Tuesday, September 8, 2009

Genuinely propositional logic

These days, it is common to develop logic by positing a logical language (e.g., First Order Logic) and then giving various rules. But there is another approach, one that I've been told was that of Russell and Whitehead. On this approach, what we are studying is the structure of the space of propositions, which we understand realistically as good Platonism.

If I were enough of a Platonist, I would want to do this myself. Here is how I would do it. First step, ontology. I take as the basic kind of entity an n-ary relation, where n is any non-negative integer. You might wonder what 0-ary relations and unary relations are: they are more familiarly known as propositions and properties, respectively. Nonetheless, they seem pretty clearly a part of the same range of entities as binary, tertiary, quarternary, ... relations.

Now I distinguish certain functions. I shall consider a (partial—but I shall suppress that word) function f to be an n-ary relation where n>1 such that if f(x1,...,xn−1,xn) and f(x1,...,xn−1,xn*), then xn=xn*. I shall write xn=f(x1,...,xn−1) for short. The first three functions I distinguish are Conj, Disj and Neg. (Or maybe just a Nand.) These satisfy the formal relations:

  1. If Conj(p,q)=Conj(p*,q*), then p=q* and q=p*, or p=q* and q=p*. If Disj(p,q)=Disj(p*,q*), then p=p* and q=q*, or p=q* and q=p*. If Neg(p)=Neg(p*), then p=p*.
(I.e., Conj and Disj are one-to-one except perhaps for order, and Neg is one-to-one.) There are some other special relations. For any function z from {1,...,n} to {1,...,m}, there is a function Pz from the m-ary relations to the n-ary relations. These have the formal property that
  1. Pz(Pw(p))=Pwz(p) where wz is the composition of w and z, and Pz(p)=p if w is the identity function.
For any n>0, there are one-to-one functions Un and En from the m-ary to the (m−1)-ary relations for any m greater than or equal to n.

Finally, for every n, there is an (n+1)-ary relation Sn, which relates n objects with one n-ary relation. This relation has formal properties like these:

  1. Sn(x1,...,xn,Conj(p,q)) if and only if Sn(x1,...,xn,p) and Sn(x1,...,xn,q).
  2. Sn(x1,...,xn,Disj(p,q)) if and only if Sn(x1,...,xn,p) or Sn(x1,...,xn,q).
  3. Sn(x1,...,xn,Neg(p)) if and only if not Sn(x1,...,xn,p).
  4. Sm−1(x1,...,xm−1,Un(p)) if and only if Sm(x1,...,x,...,xm-1,p) for all x, where the x is in the nth position.
  5. Sm−1(x1,...,xm−1,En(p)) if and only if Sm(x1,...,x,...,xm-1,p) for some x, where the x is in the nth position.
  6. If z is a function from {1,...,n} to {1,...,m} and p is an m-ary relation, then Sm(x1,...,xm,Pz(p)) if and only if Sn(xz(1),...,xz(n),p).
Next, we have a crucial non-formal condition:
  1. Sn(x1,...,xn,p) if and only if x1,...,xn stand in the relation p. (If n=0, we have: S0(p) if and only if p is true.)
Finally, if we so wish, we can add some relations with language, such as that when sentences "s" and "t" express p and q respectively, then "(s) and (t)" expresses Conj(p,q). Etc. But this is just an afterthought, and should not be taken as a definition of Conj, since there presumably are propositions that are not expressed by sentences, or at least sentences of any human language.

Open wffs correspond to n-ary relations with n>0. Sentences correspond to 0-ary relations, or propositions.

Here is one interesting corollary of this way of seeing logic. Because propositions are just 0-ary relations, it would be weird to have a metaphysics with sparse relations and properties, but with propositions—which are surely abundant! If we have abundant propositions that correspond to sentences, surely we want something abundant that corresponds to wffs.

Another advantage of doing things this way is that we can uniformly handle infinite propositions.

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