There is a very common misconception that even a lot of professional mathematicians have, which is the idea that implication is the same thing as conditional. The purpose of this post is to clear up that misconception, and explain the difference between implication and conditional.
The relationship between implication and conditional can be summarized in this short phrase: Implication is tautologyhood of the conditional. Let me explain what I mean.
Consider the system of propositional logic with a countably infinite set of propositional variables , a pair of parentheses and , as well as the logical operators , , and , which represent negation, disjunction, conjunction, and conditional, respectively.
We define the set of well-formed formulas inductively as follows: first, every propositional variable by itself is a well-formed formula. Next, if is a formula, then so is . And finally, if and are formulas, so is , where can represent either , , or .
A tautology of the propositional logic is defined as follows: First, we define a Boolean valuation as a function from the set of well-formed formulas to the set that respects these properties, for any well-formed formulas and :
1.
2.
3.
4.
A tautology is then defined as a well-formed formula which is mapped to under every Boolean valuation.
Now we come to the definition of implication. Implication is a binary relation on the set of well-formed formula that is defined as follows: for any well-formed formulas and , the relation , read “ implies ”, holds if and only if the well-formed formula is a tautology.
Let me give some examples and also some nonexamples. implies , because the well-formed formula is a tautology. Also, for any well-formed formula , implies . As a nonexample, does not imply , because the formula is not a tautology: There is a Boolean valuation that maps to . The last point is precisely the reason why I wrote “ does not imply ” in the title: If we write as , and as , then indeed does not imply .
That said, there is still one more subtle point to clear up. In most mathematical contexts, and are variables ranging over arbitrary propositions. In those contexts, whether implies depends on what the content of and are. I used and as variables that range over arbitrary well-formed formulas, but of course one can use and instead. So, basically, under my convention, does not imply , but the question of whether the formula implies the formula is indeterminate until we know what and are. Some other mathematician could have the opposite convention, and use and as two distinct propositional variables, while using and as variables ranging over arbitrary formulas. So perhaps I should have written “ does not imply ” in the title. But in any case, I hope that now I have clearly explained the distinction between implication and conditional.