Shantila's Inside Logic #14

Disjunctive Syllogism

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"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 said.

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This is valid reasoning. Congratulations, Joe; you did right. This reasoning has the form

GvS, ~G } S

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 valid 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.

Disjunctive Syllogism (DS)

Given any statements PvQ and ~P as premises, DS permits us to derive Q as a conclusion, where Q depends on all of the assumptions of the two premises.

Likewise given PvQ and ~Q as premises, DS permits us to derive P as a conclusion, where P depends on all of the assumptions of the two premises.

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.

76 PvQ, ~P } Q

  1 1. PvQ A            
  2 2. ~P A            
  1,2 3. Q 1,2 DS            

77 PvQ, ~Q } P

  1 1. PvQ A            
  2 2. ~Q A            
  1,2 3. P 1,2 DS            

Also notice that we can use DS even when one or both disjuncts are negations or conjunctions, as in the following proofs.

78 Pv~Q, ~P } ~Q

  1 1. Pv~Q A            
  2 2. ~P A            
  1,2 3. ~Q 1,2 DS            

79 Pv~Q, Q } P

  1 1. Pv~Q A            
  2 2. Q A            
  2 3. ~~Q 2 DN            
  1,2 4. P 1,3 DS            

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.

80 (P&Q)vR, ~R } P&Q

  1 1. (P&Q)vR A            
  2 2. ~R A            
  1,2 3. P&Q 1,2 DS            

The following two valid sequents reveal interesting connections between disjunctions and conditionals.

81 ~PvQ } P>Q

  1 1. ~PvQ A            
  2 2. P A (for CP)            
  2 3. ~~P 2 DN            
  1,2 4. Q 1,3 DS            
  1 5. P>Q 2,4 CP            

82 PvQ } ~P>Q

  1 1. PvQ A            
  2 2. ~P A (for CP)            
  1,2 3. Q 1,2 DS            
  1 4. ~P>Q 2,3 CP            

The validity of 81 and 82 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 (see again the proof of sequent 76). 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 82.

*Practice

14.1 Give a real-life example involving the use of DS in reasoning that corresponds to sequent 76.

14.2 Use the Rules A, MP, DN, MT, &E, &I, CP, <>I, <>E, vI and DS as needed to construct proofs for each of the following sequents. Several of these sequents obviously were just proved in the text; no matter, do them yourself without looking (and then check your work).

76 PvQ, ~P } Q

77 PvQ, ~Q } P

78 Pv~Q, ~P } ~Q

79 Pv~Q, Q } P

80 (P&Q)vR, ~R } P&Q

81 ~PvQ } P>Q

82 PvQ } ~P>Q

83 } (~PvQ) > (P>Q)

84 PvQ, P>R, ~Q } R

85 P>R, Qv~R, ~Q } ~P

86 ~R>Q, Pv~Q, ~P } R

87 P<>Q, SvP, ~Q } S

14.3 The BGSU women's basketball team played last March in the national tournament against the Oklahoma State ("the Cowgirls"). The following argument is valid. Unfortunately it did not turn out to be sound as well.

Symbolize the argument and construct a derivation of the conclusion. F: The Falcons won. C: The Cowgirls won. S: The Cowgirls were able to slow down Kate Achter.

The Falcons won or the Cowgirls won. The Cowgirls were not be able to slow down Kate Achter; but they did not win unless they were able to slow her down. So the Falcons won.

14.4 Each of the following arguments in English is valid, and in each case the sentences can by symbolized so that the conclusion can be derived from the premises using the Rules of our proof system. (a) Symbolize the premises and the conclusion for each of the following arguments, and (b) construct a derivation of the conclusion. Use these symbols. S: Sally will go. J: Joe will go. A: Anna will go.

(a) Sally will go or Joe will. Sally won't go. So Joe will.

(b) Sally will go or Joe will not go. Sally won't go. So Joe won't either.

(c) Sally will go if Joes goes. Anna or Joe will go; but Anna won't go. So Sally will.

(d) Sally will go unless Anna goes. Joe will go or Sally won't. Joe actually won't go. So Anna will go.

(e) Sally will go if Joe goes; and he will. So Sally or Anna will go.

(f) Sally will go or Anna will. So Anna will go if Sally doesn't.

(g) Joe will go if either Sally or Anna goes. So Joe will go if Anna goes.

(h) Sally won't go or Anna will. Therefore Anna will go if Sally goes.

(i) If Sally or Joe will go, then Joe will go if Sally doesn't. (Note: this is a logical truth.)

(j) If I hear anything more about these stupid people I will either go crazy or move to Siberia. If I go crazy I just might find myself running naked across campus, and if I do that my mother might see me on tv. You never know. Anyway I definitely am never going to move to Siberia. Neither will Sally nor Joe. Oh no! I heard more about those stupid people! So my mom just might see me on tv.