FORMAL LOGIC I: MORE PROPOSITIONAL LOGIC

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FORMAL LOGIC IMORE PROPOSITIONAL LOGIC
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The Truth-Table Test for Validity (again)

• a decision procedure for determining whether a
  propositional argument is valid
• isolate the form of the argument (translation)
• do the truth table (for the entire argument)
• determine by inspection whether there are any
  cases in which all of the premises are true but
  the conclusion is false

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A Version of Goulds Orchid Argument

• The Argument (after deep analysis)
• (P1) If God made the orchid, then the orchid is
  designed
• to reflect Gods wisdom and power.
  SP
• (P2) If the orchid is designed to reflect Gods
  wisdom and
• power, then the orchid is not made from
  parts
• fashioned for other purposes.
• (P3) The orchid is made from parts fashioned for
  other
• purposes.
• (P4) Either God made the orchid or the orchid
  evolved
• from ordinary flowers. SP
• (C) Therefore, the orchid evolved from ordinary
  flowers.

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The Argument Symbolized (Translation)

• M God made the orchid
• D the orchid is designed to reflect Gods
  wisdom and power
• P the orchid is made from parts fashioned for
  other purposes
• E the orchid evolved from ordinary flowers
• argument w/ prop. constants argument form
• (P1) If M, then D. M ? D p ? q
• (P2) If D, then not-P. D ? P q ? r
• (P3) P. P r
• (P4) Either M or E. M v E p v s
• (C) Therefore, E. E s

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Propositional Logicand Ordinary Language

• we need to be able to translate the propositions
  of an argument into truth-functional propositions
  i.e., we need to be able to isolate the form of
  the argument
• Q Do the logical connectives have much to do
• with ordinary language propositions?
• A Yes, but the connectives are not perfect.

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Conjunction

• non-propositional conjunctions
• Grover and Frances are a couple now. ? G F
• arbitrary conjunctions
• Frances is beautiful and I hate Jell-O. B
  H
• (Therefore, Frances is beautiful.) valid
• conjunctions with temporal import
• Warren had an embolism and died. D E / E
  D
• Warren had an embolism before he died. W
• discounting conjunctions
• Frances is beautiful, but irascible. B I
• Frances is irascible, but beautiful. I B

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A General Lesson

• Propositional logic, like all formal analysis,
  investigates only the validity of an argument.
• And considerations about temporal import and
  discounting and such typically do not affect
  validity.

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Disjunction

• arbitrary disjunctions
• Either Grover is dead or Oldsmobiles are sexy
  cars.
• (Oldsmobiles are not sexy cars.)
• (Therefore, Grover is dead.) valid
• exclusive v. inclusive disjunction not a
  problem!
• troubles with unless
• sometimes OK as inclusive
• I will go home unless you settle down. H v S
• sometimes must be exclusive
• Tomorrow you will be glad you jumped, unless you
  die today. (G v D) (G D)

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Negation

• You might not survive the parachute jump.
• ? S
• ? M

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Material Conditional

• what it does capture and why it is OK there
• propositional conditionals in which both the
  antecedent and the consequent are in the
  indicative mood
• p ? q (p q) p v q
• If Grover is decapitated, then Grover is dead.
• arbitrary conditionals
• If Grover is dead, then I hate Jell-O.

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what the material conditionaldoes not capture

• subjunctive / counterfactual conditionals
• If Caesar were the current president of the U.S.,
  then Caesar would speak English.
• If Caesar were the current president of the U.S.,
  then presidential press conferences would be
  conducted in Latin.
• conditional imperatives
• If Im not back here in an hour, call the cops.

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what the material conditionaldoes not capture

• conditional promises
• If you give me 100 today, I promise to repay you
  tomorrow.
• (I promise to make it true that, if you give
  me 100 today, then I repay you tomorrow.)

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Paradoxes of theMaterial Conditional (UA, pp.
172-173)

• any (material) conditional with a false
  antecedent is a true conditional
• If the Parthenon is made out of iron, then it is
  lighter than air. I ? L
• If Bush is a crack dealer, then Bush is an
  excellent president. C ? E
• p
• therefore, p ? q

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• any (material) conditional with a true consequent
  is a true conditional
• If the decimal expansion of Pi contains seven 4s
  in a row, then Grover is dead. S ? G
• q
• therefore, p ? q

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Answers / Solutions / Responsesto the Paradoxes

• we had good reasons for dealing with the ?
  operator in such a way that
• p ? q p v q
• p q
• therefore, p v q therefore, p v q
• 2) the Gricean Response the oddness consists
  only in defects with respect to certain
  conversational implications validity is
  unaffected
• If you give me some bourbon, then I will tell you
  your grade.

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Biconditionals andConversational Implication

• p only if q p ? q
• One is president of the U.S. only if one is 35 or
  older. P ? O
• Ernie is beheaded only if Ernie is dead.
• B ? D

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• often, though, the use of only if
  conversationally implies if and only if
• I will marry you only if you agree to shower at
  least once every day.
• (M ? A) (A ? M)
• ( M v A) (A M)

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unless constructions

• bare conditional / inclusive disjunction
• One is not president of the U.S. unless one is 35
  or older.
• P v O O ? P P ?
  O
• One is president of the U.S. only if one is 35 or
  older.
• P ? O O ? P P v
  O

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unless constructions

• biconditional / exclusive disjunction
• These logs will make a good fire unless they are
  wet.
• (F v W) (F W)
• These logs will make a good fire if and only if
  they are not wet.
• ( W ? F) (F ? W)

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(F v W) (F W) ( W ? F) (F ?
W)