Defending Williamson

“Dennis Des Chene”:http://tlonuqbar.typepad.com/phfn/2004/11/modal_glitch.html criticises a comment Timothy Williamson makes in “Must Do Better”:http://www.philosophy.ox.ac.uk/faculty/members/docs/Must%20Do%20Better.pdf.

bq. The principle that every truth is possibly necessary can now be shown to entail that every truth is necessary by a chain of elementary inferences in a perspicuous notation unavailable to Hegel.

Des Chene says this isn’t right, because it’s not a theorem in most modal logics that MLp entails Lp. (Des Chene uses boxes and diamonds, not Hughes + Cresswell notation, but I can’t do that in HTML. M is diamond and L is box.) He also notes that it is a theorem in S5, and wonders whether Williamson is using that logic. In fact Williamson’s argument, which is just a variant on “Fitch’s paradox”:http://plato.stanford.edu/entries/fitch-paradox/ needs only KT, as I’ll show.

Rather than appeal directly to K, I’ll use three consequences of K. (In what follows -> is material implication and > is entailment.)

K1. If A > B then LA > LB
K2. If A > B then MA > MB
K3. It is a theorem that ~M(A & ~A)

Here’s the proof. (Rule R is the rule that whatever is true is possibly necessary.)

1. p & ~Lp Assumption for reductio
2. ML (p & ~Lp) (1), Rule R
3. L(p & ~Lp) > Lp K1
4. L(p & ~Lp) > L~Lp K1
5. L(p & ~Lp) > Lp & L~Lp (3), (4), &-intro
6. ML (p & ~Lp) > M(Lp & L~Lp) (5), K2
7. L~Lp > ~Lp T
8. Lp & L~Lp > Lp & ~Lp (7), truth functional logic
9. M(Lp & L~Lp) > M(Lp & ~Lp) (8), K2
10. M(Lp & ~Lp) (2), (6), (9)
11. ~M(Lp & ~Lp) K3
12. ~(p & ~Lp) (1)-(11), reductio
13. p -> Lp (12), truth tables

Since p is arbitrary it follows that whatever is true is necessarily true, as required.

The proof is obviously not original, but I think it’s so clever that it’s worth posting here, just for fun.

I don’t know whether you can significantly weaken the assumptions below KT and still get the proof to work. There’s an “expert on Fitch’s paradox”:http://www.arts.cornell.edu/phil/faculty/fara.html in the adjacent office who’d probably know, but he’s gone home for the evening. Anyone with suggestions can leave them in the comments.

Having said all that, Professor Des Chane “goes on to say”:http://tlonuqbar.typepad.com/phfn/2004/11/modal_glitch.html#add01 some interesting things about the conception of philosophy Williamson is advocating. He also, I’m pleased to note, includes a shout-out to “Matt Jones’s”:http://www.columbia.edu/cu/history/htm/h_faculty_profile_jones.htm very interesting work on the connection between mathematics and philosophy in the early modern period. There’s some interesting stuff there that you should pop over and read.