-order logic may require staggering ontology (p. 151, [24]). For second-order logic, Parsons’ entanglement view is more reasonable.
- 28.
The concept is so fundamental that it may be difficult to reduce it to something more fundamental.
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Kurokawa, H. (2020). On the Semantic Concept of Logical Consequence. In: Sakamoto, M., Okazaki, N., Mineshima, K., Satoh, K. (eds) New Frontiers in Artificial Intelligence. JSAI-isAI 2019. Lecture Notes in Computer Science(), vol 12331. Springer, Cham. https://doi.org/10.1007/978-3-030-58790-1_17
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