0

An implication is the compound statement of the form “if p , then q .” It is denoted $p \rightarrow q$.

If $p$ is false, the implication $p \rightarrow q$ is always true.

So the implication $0 =1 \rightarrow 3=6$ is true. Indeed, by assuming $0 =1$, one can SHOW $3=6$ as follows:

$0=1 \Rightarrow 0+1=1+1 \Rightarrow 1=2 \Rightarrow 3 \times 1= 3 \times 2 \Rightarrow 3=6$.

Here is my question: by assuming $0=1$, can one ACTUALLY PROVE any given mathematical statement?

For example, by assuming $0=1$, can one SHOW that Fermat's last theorem is false?

Basics
  • 321
  • Yes. Taking false premise whole implication becomes true for any conclusion. – zkutch Mar 06 '24 at 10:38
  • "For example, by assuming 0=1, can one SHOW that Fermat's last theorem is false?" Obviously YES; but in order to PROVE that Fermat's last theorem is false you have to PROVE that 0=1. Are you able to do this? – Mauro ALLEGRANZA Mar 06 '24 at 10:47
  • Wrong. If the antecedent of a implication is false, we CANNOT infer that the consequent is necessarily true (or necessarily false). We CAN, however, infer that the IMPLICATION itself is true. In daily discourse, we rarely if ever consider the truth value of implications with false antecedents. It is not, however, uncommon in very technical arguments such as mathematical proofs. – Dan Christensen Mar 06 '24 at 15:22
  • The implication "$0=1$ implies FLT is FALSE" is true. But so is the implication "$0=1$ implies FLT is TRUE." – Dan Christensen Mar 06 '24 at 15:31
  • The Principle of Vacuous Truth can be stated as $\neg A \implies (A\implies B)$ for any logical propositions $A$ and $B$, regardless on their truth values. – Dan Christensen Mar 06 '24 at 15:36

0 Answers0