Logic

What is the truth maker for logic? In other words, why should I take logical truths (e.g., material implication) as true?

Comments

A few points need clarification before I can begin to answer your question.

First, logic is not concerned with truth in the way that, say, the sciences are. Logic is concerned with relationships among sentences that have truth value, not with the actual truth values of the (atomic) sentences. The only apparent exception to this might be those sentences that "must" be true (tautologies) and those that "must" be false (contradictions). But tautologies and contradictions are not atomic sentences; they are "molecular" sentences, and what makes them tautologous or contradictory are the relationships among their atomic constituents. So, for instance, "(p & ¬p)" is a contradiction because—no matter what the actual truth value of p—the truth value of "(p & ¬p)" must be false (because of the truth tables for conjunction (&) and negation (¬)). Logic isn't concerned with p's actual truth value.

Second, material implication (&rightarrow;) is not a "logical truth" nor is it even a sentence. It's a logical connective like & and ¬. Each connective has a truth table that tells you what the truth value of a molecular sentence constructed with the connective is, no matter what the actual truth values of the constituent sentences are. Again, logic isn't concerned with what those truth values are, only how they relate to each other when combined with a given connective.

So, what you may be thinking of when you said "material implication" is the rule of inference called "modus ponens":

From p

and (p &rightarrow; q),

you may infer q.

A rule of inference like this (you can also think of it as an "atomic" pattern of reasoning) is neither true nor false (any more than the number 7 is true or false). Rules of inference (patterns of reasoning, arguments, proofs—whatever you want to call them) are either "valid" or "invalid". And a rule of inference is valid if and only if it is "truth preserving", that is, if and only if, whenever the premises (in our example, these are p and "(p&rightarrow;q)") are true, then the conclusion (in our example, that's q) must be true. Logic doesn't tell you whether the premises actually are true (that's something that, say, science might be able to tell you), but it does tell you about the relationship of the premises to the conclusion, namely, that if the former are true, the latter have to be true.

So, maybe your question is this: Why should you take arguments like this as being valid? Well, consider the truth table for material implication (&rightarrow;):

p q (p&rightarrow;q)

T T T

T F F

F T T

F F T

Look at the rows that correspond to a modus ponens inference with both premises being true. There's only one such row: the first one. And in that row, q is also true. That's why modus ponens is valid.

Now, your next question might be: Why should we accept that truth table for &rightarrow;?

Here, there are two issues. First, not all logicians do accept it, or, to put it a better way, some logicians think that this connective doesn't really capture the meaning of ordinary English "if-then". One example of a different kind of logic that's been devised to deal with this is "relevance logic". Second, a more general question is: Why should we accept any truth table? Here, the answer I'll give you is that, for any combination of 1 or 2 atomic sentences with two truth values, there are only 16 possible truth tables, that is, only 16 possible logical connectives. Logic (more precisely, classical sentential or propositional logic) can be thought of as the study of them, independently of whether any of them capture the meaning of ordinary English connectives.

So, perhaps the answer to your question is: You don't have to take them as "true" or "valid". You're free to create other logics! To see some of the possible variety in logics, take a look at all the articles in the Stanford Encyclopedia of Philosophy under the heading "logic" in the index.

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