Why do we need formal logic proofs when we have truth tables?

Question

Truth tables are capable of demonstrating the validity of a formal logic statement. This can be done extremely quickly using technology and it doesn't require any high-level proof rules (everything can be calculated on a fundamental level).

In comparison, proofs require a human (or a much more sophisticated algorithm) and they can be complicated to develop.

Assuming a computer is available, why do we need to prove validity with proofs when we could just use computers to quickly generate truth tables?

Just to be clear, I'm not complaining about logic proofs. I think that they are beautiful and I want to know if they are better than truth tables in any practical way. — DivideByZero, Mar 02 '20 at 11:17
Define practical: Do you mean 'it gets me an answer' or do you mean 'it enhances our understanding of mathematics' — tomi, Mar 02 '20 at 11:23
Boolean logic (albeit useful) is only a tiny part of the huge world of mathematical logic : https://www.encyclopediaofmath.org/index.php/Mathematical_logic — Jean Marie, Mar 02 '20 at 11:33

score 7 · Accepted Answer · answered Mar 02 '20 at 11:32

In addition to the other answers, which correctly point out that truth tables only work for weak types of logic, let me address

This can be done extremely quickly using technology

In fact truth tables get very large, very quickly: if you have $n$ propositional variables, then there are $2^n$ rows in the table to check. If you have a sufficiently complex statement, with many variables, this will be too many to feasibly do on a computer. (You would start hitting this limit before $n$ hits 100.) In that case, using human insight into the actual problem encoded in the logical statement to produce a proof can very well be much faster than providing a truth table.

(Although it also deserves note that, unless $\mathsf{P}= \mathsf{NP}$, there is no fast algorithm for producing proofs in general, so they are not fundamentally better than truth tables; and if you believe (as many, including me, do) that humans are bound by computability restrictions, then no human could asymptotically outperform truth tables in all cases either.)

score 5 · Answer 2 · answered Mar 02 '20 at 11:18

5

How would you use a truth table to prove the existence of an infinite number of primes?

answered Mar 02 '20 at 11:18

tomi

9,594

score 5 · Answer 3 · 2020-03-02T11:30:18.413

5

Propositional calculus theorems can indeed by proven by truth tables, provided the number of variables is not too large (this is the normal case).

First-order logics cannot.

edited Mar 02 '20 at 11:30

answered Mar 02 '20 at 11:23

@MeesdeVries: you are right, my bad. – Mar 02 '20 at 11:30

score 4 · Answer 4 · answered Mar 02 '20 at 11:19

4

How do you propose you prove that the function $f(x)=x$ is continuous using only truth tables? In other words, how can you prove

$$\forall x_0\forall \epsilon > 0\exists \delta\forall x: |x-x_0|<\delta\implies |f(x)-f(x_0)|<\epsilon$$ using nothing but truth tables?

answered Mar 02 '20 at 11:19

5xum

123,496
6
128
204

3

It suffices to try all quadruples $(x_0,\epsilon,\delta,x)$. :-) – Mar 02 '20 at 11:25
1

@YvesDaoust I'd challenge the op to count the number of rows in that truth table :) – 5xum Mar 02 '20 at 11:26
He can do it "in the end". – Mar 02 '20 at 11:28

score 2 · Answer 5 · answered Mar 02 '20 at 11:22

2

A good proof will give you an insight into the problem. A truth table will not do that very easily. Sure, you might see a row of 0s or 1s that indicates some underlying truth, but it is not beautiful.

answered Mar 02 '20 at 11:22

tomi

9,594

Why do we need formal logic proofs when we have truth tables?

5 Answers5