# A truth table method of decision for modal logic

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Wed 23 Dec, 2009 04:05 am
If we grant that there are 4 truth values: T = logical truth, t = empirical truth, f =empirical falsity, F = logical falsity.

Not:
~T=F, ~t=f, ~f=t, ~F=T.

Or:
TvT=T, Tvt=T, Tvf=T, TvF=T, tvT=T, tvt=t, tvf=T, tvF=t
fvT=T, fvt=T, fvf=f, fvF=f, FvT=T, Fvt=t, Fvf=f, FvF=F.
Necessity:
[]T=T, []t=F, []f=F, []F=F.

Definitions:
<>p =df ~[]~p.
(p -> q) =df ~p v q.
(p & q) =df ~(~p v ~q).
(p <-> q) =df ((p -> q) & (q -> p)).

All of the theorems of the 2-valued logic are tautologous here.
The aditional axioms of S5 are given..

http://mally.stanford.edu/S5.html

1. [](p -> q) -> ([]p -> []q).
2. []p -> p.
3. <>p -> []<>p.

2. []T -> T, []t -> t, []f -> f, []F -> F. Each are tautologous by the above.
3. <>T -> []<>T, <>t -> []<>t, ,<>f -> []<>f, <>F -> []<>F. Each of these are also tautologous.

1. p q | [](p -> q) -> ([]p -> []q))
....T.T...T.....T.......T....T....T..T
....t..T...T.....T.......T....F....T..T
....f..T...T.....T.......T....F....T..T
....F.T...T.....T.......T....F....T..T
....T.t....F.....t.......T....T....F...F
....t..t...T.....T.......T....F....T...F
....f..t...F......t.......T....F....T...F
....F..t...T.....T.......T....F....T...F
....T..f...F.....f........T....T....F...F
....t...f...F.....f........T....F....T...F
....f...f...T.....T.......T....F....T...F
....F..f...T.....T.......T....F....T...F
....T..F...F.....F.......T....T....F...F
....t..F....F.....f.......T....F....T...F
....f..F....F.....t.......T....F....T...F
....F..F...T.....T......T....F....T...F

That is, all of the axioms of S5 are tautologies.

Doesn't it follow that all theorems (tautologies) of S5 are show to be the case by this method?

mickalos

Wed 23 Dec, 2009 12:34 pm
@Owen phil,
Assuming that 'tvF=f' is a mistake, you could show the theorems of the system to be true using truth tables like that, but why on earth would you want to? Truth tables are cumbersome and unwieldy enough enough with two truth values, never mind four. Semantics like this also raises the question of how one is to define validity. You can't simply say an argument is valid if there is no interpretation under which it's premises can be true while the conclusion is false, because then we get fallacies like the one in your other thread about the number of planets. Nor does it provide us with a very intuitive way of thinking about logical truth, or modality. For example, we don't normally think of logical truths as being primary, but as being a subset of all true formulas (the subset that is true under any assignment of truth values). Nor do we normally consider 'It is necessary that...' and 'It is possible that...' to be truth functions.

Much better to use possible world semantics. Massively simplified, we think of the ordinary first order formula as being true or false in a given possible world, and the modal operators act as quantifiers over the set of possible worlds.

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