Finding the WFF of a Truth Table: Strategies Explained

[E92M3]

Hi, I'm new to logic. I can fill in a truth table if given a wff but can't do it the other way around. What is the general strategy in finding the wff of a truth table?

 

[Werg22]

There might be more than 1 wff for a given truth-table, but here's how to find one:

For each row where the wff must evaluate to true, form the sentence P1 ^ P2 ^ ... ^ -Q1 ^ -Q2 ^ ... where P1, P2, ... are the atomic propositions set to true in that row, and Q1, Q2, ... those set to false. Then take all these sentences and "or" them, e.g.:


P Q R wff
T T T T -----> P ^ Q ^ R
T T F F
T F T T -----> P^ -Q ^ R
F T T F
T F F F
F T F T -----> -P ^ Q ^ -R
F F T F
F F F T -----> -P ^ -Q ^ -R

Then we get (P ^ Q ^ R) V (P^ -Q ^ R) V (-P ^ Q ^ -R) V (-P ^ -Q ^ -R)

Hopefully you can see why this works.

 

[E92M3]

But what do you mean by "^"? is it &? How can you have (P^Q^R)? I Thought this can't be a wff. Shouldn't it be like ((P^Q)^R) since two two place connective will require two sets of breckets. Also does -Q mean ~Q?

 

[Werg22]

Yes, that's what I mean, ^ is &, - is ~.

Note that (A & B) & C = A & (B & C) for all valuations, so A & B & C is unambiguous, semantically speaking. But if you wish, you can put brackets to make it a syntactically correct wff (it's the same story with the or's).

 

[E92M3]

Then how can I shorten those wff to make them cleaner?

 

[Werg22]

Well, for example in

(P ^ Q ^ R) V (P^ -Q ^ R) V (-P ^ Q ^ -R) V (-P ^ -Q ^ -R)

you can see that whenever P and R are true, the sentence will be true (the truth value of Q doesn't matter). It's the same whenever -P and -R are true. So I can shorten this as:

(P ^ R) V (-P ^ -R)

Those are the kind of simplifications that you can hope for when dealing with these things.

Edit:

Another way to find wff, is: for each row where the wff must be false, "and" the atomic sentences (negating them if false), "or" all the sentences you obtain, then finally negate the sentence you get. This is preferable if you have more occurrences of true than false, for example

P Q wff
T T T
T F T
F T F
F F T

you get -(-P ^ Q).

You can always try to play with De Morgan's laws to simplify the expressions. You can also try expressing things with other conectives e.g. -(-P ^ Q) = -P -> -Q