# Can we logically analyze mathematical theorems as if-then statements?

Many theorems in math have an if-then form. For example: "If a polynomial is of $n^{th}$ degree, then it has $n$ roots. In my other question, I learned that in order to analyze statements using truth tables, the statements must be completely independent. However, I'm not sure that anything in math is independent. Everything can be proven from the axioms as far as I know, so if $p \implies q$, the truth value of $p$ automatically determines the truth value of $q$. So if the statement "$f$ is polynomial is a degree $5$" automatically makes the statement $f$ has $5$ roots", the truth table seems nonsensical. Do mathematical if-then statements have anything to do with the classical if-then statements from logic?

$$p \implies q$$

$$\begin{array}{|c|c|c|} \hline p&q&p\implies q\\ \hline T&T&T\\ T&F&F\\ F&T&T\\ F&F&T\\\hline \end{array}$$

• Why do you think that the truth table is non-sensical ? It shows false, if $p$ is true AND $q$ is false. In the other cases, it shows true. What should be non-sensical ? – Peter Jun 4 '16 at 18:12
• Additionally, there are true, but unprovable statements in mathematics. – Peter Jun 4 '16 at 18:14
• yes , if a then b else c; is translated to a*b+not(a)*c – Abr001am Jun 4 '16 at 18:27
• @Peter Well from the answers of the question linked in the OP, I understood that my truth table was nonsensical because $p$ and $q$ were not independent. $p \implies q$ is true when $p=F$, $q=T$, but in the example that I gave, $(F \implies T)=T$ was nonsensical, and the reason was that $p$ and $q$ were not independent. – Ovi Jun 4 '16 at 18:30
• @Agawa001 Does "*" mean "$\implies$" and the $+$ mean "or"? – Ovi Jun 4 '16 at 18:34

## 1 Answer

Do mathematical if-then statements have anything to do with the classical if-then statements from logic?

Yes; they use "the same" if-then.

Everything can be proven from the axioms

or, more precisely, in a mathematical theory every theorem is proved from the axioms.

This amount to:

if the axioms are true, then also the theorems are.

What happens if the axioms are not true? Well, the conditional still holds, but we are losing our time with a "wrong" mathematical theory.

A well-know quote from Bertrand Russell, The Principles of Mathematics page 3:

1. Pure Mathematics is the class of all propositions of the form “$p$ implies $q$”, where $p$ and $q$ are propositions containing one or more variables, the same in the two propositions, and neither $p$ nor $q$ contains any constants except logical constants.

Examples of arithmetical theorems that are conditionals with false antecedent.

1) "if $2$ is odd, then $2=1$".

Proof

We define : $Even(n) := \exists z (n = z \times SS0)$.

Finally, we define: $Odd(x) := \lnot Even(x)$.

From Peano axioms it is easily proved that: $Even(SS0)$, i.e. $Even(2)$.

Thus, having proved $\lnot Odd(2)$, using the tautology:

$\lnot A \to (A \to B)$,

with $Odd(2)$ in place of $A$ and $2=1$ in place of $B$, by modus ponens we conclude with:

$Odd(2) \to 2=1$.

2) "if $2$ is odd, then $2=2$".

Proof

From equality axiom: $\forall x (x=x)$, we get: $2=2$.

Thus, using the tautology:

$A \to (B \to A)$,

with $2=2$ in place of $A$ and $Odd(2)$ in place of $B$, by modus ponens we conclude with:

$Odd(2) \to 2=2$.

• So my understanding is that when we are given a theorem in math as an $p \implies q$ statement, we are being told "This $p \implies w$ statement is always true, meaning that we can possibly find every combination of $p$ and $q$ except for $p=F$ and $q=T$"? I guess what I'm not really sure I get why my example here (math.stackexchange.com/questions/1793713/…) does not work (would really appreciate it if you could take a look). As an answer to that question, people said $p$ and $q$ were not independent. But what does that mean? – Ovi Jun 4 '16 at 20:43
• ...Does it mean that it's not valid to put $p$ and $q$ in a $p \implies q$ relationship? Does it mean it's to put in in $p \implies q$ as long as $q$ as long as $q$ is true? – Ovi Jun 4 '16 at 20:44
• @Ovi - in math usually we are not interested into conditional whatever like "if $0=1$, then the moon is round", but to theorems proved from axioms. Thus, the conditional we are using are like: "if axiom $A$ holds, then theorem $T$ holds also". The fact that the conditional is true also when $A$ is false is a fact than we accept but of course is of no relevance for the mathematical theory we are workin whit. – Mauro ALLEGRANZA Jun 5 '16 at 7:36
• Thanks a lot, your last comment (especially the last sentence) helped A LOT! – Ovi Jun 5 '16 at 15:50