Proof and Truth
There's another way of thinking about the difference between proof theory and semantics. Semantics provides us with rules of truth, that is with the conditions under which propositions, sets of propositions, and propositions in arguments are true and false. We then use our semantic notions, such validity and invalidity to evaluate arguments and propositions. Proof theory, in contrast, makes no reference to truth or truth conditions. The rules of proof theory, the rules of inference, specify what can be inferred, and they do that by referring only the the syntactic features of the sentences under consideration.
To make the difference clearer, let's consider how we evaluate an argument from the standpoint of semantics, and how we will treat that same argument from the standpoint of proof theory.
Semantics:
| If Ignat passes his history class, he will graduate. Ignat does not graduate. Therefore, Ignat does not pass his history class. |
Is this a valid argument? If it is, then it's not possible for the
premises to be true and the conclusion false. The argument, translated into
Propositional Logic is:
| P
⊃ Q ~Q ~P |
where "P" is "Ignat passes his history class" and "Q" is "Ignat will graduate." The full truth-table method, the gappy truth-table method, or the truth-tree method shows that this argument is valid.
Proof Theory:
Now consider the following sentences:
| If Mathilda graduates, she'll go to law school and get rich. Mathilda graduates. |
First, we translate:
| P
⊃ (Q & R) P |
where "P" is "Mathilda graduates," "Q" is "She'll go to law school" and "R" is "She'll get rich." We ask a proof-theoretic question, namely, what follows from these two propositions? We'll see that our proof theory contains a rule which says that whenever we have a conditional as one starting premise, and the antecedent of that conditional as a second premise, we can infer the consequent of that conditional. The rule is called "modus ponens." So we can infer as follows:
| 1. | P ⊃ (Q & R) | premise |
| 2. | P | premise |
| 3. | (Q & R) | mp 1, 2, |
Notice that we've numbered the lines in the first column, written the wffs in question in the second, and in a third column we give the justification or reasons supporting what we've written. Lines 1 and 2 are our premises. Line three contains the wff we inferred from 1 and 2, and the name of the rule "MP" - short for "modus ponens". Can we make any other inferences? The answer depends on what our rules of inference allow. We'll see shortly that we have another rule, called "simplification" which tells us that whenever we have a conjunction on a line, we can infer either conjunct. So we can add another line to what we have so far:
| 1. | P ⊃ (Q & R) | premise |
| 2. | P | premise |
| 3. | (Q & R) | mp 1, 2 |
| 4. | Q | simp, 3 |
The rules of inference don't refer to truth or validity. They are just a set of rules which tell us what inferences we may make from the wffs we start out with. That said, we will see later in this chapter that there's a very important relationship between proof theory and semantics. We will choose rules of inference in such a way, that truth is preserved. That is, if we were to begin our proof theory transformations with premises which were true, the wffs we infer from them would also be true.
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