Abstract
In this chapter, we relate the different formal logical systems characterized in this work to nominalist and realist approaches to sortals. We also discuss the use of the absolute identity sign and the absolute quantifiers in the formal semantics. We argue that the way those two logical expressions occur in the semantic clauses cannot be necessarily interpreted as a reduction, by the formal semantics, of sortal identity and first-order sortal quantification.
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Notes
- 1.
This is important given the fact that formal elements of a particular conceptualist second-order logic, which represents holistic conceptualism, can also be used to represent logical realism. For details, see Cocchiarella (1986).
- 2.
- 3.
We are not here specifying, and it is not necessarily to do so, whether by a predicate constant we mean a type or a token. Nominalism might favor the latter, given its strong commitment to individuals and rejection of abstract entities.
- 4.
In this tentative definition of satisfaction, the clause for relative identity may be the following: \(\models _{\mathfrak {A}}x=_{S}y\) iff \( \mathbb {A}(x)=\mathbb {A}(y)\) and \(\mathbb {A}(y)\in \ast (S),\) where ∗ = f \(\circ \mathbb {A}\) (i.e., the composition of f and \(\mathbb {A})\).
- 5.
For a representative theory of natural realism, see Armstrong (1980).
- 6.
Although we should keep in mind that a version of conceptualism exists that would adhere to clause 4, a version that we have denominated conceptual sortalism about individuals. No variant of conceptualism would exclude the empty from \(\mathbb {S}\). Thus, clause 3 will never apply to a conceptualist semantic model.
- 7.
- 8.
See Plato’s Phaedo 73A–81A and Republic 507B–507C in Hamilton and Cairns (1961).
- 9.
See Aristotle’s Categories 11 (14a8-10) in McKeon (1941). For a recent version of Aristotelian realism, see Lowe (2009). As far as concrete individuals are concerned, two of the most important theses of Lowe’s theory are that (1) every kind (or sort) is instantiated by some individual, and (2) every individual is of some kind (or sort) or other.
Kinds or sorts of individuals might be natural and non-natural in Lowe’s theory. Thus, the above theses apply to sortal predicates for natural kinds as well as for artifacts and abstract objects. Clearly, Lowe ´ s theory excludes the possibility of bare particulars.
- 10.
Lowe (2009) illustrates this form of a realism.
- 11.
- 12.
- 13.
- 14.
We should add that an important philosophical problem would be encountered if one were to attempt a reduction, on the basis of the above equivalences, of sortal identity to set-membership cum absolute identity. The reduction could not be carried out unless one would be willing to concede an infinite regress as a rationally acceptable consequence. This regress will be propelled by the fact that an assertion of set-membership will presuppose the notion of predication itself, that is, in order for the assertion to be true, one would have to predicate set-membership of one entity into another. This sort of infinite regress can also be generated by a reduction of predication to instantiation, application or falling under a concept. See Loux (1998) for a discussion on the infinite regress arguments in connection with the theory of universals.
- 15.
This is usually called pure ZF-theory.
- 16.
- 17.
This is sometimes called ZF with urelements.
- 18.
- 19.
- 20.
Jean Piaget’s theory of cognitive development points in this direction.
- 21.
Following Wittgenstein’s Tractatus metaphor, we might view the set-theoretical framework together with its absolute notions as a sort of theoretical ladder, whereby we can come to grasp the content of the formal logics for sortals. Once we are able by their means to climb to this level of understanding, we can kick such a ladder. See Wittgenstein (1998), Proposition 6.54.
- 22.
Such as Galileo’s comparison of a moon falling out of its orbit to a rock dropped from the mast of a moving ship or Rutherford’s comparison of the atom to the solar system, the analogy of sound propagation in air to the propagation of waves in water, and the hydraulic model of electric circuitry.
- 23.
Think of the concept of absolute identity itself. Its extension is intended to cover all objects. However, there is not a set of all objects in the ZF’s universe. We might want to extend ZF by adding classes, such as it is done in Von Neumann (1925). Nevertheless, we shall still find concepts whose extensions are not classes. For example, the extension of the concept class is not itself a class. Nor will the extension of absolute identity, understood as covering all objects, be a class (in Von Neumann’s sense) since if it were, it will be a member of itself, which is impossible. That is, we know that such an extension is not a set. Then, it will have to be a proper class, which means that it cannot be a member of a class, in particular, of itself. For a details on proper classes, see Maddy (1983).
- 24.
Although it is controversial the extent to which concepts for numbers are sortals, consider, for example, the concepts of ordinal and cardinal number. If they were sortal concepts, they could not be considered in their entire extensions since these extensions are not sets, and so cannot be subsets of a given universe of discourse. For a discussion of the concept of number as a sortal, see, for example, Wright (1983) and Grandy (2016).
- 25.
For a discussion on this topic, see Fitting (2015).
- 26.
As already pointed out in Sect. 8.1, this is a problem that can be generalized to all sorts of intensional logics. While some think that, at some point, a full extensional representation of intensionality will be achieved, there are others who think otherwise. A simple example where the extensional approach is problematic is that of the meaning of tautologies. Under the classical intensional set-theoretic approach, tautologies get assigned the same set-theoretic function, even though they have different meanings. Work has been done to overcome this problem, such as that of Lewis (1970). Lewis solution, however, presents problems in belief contexts. For details on how the problem of tautologies in such contexts is dealt with by a set-theoretic semantics, see Bäuerle and Cresswell (2003), and Fox and Lapin (2005).
- 27.
The formal semantics of monadic predication sets an equivalence of such a predication to membership in the set representing the extension of the predicate. Through this equivalence, as well as the semantic clause for sortal identity, one might infer the equivalence of sortal identity with the conjunction of monadic predication and absolute identity. This is the reductive proposal that has been advanced by some, as noted in Chap. 1.
- 28.
To continue with the example of Footnote 26, think of the logic of tautologies with respect to a fine-grained representation of their meaning. The fact that more factors are needed to set-theoretically represent tautologies, so as to be able to express differences in their meanings, does not nullify the logical results achieved in the different sorts of intensional logics where such differences are not set-theoretically made, such as in standard modal logic. But also consider the case of logical validity for truth-functional propositional logic with respect to logical validity of first-order logic. The fact that the former theory does not capture aspects of the latter does not invalidate the completeness and soundness results of truth-functional propositional logic.
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Freund, M.A. (2019). Final Considerations. In: The Logic of Sortals. Synthese Library, vol 408. Springer, Cham. https://doi.org/10.1007/978-3-030-18278-6_8
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