Shantila's Inside logic #4ProofsWhen we reason to conclusions, based on premises, our reasoning has a form. The form can be expressed as a rule. When we reason, we are following rules. Of course, we normally are not aware of the rules that we are following. In logic try we discover the rules and state them clearly; and then we explore with them. We explore by constructing "proofs" (or "arguments") in which we use the rules, and this exploration deepens our understanding of logical form. Typically when we think about something, the conclusions we reach are then used as premises for further reasoning. We reason in stages. Recall the unhappy couple at the airport. After you overhear the conversation that leads you to think she is going to leave him, it might occur to you that if she leaves him, she will probably be better off; that is, L> B, where (as before) we let L stands for "she is going to leave him" and B stands for "she will probably be better off". So first you reason to L based on the two original premises: W> L (if he won't confess, she is going to leave him) W (he won't confess) And then once you have used these premises to reason to L, you use L itself as a premise, together with L>B to derive B. That is, you use the two premises L (she is going to leave him) L> B (if she leaves him, she will probably be better off) to derive the conclusion B The conclusion L in the first example (that she's leaving) is used as a premise at the second stage to reach the conclusion B (that she'll probably be better off). This example is typical of reasoning processes, in which we move from one conclusion to another as things unfold. This idea of reasoning in stages is represented in proofs insofar as we construct proofs line by line. Each line represents a stage of reasoning. The statement on each line will be either (a) a conclusion drawn from earlier lines and based on those lines as premises, or (b) an assumption of the reasoning as a whole. We will number the lines. In order to make things totally clear we also will state to the right of the statement on each line what rule was used to reach that step (as well as the line numbers, if any, used as premises). And to the left we will list the numbers of the original assumptions upon which that line depends. These points will become clear to you as you consider the examples below. Notice that the "assumptions" for an argument can come from anywhere, so far as logic is concerned. We usually have reasons for our assumptions, of course, based on what we hear and see and remember (or seem to remember). But recall that it generally is not the business of logic to investigate the truth or falsity of the assumptions made when we reason. Our interest rather is to make sure that the reasoning itself is good. Nonetheless reasoning requires assumptions and the conclusions that we reach in reasoning are based on those assumptions. The Rule of Assumptions allows us to get proofs started by stating one or more assumptions for the proof.
If we assume P on the first line of some proof we would write
This is the first line of a proof, as indicated by "1." and the assumption of this line is this very line itself, as indicated by the "1" on the left. An assumption can be introduced anywhere we like in a proof; for example, if in a certain proof we assume P on the seventh line of the proof, we would write:
The reason for having such a "liberal" rule of inference is simply to be able to get a proof started. If this rule seems too liberal, remember again that in logic we are not primarily interested in the truth or falsity of assumptions. Rather we are concerned with whether reasoning that is based on those assumptions is good. The assumptions in the reasoning about the kiss, for example, can be listed one by one in order to begin a proof.
Our second rule is more interesting. It corresponds to the form of reasoning with "if" upon which we have focussed our attention in the examples so far, as in the little stories about the people at the airport.
With this rule we can add a third line to the proof we began earlier:
Notice on the right in line 3, that "1,2 MP" indicates that the previous lines 1 and 2 were used as premises for this line and that MP was the Rule used to derive this line. And notice on the left in line 3, the numbers "1,2" indicate that the statements on lines 1 and 2 are the assumptions upon which line 3 depends (since the rule for MP states that the assumptions of the new line are all those of any line that is used in the application of MP). Here is another example, again using both of the rules A and MP. It represents the entire reasoning about the unhappy couple that was described earlier.
Line 5 indicates your conclusion B, she probably will be better off. This conclusion is "derived" (as we say) from lines 3 and 4 by MP -- and this is indicated by "3,4 MP" on the last line. The conclusion on line 5 is based on the assumptions on lines 1,2, and 4 -- and this is indicated by the numbers "1,2,4" on the left. * By the way, notice that the following two sentences mean exactly the same thing, even though the word order differs. If he kisses her nicely, then she will marry him. She will marry him if he kisses her nicely. In the second sentence, the "if" part (the antecedent) occurs in the second part of the sentence, not the first part. The word order differs but both of these sentences are represented as K>M. We are representing the meaning of the sentence.
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