Abstract
As we have concluded at the end of the section 1.7, if we interpret classical quantifiers, individual constants, and generalised quantifiers of any known kinds as subsets of P(D), then the semantically unexploited subsets of this set can still remain. Let us assume then that they can serve as interpretations of the extended (beyond quantifiers and constants) term category. Now suppose, for example, that t is a term such that I(t) = {set of idlers, set of students}. Then it turns out that t (or, more precisely, the entity represented by t) possesses, in the sense determined by the truth condition (2) in 1.6, the following properties: being lazy, being a student, being a human, etc. At the same time, it does not possess many other properties, like being a cat, being a girl, being handsome, being not handsome, etc. So t may be identified as lazy student (we do not put here any article since neither does justice to the meaning of the term). In a similar way we may create more objects, and even, in a sense, every thinkable and nameable object, and this procedure will perfectly agree with the bundle theory of objects. If we once admit that an individual may be represented by (or even identified with) a set of properties, why not allow every set of properties to represent an object?
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Notes
We can define the set of formulas in a simpler way by replacing conditions (c) and (d) with a single condition: if A is a formula then txA is a formula. However, it is important for us to mention the predicate category explicitly.
To introduce the existential quantifier “officially” we should consider it as a primitive term of M-language and, additionally, adopt the following additional axiom: -VxAD3x-A (the reverse implication, as we shall see, follows from the axioms: thesis M11).
We adopt several conventions which will be frequently used in proofs or in descriptions of proofs in order to make them more clear: (a) derived rules will be designated by respective axioms or theses; (b) numbers in parentheses will indicate numbers of proof lines to which the respective rules were applied; however, there will be no number if those were applied to the line immediately preceding the current one; (c) derivations based on the sentential calculus, truth functional derivations, will be indicated by Ml or will not be indicated at all; (d) we will not mention the use of very common rules just giving the number of line to which they were applied; (e) in order to further to shorten proofs we will use compound derivations, if e.g. FADB then according to MG-M2 we get txADtxB. In the case where the order of the application of rules is unimportant, we will use a comma rather than a dash; (f) when nothing follows a thesis it means that the proof is trivial; sometimes we mention only numbers of theses from which the proof immediately follows. The use of these conventions will be flexible, adjusted to formal complexity of a given proof.
To see how far the duality enables us to shorten proofs, we present the proof of M9 not appealing to this property of M-system:
See our discussion of monotonicity in the previous chapter.
I.e. for any set XcP(D), the maximal set is X={YcD: 3Z,XZcY}. It should be noticed that the minimal set with respect to the relation = does not alwayes exist. E.g. assume that D=N (the set of natural nukmbers) and consider the following subset of P(D): (XcD: V„, N X={n,n+1,...}}.
Although M„-formulas coincide with M-formulas we write “M„-formulas” to indicate their narrower interpretation.
In particular, the following M-formulas are theses of M*-logic: The formulas can be understood as saying that if a term t is not an individual constant (for individual constants we have: tx - A = - txA) then the term either coincides with or with `d, i.e. it is not distinguishable from the quantifiers with respect to predication (recall that the reverse implcations: txBD∃xB and VxBDtxB are theses of M-system).
We still mention here “∃” as merely an abbreviation of” -Vx n“. Instead of ”∃ we may put any other term but this one seems to be the most natural.
Since B does not contain any free variable besides x,we do not relativise I to an assignment.
Notice that generally Ct)[xBJ* - (t[xB]) and 3[xB]=3 for every xB such that 1(xB)D.
They will coincide with the classical restricted quantifiers on the ground of the free M-logic, see section 4.1.
Notice that as far as y#B then [xB] and [yB(y |x)] are indistinguishable in the sense of M5.
Cf. da Costa, Doria, Papavero [ 1991 ].
Actually tx(B1,…,B n ) is a more general syntactic and semantic notion of definite description than the Russellian one. Notice that only one formula can define the Russelian description whereas the description in M-logic is defined by a sequence of formulas and tx(B1,…,B n ) does not coincide with tx(B,A…AB„).
When This description is introduced to the free version of M-logic then the case when existence or uniqueness is lacking is the same as in the standard theory of description, i.e. all predications turn out to be false. See section 4.1.
Also, if DM2 is already present in M*-logic then the logic can dispense with one of its specific axioms: M*1 or M*2 since the two axioms, as one can easily notice, are dual one to another.
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© 1998 Springer Science+Business Media Dordrecht
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Paśniczek, J. (1998). M-Logic. In: The Logic of Intentional Objects. Synthese Library, vol 269. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8996-3_3
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