Abstract
We use a modified version of E.Beth’s concept of implicit definitions to show that all the usual logical operators as well as the first and second order quantifiers are implicitly defined—and for essentially the same reason that involves an account of the logical operators using a concept of filter conditions. An “inferential” proposal is then suggested for a Gentzen-like account as a necessary condition for the familiar logical operators. We then explore the question of whether our proposal can also be taken as a sufficient condition. To this end, we discuss whether other operators, like a truth operator, the counterfactual conditional, the identity, and the modal operators are also logical operators. The paper closes with a brief discussion of what is called the robustness of the logical operators: What happens to the logical operators when there is a shift from one logical structure to another which extends it, and what happens when there is a shift from one structure to one in which it is homomorphically embedded.
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Notes
- 1.
Nagel [22].
- 2.
Hilbert [12].
- 3.
- 4.
Shapiro [25].
- 5.
Belnap [2]. I understand Belnap as advocating the kind of schematic expression of the condition for implicit definitions that we made explicit. There may be reason to think otherwise. L. Humberstone [13, p. 267] reports that in correspondence Belnap suggested that the kind of uniqueness he had in mind might also be thought of as implicit definition as long as some kind of second order (propositional quantification) is included. Here it has been Beth who was the inspiration for using his notion of implicit definition carried over to connectives and operators rather than just relations, functions, and constants. One shouldn’t however overlook the important and unjustly neglected paper in J.Harris [10].
- 6.
Church [5, p. 39] for example, places them in the special category of operators.
- 7.
- 8.
Cf. Koslow [16].
- 9.
The account is broadly “inferential”, originating with P. Hertz, and G. Gentzen, and including N. Belnap [2]. D. Prawitz [23], M. Dummett [6], J.I. Zucker [27], R.S. Tragesser [27, 28] V. McGee [18, 19], and most recently S. Feferman [7]. I cannot specify the bibliographic detail as to dates and pagination, as the paper is included in the present. Profound apologies for the omission of the many significant semantic and proof-theoretical recent work that also deserve close study.
- 10.
For an extended study of the following paired conditions on these logical operators, cf. Koslow [16].
- 11.
Although these conditions suffice for showing that the logical operators are implicitly defined by the paired conditions, more is needed to obtain the full story: All these conditions need to be stated in a more general way: let Γ stand for any finite (possibly empty) sequence of members of the implication structure. Then, for example, the two modified (parameterized) conditions for conjunction are: for any Γ, A, and B, (1) \(\varGamma,A, (A \rightarrow B) \Rightarrow B\), and (2) It is the weakest member of the structure to satisfy the first condition. That is, for any T in S, if \(\varGamma,A,T \Rightarrow B\), then \(\varGamma,T \Rightarrow (A \rightarrow B)\). With this adjustment, if follows that for any A and B in a structure, if their conjunction exists in the structure, then \(A,B \Rightarrow A \wedge B\). The fuller account is in Koslow [16], ch. 15.
- 12.
Tarski [26, p. 191].
- 13.
Mostowski, A. On a Generalization of Quantifiers. In Mostowski [21].
- 14.
We set to one side the relatively simple proofs that each of the familiar logical operators satisfies the proper filter property.
- 15.
We have generally used versions equivalent to the treatment of the second- order quantifiers to be found in Mendelson [20, pp. 376–89] and Vann McGee [18, pp. 54–78, esp. p. 63]. Not everything however is listed (no inclusion of the comprehension schema), but enough to be able to prove implicit definability.
- 16.
Prior [24].
- 17.
We leave to one side the proofs that the truth operator, as well as the counterfactual conditional and the identity operators to be discussed, all satisfy the proper filter property (PRP).
- 18.
- 19.
Lewis [17, pp. 132–33].
- 20.
It is clear that the heavy lifting in this argument is supplied by (4) (and its mate (4∗)). These conditions, as Humberstone notes are the modal analogs of the substitutivity condition on identity, (5) \(t = u,C(t) \Rightarrow C(u)\), for which is well known to imply that any two relations satisfying it mutually imply each other.
- 21.
Cf. Koslow [16, p. 389, fn. 8].
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Koslow, A. (2015). Implicit Definitions, Second-Order Quantifiers, and the Robustness of the Logical Operators. In: Torza, A. (eds) Quantifiers, Quantifiers, and Quantifiers: Themes in Logic, Metaphysics, and Language. Synthese Library, vol 373. Springer, Cham. https://doi.org/10.1007/978-3-319-18362-6_3
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