Shantila's Inside Logic #12 Disjunctive Syllogism "Joe, listen," you said. "Listen to me. Just give him the ticket. Either you give it to him or he'll shoot us both. Suppose you don't give it to him." "Then he'll shoot us both," Joe says. * This is valid reasoning. It has the form GvS, ~G } S. This is valid because any reasoning with this form will be such that the conclusion must be true if the premises are true. Here's another example with this interesting form. Sally lives in Toledo or she lives in Cleveland. She does not live in Cleveland. So she lives in Toledo. That is, TvC, ~C } T. This pattern traditionally is called "Disjunctive Syllogism" (DS). (The word "syllogism" traditionally was used to refer to valid forms of reasoning.) It is our next valid Rule.
This rule DS is valid because of a simple fact about disjunctions: the negation of one of the disjuncts means the other disjunct must be true. This is simply how "or" works! To consider another example of DS, suppose George is either a human or George is a squirrel, HvS; and George is not a human, ~H. So S, George is a squirrel. This is valid because there is no way that the two premises can be true without the conclusion being true. Think about it -- can you describe any coherent way in which the two premises could be true but the conclusion false? (Whether or not it is sound, as you will recall, depends on whether or not the two premises are in fact true; and that would depend on who or what we are using the name 'George' to refer to.) If you are not convinced that DS is a valid rule, then try to construct a counterexample -- that is, give a coherent example where the two premises PvQ and ~P are true, but Q is not true. The claim that DS is valid means that it is impossible to do this; no matter how hard you try, you will not be able to find such a counterexample. The new rule enables us to prove each of the following sequents. 62 PvQ, ~P } Q
63 PvQ, ~Q } P
Also notice that we can use DS even when one or both disjuncts are negations or conjunctions, as in the following proofs. 64 Pv~Q, ~P } ~Q
65 Pv~Q, Q } P
Notice that in the preceding proof, ~~Q on line 3 is the negation of ~Q, one of the disjuncts in the disjunction on line 1, so we can use DS on line 4 to derive the other disjunct, P. 66 (P&Q)vR, ~R } P&Q
The following two valid sequents reveal interesting connections between disjunctions and conditionals. 67 ~PvQ } P>Q
68 PvQ } ~P>Q
The validity of 67 and 68 make sense if you think about it. Consider again the example of reasoning with which we began this chapter: "Either you give it to him or he'll shoot us both. Suppose you don't give it to him." --"Then he'll shoot us both." That is, GvS, ~G } S We can prove that this sequent is valid (as we do in the proof of sequent 62). But equally the story could have been written as follows and it would mean the very same thing. "Either you give it to him or he'll shoot us both." --"So if you don't give it to him, he'll shoot us both." That is, GvS } ~G>S We prove that this form is valid in the proof of sequent 68.
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