T4.3 4 of 6
The Material Conditional
Perhaps the most controversial symbolization is for the conditional. As we defined it, an SL conditional sentence 'P>Q' is false only if its antecedent is true and consequent false and is true otherwise. Any conditional of any language with this truth condition is called a material conditional.
Is English "if...then..." ever a material conditional? We've been assuming that we can just translate this English to the horseshoe. But perhaps that is overly simplistic?
One reason to think that it's not so simplistic comes from our old example.
(1) If you pass all the exams in this class, then I will pass you for the course.
There is one way for (1) to be false: if you pass all the exams and yet I do not pass you for the course. Thus the only way this conditional may be false is for its antecedent to be true while its consequent is false.
But don't be too quick to conclude that (1) really is a material conditional, i.e., that it really is like a horseshoe sentence. It remains possible that it is neither true nor false in some truth value assignment. (Some people's intuition is that in the case in which its antecedent and consequent are both false, (1) is untested and makes no statement.) In that case, it would not have a complete truth table, would not express a statement, and could not be well symbolized by way of our horseshoe.
But another example may help us see that English does include a material conditional.
(2) If you make over $6,000, then you must file.
Or so says the IRS. They say this, knowing it is true, and at the same time knowing that there are people to whom it applies but who fall in all categories:
For instance, row three says of people who make less than $6,000 but who must file that the statement is true. But that is exactly right. The IRS truly makes this statement about all US residents. And some of them (e.g., many children with investment income of between one and six thousand) fit this third row.
Thus, it would appear that English does include a "material conditional", does include "if...then" statement properly symbolized by the horseshoe.