Elementary Logic PHIL 105-302 Intersession 2013 MTWHF 10:00 – 12:00 ASA0118C Steven A. Miller Day 4.

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Elementary Logic PHIL Intersession 2013 MTWHF 10:00 – 12:00 ASA0118C Steven A. Miller Day 4

Formalizing review Symbolization chart: It is not the case= ~ And= & Or= v If … then= → If and only if = ↔ Therefore= ∴

Logical semantics Our interpretations are concerned with statements’ truth and falsity. Principle of bivalence: Every statement is either true or false (and not both).

Logical semantics Negation semantics “The Cubs are the best team” is true, then … what’s false? “It is not the case that the Cubs are the best team.”

Logical semantics Negation semantics Likewise, if: “The Cubs are the best team” is false, then … what’s true? “It is not the case that the Cubs are the best team.”

Logical semantics Negation semantics (truth table) P~P TFTF FTFT

Logical semantics Conjunction semantics “My name is Steven and my name is Miller.” is true when “My name is Steven Miller.”

Logical semantics Conjunction semantics “My name is Steven and my name is Miller.” is false when “My name is not Steven or Miller, or both.”

Logical semantics Conjunction semantics (truth table) P Q P & Q T TT T FF F TF F FF

Logical semantics Disjunction semantics “My name is Steven or my name is Miller.” is true when “My name is Steven or Miller, or both.”

Logical semantics Disjunction semantics “…or both”: “Soup or salad?”

Logical semantics Disjunction semantics Inclusive disjunction: this, or that, or both Exclusive disjunction: this, or that, but not both

Logical semantics Disjunction semantics For our purposes, unless stated otherwise, all disjunctions are inclusive: “or” means: this, or that, or both

Logical semantics Disjunction semantics (truth table) P Q P v Q T TT T FT F TT F FF

Logical semantics Disjunction semantics Exclusive disjunction symbolization: (P v Q) & ~(P & Q)

Logical semantics Exclusive disjunction semantics (truth table) P Q (P v Q) & ~ (P & Q) T T T F F T T F F F

Logical semantics Exclusive disjunction semantics (truth table) P Q (P v Q) & ~ (P & Q) T T T F T T F T T F F T T T F F F F T F

Logical semantics Exclusive disjunction semantics (truth table) P Q (P v Q) & ~ (P & Q) T T T F F T T F T T T F F T T T T F F F F F T F

Logical semantics Exclusive disjunction semantics (truth table) P Q (P v Q) & ~ (P & Q) T T T F F T T F T T T F F T T T T F F F F F T F

Logical semantics Material conditional semantics Follows the rules of deductive validity (in fact, every argument is an if-then statement). Is false only when antecedent (premises) is true and consequent (conclusion) is false.

Logical semantics Material conditional semantics This can be counter-intuitive, see: If there are fewer than three people in the room, then Paris is the capital of Egypt.

Logical semantics Material conditional semantics If there are fewer than three people in the room, then Paris is the capital of Egypt. Antecedent = false Consequent = false

Logical semantics Material conditional semantics (truth table) P Q P → Q T TT T FF F TT F FT

Logical semantics Biconditional semantics Biconditional is conjunction of two material conditionals with the antecedent and consequent reversed: P ↔ Q = (P → Q) & (Q → P)

Logical semantics Biconditional semantics (truth table) P Q (P → Q) & (Q → P) T TTT T FFT F TTF F FTT

Logical semantics Biconditional semantics (truth table) P Q (P → Q) & (Q → P) T TT TT T FF FT F TT FF F FT TT

Logical semantics Biconditional semantics (truth table) P Q (P ↔ Q) T T T T F F F T F F F T

Seventh Inning Stretch (“…Buy Me Some Peanuts …”)

Logical semantics Combining truth tables Always work from the operator that affects the least of the formula to that which affects the most of it. ~[(P & ~Q) v (Z ↔ Q)]

Logical semantics Combining truth tables P Q ~~ (P & Q) T T T T F F F T F F F F

Logical semantics Combining truth tables P Q ~~ (P & Q) T TF T T FT F F TT F F FT F

Logical semantics Combining truth tables P Q ~~ (P & Q) T T T F T T F FT F F T FT F F F FT F

Logical semantics Combining truth tables P Q ~~ (P & Q) T T T F T T F FT F F T FT F F F FT F

Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T T T T F T F F T F T F T T F F F F

Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T T F T T T T T F F T F F T F T T F T T F F F T F F F F

Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T T F T F T T T T F F T F F F T F T T F T T T F F F T F F F F F

Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T T F T F T T T T T F F T F F F T T F T T F T T T T F F F T F F F F F F

Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T T F T F T F T T T T F F T F F F F T T F T T F T T F T T F F F T F F F T F F F

Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T T F T F T T F T T T T F F T F F T F F T T F T T F T T F F T T F F F T F F F T T F F F

Logical semantics Combining truth tables P Q (~P & Q) → ~ (Q v P) T T F T F T T F T T T T F F T F F T F F T T F T T F T T F F T T F F F T F F F T T F F F

Three kinds of formulas Tautologies – true in all cases PP v ~P TTFTTF FFTFFT

Three kinds of formulas Tautologies – true in all cases PP v ~P TT TF FF TT

Three kinds of formulas Tautologies – true in all cases PP v ~P TT TF FF TT

Three kinds of formulas Contradictory (or truth-functionally inconsistent) – false in all cases PP & ~P TTFTTF FFTFFT

Three kinds of formulas Contradictory (or truth-functionally inconsistent) – false in all cases PP & ~P TT FF FF FT

Three kinds of formulas Contradictory (or truth-functionally inconsistent) – false in all cases PP & ~P TT FF FF FT

Three kinds of formulas Contingent – can be both true and false Z R Z & R T T T T F F F T F F F F

Putting it all together Either Peter or Saul went to the bar. Peter did not go. Therefore, Saul went. 1) P v S 2) ~P 3) ∴ S

Putting it all together 1) P v S 2) ~P 3) ∴ S What’s this argument’s form? Disjunctive syllogism.

Putting it all together 1) P v S 2) ~P 3) ∴ S [(P v S) & ~P] → S

Putting it all together P S[(P v S) & ~P] → S T T T T F T F T F F T F T F T F F F F F F

Putting it all together P S[(P v S) & ~P] → S T T T T F T T T F T F F T F F T F T T F T F F F F T F F

Putting it all together P S[(P v S) & ~P] → S T T T T T F T T T F T T F F T F F T F T T T F T F F F F F T F F

Putting it all together P S[(P v S) & ~P] → S T T T T T F F T T T F T T F F F T F F T F T T T T F T F F F F F F T F F

Putting it all together P S[(P v S) & ~P] → S T T T T T F F T T T T F T T F F F T T F F T F T T T T F T T F F F F F F T F T F This argument is valid; there is no line where the premises are all true and the conclusion is false.

Putting it all together A truth table that has no lines where the premises are all true and the conclusion false presents a valid argument. A truth table that has at least one line where the premises are all true and the conclusion false presents an invalid argument.

Things we’re skipping - Truth / refutation trees, S. pp identical in purpose to tables - more efficient - but no time = no need