imdefMaterial implication P -> Q is equivalent to ~P v Q. It is generally agreed that the "if P, then Q" construction in ordinary language is not always the same as material implication. However, when you study mathematics, you're trained to think that, in mathematics, "if P, then Q" really is material implication. Here is an in many ways careful explanation: (1/n)
https://gowers.wordpress.com/2011/09/28/basic-logic-connectives-implies/
I was never sure what to make of this, because I have yet to read a discussion of why material implication is a better model of mathematicians' "if P, then Q" than other alternatives. For example, why not understand "if P, then Q" in mathematics as "necessarily, if P, then Q" and take it to correspond to [](P -> Q), where [] is an operator of modal logic? I'm sure people already thought of this, I just haven't seen the pros and cons of this alternative (and other alternatives) compared to the pros and cons of the material implication. E.g., what about the implication in relevance logic? (2/n)
Because of the thoughts like the above, I found the following paper quite interesting:
https://www.tandfonline.com/doi/abs/10.1080/11663081.2014.911540
Vidal points out that (P ^ Q) -> R is equivalent to (P -> R) v (Q -> R). Both these forms can be seen to be equivalent to ~P v ~Q v R. Specific instances of this equivalence can be awkward/counterintuitive:
Example 1:
("x is a rhombus" ^ "x is a rectangle") -> "x is a square"
("x is a rhombus" -> "x is a square") v ("x is a rectangle" -> "x is a square")
(3/n)
Example 2: write a|x for "x is divisible by a" or "a divides x". Then
(2|x ^ 3|x) -> 6|x
(2|x -> 6|x) v (3|x -> 6|x)
In both cases, the first form is natural and obvious and the second is something you'd normally never write. But, if pressed, maybe you'd bite the bullet and agree it's an equivalent form. I'm still undecided but I enjoyed the paper.
#logic #implication #conditional (3/3)
I should have been more precise. The two formal expressions
(2|x ^ 3|x) -> 6|x
(2|x -> 6|x) v (3|x -> 6|x)
are equivalent. However, it is less clear cut with their ordinary language translations:
"If x is divisible by 2 and x is divisible by 3, then x is divisible by 6."
"If x is divisible by 2, then x is divisible by 6, or if x is divisible by 3, then x is divisible by 6."
#logic #implication #conditional