Abstract
An extensional formal language for sortals, its formal semantics as well as a formal system for the language are characterized. The system is shown to be sound and complete relative to the semantics. The restriction-relation among sortal concepts is introduced and discussed in connection with other related important concepts, such as those of a phase sortal, a substance sortal and an ultimate sortal. A formal representation of the relation is provided. The view that every individual must fall under a sortal is considered and formalized as well. The formalizations in question are extensions of the conceptual framework provided by the extensional logic of sortals.
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Notes
- 1.
Among other reasons, second-order sortal quantifiers are introduced at this extensional stage to make possible the expression of several theses involving the notion of ultimate sortals. For details on these, see Sect. 2.8.
- 2.
That is,
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p & q =Def ∼ (p →∼ q),
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p ∨ q =Def ∼ (∼ p & ∼ q)
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p ↔ q =Def((p →∼ q) & (q → p)).
As the reader may have noticed, we have used in the metalanguage the primitive symbols or the defined ones as names of themselves. This in order to avoid the use of quotation marks, which would have made their listing more difficult to read.
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- 3.
The fact that we are assuming relative identity and not mentioning at all absolute identity in the set of primitive notions of the language does not mean that we are adopting thesis D of the sortal dependency of absolute identity. Instead, the goal was to explore the logical properties of relative identity independently of absolute identity. As explained in Sect. 1.7, we are not committing to or rejecting the validity of D, or its opposite.
- 4.
The semantics of this section, as well as that of Sect. 3.1 of Chap. 3, is derived from a semantics originally formulated in Cocchiarella (1977), for a language for sortals with the future, past, and the now temporal operators. The characterization of a formal system for such a semantics, as well as proof of its soundness and completeness regarding the logical validity of the semantics, were left as open problems. In this chapter, we prove soundness and completeness when the semantics in question is restricted to an extensional language for sortals, and in Chap. 3, when the language includes the future and past tense temporal operators. The last proof can easily be extended to cover the now operator, as well as some other temporal operators. We should finally note that the different semantic systems of Chaps. 4, 5, 6 and 7 and Sect. 2.7 are originally ours.
- 5.
- 6.
See Chap. 8, Sects. 8.2, 8.3, 8.4 and 8.5, for a discussion of how a theory of universals determines the range of the sortal term variables, and why, in the case of conceptualism, such variables could only be assigned, in a set-theoretic semantics, set-theoretic representations of the extensions of concepts.
- 7.
- 8.
- 9.
An alternative extensional formal logic of sortals has been stated in Stevenson (1975). This system is less general than the one we are characterizing in this chapter. For one thing, Stevenson’s system applies only to non-vacuous sortal concepts, that is, to sortal concepts whose extensions are not the empty set. For another, the system also assumes both that every possible object of the domain of discourse falls under a sortal concept and that there are ultimate sortals under which such objects must fall. Our extensional logical system ES together with its formal semantics is not committed to Stevenson’s assumptions. Since these assumptions are consistent with our logical system, Stevenson’s system can be easily obtained by adding to the semantics of ES set-theoretic conditions representing such assumptions, as well as by adjoining their syntactic representations to the set of axioms of ES.
- 10.
Clearly, this idea will be coherent with a view that excludes bare particulars from our ontology.
- 11.
John Locke might be interpreted as a conceptualist who assumes the existence of bare particulars, given his position regarding the substrata of common sense objects. For its part, Wiggins (2001) is a form of conceptualism that adheres to the opposite position.
- 12.
Relative to a given view of universals, a logic of sortals by just being a logical theory should not exclude any logical possibility regarding sortals and individuals. Given the conceptualist background theory, either we have cognitive access through sortal concepts to all individuals or not. If we do not have such access, either we have access to some individuals but not to all of all them, or to none of them. Both of these possibilities might be temporal or in principle. In case it is in principle, it might be due to our cognitive limitations or because there are bare particulars. All of the above possibilities are being taken into account in the formal semantics. The case where we have access through concepts to all individuals of the universe of discourse is considered in the semantics since it allows the possibility that S might be equal to the power set.
- 13.
For more on this stance, see Wiggins (2001).
- 14.
- 15.
We should point out that the entire idea of positing covering sortal concepts either for transitions of one life-cycle to another or for the entire career of an individual has been questioned. Elaborate arguments have been developed that purport to show that no concept is needed to keep track of an individual during its changes or modifications. Other cognitive mechanisms are thought to enable us to achieve the required re-identification. For more on this, see Ayers (1997) and Campbell (2002).
- 16.
The notion of a restriction of a concept in this sense was originally introduced by Peter Geach. See, for example, Geach (1980).
- 17.
For this reason, the extensional logic for sortals proposed in Stevenson (1975) focuses on non-vacuous sortal concepts only.
- 18.
This approach is adopted, for example, in Cocchiarella (1977).
- 19.
Conceptual necessity might be one of such possible interpretations.
- 20.
Clearly, as a way out, one might deny that the concept square circle and similar ones convey criteria of identity and so to be sortals. In any case, a general explanation would be needed to show the conceptual impossibility of necessarily vacuous sortal concepts before one can adopt the intensional definition in question.
- 21.
References
Ayers, M. (1997). Is Physical Object a sortal concept? A reply to Xu. Mind and Language, 12, 393–405.
Campbell, J. (2002). Reference and consciousness. New York: Oxford University Press.
Church, A. (1958). Introduction to mathematical logic. Princeton, MA: Princeton University Press.
Cocchiarella, N. (1977). Sortals, natural kinds and re-identification. Logique et Analyse, 80, 439–474.
Cocchiarella, N. (1986). Logical investigations of predication theory and the problem of universals. Naples: Bibliopolis Press.
Geach, P. (1980). Reference and generality. Ithaca: Cornell University Press.
Stevenson, L. (1975). A formal theory of sortal quantification. Notre Dame Journal of Formal Logic, XVI, 185–207.
Wiggins, D. (2001). Sameness and substance renewed. Cambridge: Cambridge University Press.
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Freund, M.A. (2019). An Extensional Logic for Sortals. In: The Logic of Sortals. Synthese Library, vol 408. Springer, Cham. https://doi.org/10.1007/978-3-030-18278-6_2
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