Using 0/1 instead of T/F in propositional logic. Is there any interest in doing so? ( either at the language level or at the metalogical level)

Question

Is there any interest in using 0/1 instead of T/F in propositional logic?

Does it allow things the T/F notation doesn't?

Does it make easier or simplyfy in any way the exposition of logical theory?

Answer 1

I partially disagree with Arthur. If you are using $0/1$ instead of $T/F$. Then so long as you are in two valued logics then both are same for practical purposes.

However, using $0/1$ instead of $T/F$ has some other advantages. I will mention only two.

• First, whenever you use $0/1$, a natural idea would be to extend this truth-value set to more than two values. The idea of having more than two values comes arguably more naturally if you are working with integers values than symbols.

• Second, notice that whenever you use the notation $0/1$ , you immediately get a natural order between them. This is not so obvious in case of $T/F$. So this also naturally rises the question whether the "truth-value set" can be seen as a poset.

Answer 2

Changing the names of variables rarely affects anything. It may help you to think about what happens, but it doesn't change any formal properties.

Answer 3

Using $0$ and $1$ as truth values allows us to reduce logic to arithmetic. So if $t(P)$ is the $0/1$ truth value of proposition $P$ then

$t(\lnot P) = 1-t(P) \mod 2\\t(P \land Q) = t(P)t(Q) \mod 2 \\ t(P \oplus Q) = t(P) + t(Q) \mod 2\\t(P \lor Q) =t(P)+t(Q)+t(P)t(Q) \mod 2$

Then if we know that properties such as double negation, commutativity and associativity hold in the realm of arithmetic, we can immediately derive equivalent properties in the realm of logic.