sourgrass wrote:Yes, also true. But the point of the principle of explosion is that when you have a contradiction in the system then you can show any statement to be true. Thus any of those numbers is in fact your Mom's number.

No, that isn't the point of the principle of explosion (although it may be the point of the comic by that title). The point of the principle of explosion is that any state of affairs that satisfies the contradiction will satisfy any other statement as well.

This does not mean that "any of those numbers is in fact your Mom's number", rather it means that the original proposition, which was (P AND NOT P), is not a possible in any state of affairs at all. It is allowed in symbolic logic because symbolic logic is allowed to express propositions that cannot be satisfied by any states of affairs.

So you make a truth table. The original propositions will be satisfied by some (possibly empty) set of rows in the truth table. The rules of symbolic logic allow you to extend those propositions in any way such that the set of rows in the truth table that satisfy the extended set of propositions is identical to the set of rows that satisfy the original propositions.

If the truth table contains a column for P and a column for "555-1234 is your Mom's Phone number" and another column for "555-6789 is your Mom's Phone number", then any rows that satisfy (P AND NOT P) will also satisfy both of those other statements, and they will also satisfy the negation of those other statements, simply because there are no rows at all that satisfy (P AND NOT P) and making the other assertions doesn't change that. Whether the entry in the truth table contains a T or F for P, the row will get excluded by (P AND NOT P) regardless of whether it has a T or F in any other column.

The trick is that having a proposition in symbolic logic does not mean that the proposition is true, it only means that it is asserted to be true, in other words that it will be true in any state of affairs that satisfies the proposition. But if there is no row in the truth table that satisfies the proposition, it is still allowed to be asserted in symbolic logic, even though it will never be true.