Page 307 - Encyclopedia of Philosophy of Language

a range of proposals from conventionalism at one extreme (Wright 1980; Sidelle 1989) to unqualified realism at the other (Lewis 1986b).
2. Mathematical and Logical Necessity
According to the mathematical Platonist, math- ematics is the study of certain abstract objects such as numbers and functions. These objects exist necessarily and their intrinsic properties, as well as their relations amongst themselves, are also necessary: it is necessary that the number 3 exists, necessary that it is prime, and necessary that it is greater than the number 2. For a mathematical Platonist, mathematical necessity is sui generis, deriving from the peculiar nature of the subject matter of mathematics.
The most important movement in modern phil- osophy of mathematics has been the logicist move- ment, founded by Frege and Russell (Frege 1986; Russell 1919). The logicists proposed to reduce math- ematics to logic (so they were not 'mathematical' Pla- tonists), but the actual reductions which Frege and Russell produced were to set theories and type theories which it would require exceptional generosity to class- ify as logic. For these theories have substantial exis- tential import (a set theory to which number theory reduces has only infinite models), and it is hard to see how logic alone, or logic and definitions, could give rise to any existential consequences whatsoever; even classical logic's theorem 'something exists' is some- what objectionable, and a free logic where it fails is preferable. So the logicism of Frege and Russell does not provide a good reason for thinking that math- ematical necessity is a species of logical necessity.
However, there is an interesting contemporary vari- ant of logicism to which the objection from unwanted existential import does not arise. Hartry Field has argued that mathematical necessity is a species of logi- cal necessity, but not because mathematical truth reduces to logical truth. Field is an instrumentalist about mathematics, and holds that the statements of mathematics are not bearers of truth and falsity at all. However, there is such a thing as mathematical knowledge. Mathematical knowledge is knowledge of the form // is logically consistent (possible) that A where A is the conjunction of the axioms of a math- ematical theory, or of the form /'/ is logically necessary that if A then B, where B is some sentence which follows from the axioms A. Field elaborates these basic ideas to accommodate nonfinitely axiomatized theories and develops a modal logic for the logical possibility and necessity operators (Field 1989). On this approach, then, one can trace the appearance of necessity in a theorem B to the necessity of the conditional if A then B. But clearly, there is a con- troversial question whether this approach is successful in the special case where B is one of the conjuncts of A (i.e., an axiom). That is, can the apparent necessity of ifx andy have identical successors then x andy are
identical be explained merely by its being a conse- quence by &-Elimination of the collection of logically consistent statements that make up Peano's axioms for arithmetic?
3. Some Metaphysical Necessities
Logical necessity narrowly construed allows as possi- bilities some things which seem in a substantial sense to be impossible; one says they are metaphysically impossible. For example, it is impossible that an object which is red all over at a time t is also green all over at t, but logic alone cannot discover a contradiction in the hypothesis that it is both. More interestingly, it also seems impossible that I, the present writer, could have been a musical score, or a tree, or an insect, rather than a human being: the kind of thing I am seems to be 'essential' to me (Wiggins 1980). An essen- tial property of an object x is a property x could not have lacked except by failing to exist; so it is necessary that if x exists, it has the property. The interesting examples of impossibilities not precluded by logic (or mathematics) are examples of this sort, where we have the intuition that there is no genuinely possible way things could have gone (no genuinely possible world) in which the relevant object x lacks the property. These examples involve specific objects or kinds of objects, and for that reason the necessities have been traditionally known as 'de re' necessities, necessities concerning objects.
Following on work of Kripke (1980), two of the most widely discussed de re necessities are the 'necess- ity of identity' and the 'necessity of origin.' Suppose the Superman story is fact. Then Clark Kent = Super- man. Is this a necessary or contingent fact? There is some impulse to say it is contingent, but Kripke argues that this is a confusion. It is true that, in advance of detailed investigation, what someone knows may be 'consistent' with Superman being someone other than Clark Kent, but this is merely an epistemic possibility and is therefore irrelevant to the question whether, granted that Superman in fact is Clark Kent, there is a genuinely possible world in which Superman is not Clark Kent. There is certainly a world in which Super- man is not the person called 'Clark Kent,' but that is quite different from not being Clark Kent; after all, there is a world in which Clark Kent is not the person called 'Clark Kent.' The strongest argument that there is no possible situation in which Clark Kent is not Superman is that such a situation would be one in which, per impossibile, Superman is not Superman. For if Superman and Clark Kent are the same person, anything possible for one is possible for the other, so if not being Clark Kent is possible for Superman, not being Clark Kent is possible for Clark Kent. And such a possibility is hard to understand.
There is no comparably simple argument for the necessity of origin, according to which a particular organism O which actually originates from a cell C
Necessity
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