Q&A: Axioms and Self-Contradiction
Axioms and Self-Contradiction
Question
Logic describes a valid argument as follows: "If there is no situation in which all the premises are true and the conclusion is false, then the proof is valid."
In your opinion, is an argument like this:
Premise 1: Kobi and Ofra are married, but they never got married.
Premise 2: Yossi went to the sea.
Conclusion: Yossi knows how to fly
valid?
After all, it meets the definition (there really is no situation in which the premises are true but the conclusion is false—because the first premise is self-contradictory, which makes the premises not true!). Or is the first statement, with its self-contradiction, rejected from entering the logical system in the first place?
Answer
That is not a true statement but a valid argument. An argument whose premises are contradictory is always valid (this is what people mean when they say that from a contradiction, anything can be inferred). The validity is, of course, purely formal (what is called "material").
Discussion on Answer
This is logical formalism. Follow the definition: there cannot be a situation in which the premises are true and the conclusion is false.
So the argument is valid only because of a falsehood? When there is a contradiction (which we regard as falsehood) in the premises, does that teach me every falsehood there could be? But the content of the falsehood also has to be valid, doesn't it?
Thanks for the quick and courteous reply at this late hour!
As I wrote, formally, from a contradiction anything can be inferred.
Could the Rabbi please explain how, formally, everything can be inferred from a contradiction?
It was explained above. I'll summarize briefly: a valid argument is one in which it is impossible for the premises to be true and the conclusion false. In such a case we say that the conclusion follows from the premises.
In any argument where one of the premises is a contradiction (= always false), it is impossible for the premises to be true. And from that it follows that it is also impossible for the premises to be true and the conclusion false.
Hence the conclusion follows from the premises.
A shy person does not learn, so I'll dare ask: how do we determine that a certain statement is a conclusion of the premises that preceded it? Thank you very much for the patience.
It was explained. If it is impossible for them to be true and for it to be false.
How on earth does the conclusion follow from the premises in Orel's example?