Reason and Argument

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Reason and Argument

• Chapter 6 (2/3)

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A symbol for the exclusive or

• We will use ? for the exclusive or
• Strictly speaking, this connective is not
  necessary. We could just as easily use ((p v q)
  (p q)) for p ? q
• I shall generally avoid use of the exclusive
  or, though it be in the book.

p q p ? q
T T F
T F T
F T T
F F F
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Conditionals

• Conditionals are statements of the form If
  _______ then _______ where the blanks are filled
  with sentences that express propositions.
• Again, the first blank is called the antecedent
  and the second blank is called the consequent.

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The kinds of conditionals that are allowed to be
?

• There is some dispute as to whether there are any
  material conditionals (the kind that are
  expressed by ?) in English.
• In any case, we shall use ? for any and all
  conditionals in the PRESENT tense, ACTIVE voice,
  INDICATIVE mood.

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Subjunctive Conditionals

• Consider the sentence If the Germans had won
  WWII, then we would all be speaking German
• A statement of the form p ? q would involve two
  separate propositions, and would connect them in
  the appropriate way.
• The Germans had won WWII does not really
  express a proposition all by itself, nor does we
  would all be speaking German.
• Further, in the kind of conditional we are
  concerned with, someone does not commit
  themselves to the truth or falsity of the
  antecedent. In a counterfactual conditional as
  above, one is committed to the falsity of some
  particular state of affairs (in this case that
  the Germans do not win WWII).
• The above, therefore, is simply p, not p ? q

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The (seemingly) wacko truth table for
conditionals

• Assume p is The pitcher throws a fastball and q
  is The batter hits a home run
• Line 2 is very straightforward. If Bob bets you
  that if the pitcher throws a fastball then the
  batter will hit a home run, Biff will lose his
  bet if things turn out as on Line 2. But what
  about the others?

p q p ? q
T T ?
T F F
F T ?
F F ?
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Conditional Truth Table

• Line 1 seems equally straightforward. Biff wins
  his bet by virtue of saying something true, just
  as on line 2 he would lose his bet by virtue of
  saying something false.
• But what happens when the antecedent is false?

p q p ? q
T T T
T F F
F T ?
F F ?
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Conditional Truth Table

• Imagine youre watching the game and Biff makes
  his bet. You accept, and you see the pitcher
  throw a curve ball (i.e. NOT a fastball, making
  the proposition p false) yet the batter still
  hits a home run (making the proposition q true,
  as on line 3).
• The best way to interpret this is that the bet is
  neither won nor lost, and no money changes hands.
  This would also be the case if the batter has
  swung at and missed the curveball (line 4). But
  since we still have to assign one or the other
  truth values to if p then q what do we do?

p q p ? q
T T T
T F F
F T ?
F F ?
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Do I really need to type the title for this slide
again?

• We give the conditional phrase the benefit of the
  doubt. We have better reasons for saying that
  the conditional is not false than we have reasons
  to say its not true.
• Also there are more weird problems that result
  from taking the conditional to be false in lines
  3 and 4 than result from taking it to be true.
• This comes through more clearly when dealing with
  conditionals that are not predictions. If it is
  raining then the ground is wet is a true
  conditional even if its not raining.

p q p ? q
T T T
T F F
F T T
F F T
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One of those reasons Material Implication

• If we focus on the second line of the truth table
  for conditionals, which was a clear case, we can
  see that having a true conditional must mean that
  it is not the case that the antecedent is true
  and the consequent false. Formalized, that looks
  like this
• a true conditional (p ? q) implies that it is not
  the case that (p is true and q is false)
• or (p q)
• By DeMorgans Law, p v q (read as It is not the
  case that p unless q is the case) is equivalent
  to the above

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If p ? q, (p q), and p v q are equivalent
p q p ? q (p q) p v q p q q p
T T T T T F F F
T F F F F T T F
F T T T T F F T
F F T T T F T T
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meet the MODI

• Modus Ponens
• p ? q
• p____
• q

• Modus Tollens
• p ? q
• q___
• p

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Valid Moduses
P2 C P1
p q p ? q
T T T
T F F
F T T
F F T
P1 P2 C
p q p ? q q p
T T T F F
T F F T F
F T T F T
F F T T T
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A couple common fallacies (and trouble with
conditionals in general)

• Affirming the consequent
• p ? q
• q____
• p

• Denying the antecedent
• p ? q
• p___
• q

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Hypothetical Syllogism (Chain Argument)

• p ? q
• q ? r
• p ? r
• Valid?

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Another reason for conditionals to be true when
antecedent is false

• If lines 3 and 4 of the truth table for
  conditionals are replaced by either the value F
  or N for neither, we get some highly
  counterintuitive results
• Modus Tollens is such that the premises are never
  true at the same time
• Denying the Antecedent has the same status as
  Modus Tollens
• Affirming the Consequent comes out valid!

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Procedure for using truth tables to find validity

• Create the reference columns (one per
  propositional variable, in alpha. order)
• Create one column for each logical connective
  (v,,, ?)
• Fill in reference columns
• of rows 2n where n of propositional
  variables
• Fill first half of leftmost columns rows with
  value T, the rest with value F
• Fill next column with T and F, beginning with T
  and having exactly half as many consecutive
  iterations of T and F as occurs in column
  leftward.
• Repeat (c) until reference columns are filled in.
• Fill in the rest of the table
• Check for any case in which the premises are all
  true and the conclusion is false. If such a case
  is on the table, the argument form is invalid,
  valid otherwise.

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Example Ex. 24 5

• Argument is
• p ? q
• q ? r
• r__
• p
• Step 1 Create the reference columns

p q r
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Step 2, One column for each connective
p q r p ? q q ? r r p
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Step 3 Fill In Reference Columns
p q r p ? q q ? r r p
T T T
T T F
T F T
T F F
F T T
F T F
F F T
F F F
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Step 4 Fill in remainder
p q r p ? q q ? r r p
T T T T T F F
T T F T F T F
T F T F T F F
T F F F T T F
F T T T T F T
F T F T F T T
F F T T T F T
F F F T T T T
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Check for Validity
Premise 1 Premise 2 Premise 3 Conclusion
p q r p ? q q ? r r p
1 T T T T T F F
2 T T F T F T F
3 T F T F T F F
4 T F F F T T F
5 F T T T T F T
6 F T F T F T T
7 F F T T T F T
8 F F F T T T T