No, I am doing logic, but correctly.Originally Posted by steve bank
Sure, though what conclusions you can derive depends on the logic (there are different logics, e.g., intuitionistic is weaker than classic), but I'm doing classical logic here, which is by far the most common and the most widely accepted - and the strongest in terms of what one can prove.Originally Posted by steve bank
I would also defend it as the better fit for our language, but that would not strictly be logic.
No, I am telling you that your application of logic is mistaken.Originally Posted by steve bank
That is easy. I only need to show that there is no assignment of values on which all of the premises are true, but the conclusion is false. Let us formalize:Originally Posted by steve bank
P1: P
P2: ¬P.
C: P∧¬P.
The possible assignments of value for P are T or F. So, we have:
P:T
P1:T
P2:F
C:F
Not a problem, because on this assignment, not all of the premises are true (P2 is false), and thus, it is not the case that all of the premises are true but the conclusion is false.
Let us try the other possible assignment:
P:F
P1:F
P2:T
C:F
Not a problem, because on this assignment, not all of the premises are true (P1 is false), and thus, it is not the case that all of the premises are true but the conclusion is false.
Since there is no other possible assignment, this proves on classical logic that the conclusion follows from the premises.
The same principles of classical logic that are used widely in math, logic, philosophy, and other fields, also imply that. It can be useful when doing math exercises, but also being familiar with that may be helpful to better understand logic (e. g., just look at some of the exchanges in this thread) . At any rate, the OP arguments do not involve the Principle of Explosion.