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# Elementary Logic - PowerPoint PPT Presentation

Elementary Logic. PHIL 105-302 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.

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### Elementary Logic

PHIL 105-302

Intersession 2013

MTWHF 10:00 – 12:00

ASA0118C

Steven A. Miller

Day 4

Symbolization chart:

It is not the case = ~

And = &

Or = v

If … then = →

If and only if = ↔

Therefore = ∴

Our interpretations are concerned with statements’ truthand falsity.

Principle of bivalence: Every statement is either true or false (and not both).

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.”

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.”

Negation semantics (truth table)

P ~P

T F

F T

Conjunction semantics

“My name is Steven and my name is Miller.”

is true when

“My name is Steven Miller.”

Conjunction semantics

“My name is Steven and my name is Miller.”

is false when

“My name is not Steven or Miller, or both.”

Conjunction semantics (truth table)

P Q P & Q

T TT

T F F

F T F

F FF

Disjunction semantics

“My name is Steven or my name is Miller.”

is true when

“My name is Steven or Miller, or both.”

Disjunction semantics

“…or both”:

Disjunction semantics

Inclusive disjunction:

this, or that, or both

Exclusive disjunction:

this, or that, but not both

Disjunction semantics

For our purposes, unless stated otherwise, all disjunctions are inclusive:

“or” means:

this, or that, or both

Disjunction semantics (truth table)

P Q P v Q

T TT

T F T

F T T

F FF

Disjunction semantics

Exclusive disjunction symbolization:

(P v Q) & ~(P & Q)

Exclusive disjunction semantics (truth table)

P Q (P v Q) & ~ (P & Q)

T TTT

T F T F

F T T F

F FFF

Exclusive disjunction semantics (truth table)

P Q (P v Q) & ~ (P & Q)

T TT F T

T F T TF

F T TTF

F FF T F

Exclusive disjunction semantics (truth table)

P Q (P v Q) & ~ (P & Q)

T TT F FT

T F TTTF

F T TTTF

F FFFTF

Exclusive disjunction semantics (truth table)

P Q (P v Q) & ~ (P & Q)

T TTFFT

T F TTTF

F T TTTF

F FFFTF

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.

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.

Material conditional semantics

If there are fewer than three people in the room, then Paris is the capital of Egypt.

Antecedent = false

Consequent = false

Material conditional semantics (truth table)

P Q P → Q

T TT

T F F

F T T

F FT

Biconditional semantics

Biconditional is conjunction of two material conditionals with the antecedent and consequent reversed:

P ↔ Q = (P → Q) & (Q → P)

Biconditional semantics (truth table)

P Q (P → Q) & (Q → P)

T TTT

T F F T

F T TF

F F T T

Biconditional semantics (truth table)

P Q (P → Q) & (Q → P)

T TTTT

T F FFT

F T T F F

F FTTT

Biconditional semantics (truth table)

P Q (P ↔ Q)

T TT

T F F

F T F

F F T

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)]

Combining truth tables

P Q ~~ (P & Q)

T TT

T F F

F T F

F FF

Combining truth tables

P Q ~~ (P & Q)

T T F T

T F T F

F T TF

F F T F

Combining truth tables

P Q ~~ (P & Q)

T TTFT

T F FTF

F T F TF

F FFTF

Combining truth tables

P Q ~~ (P & Q)

T TTFT

T F FTF

F T FTF

F FFTF

Combining truth tables

P Q (~P & Q) → ~ (Q v P)

T TTTTT

T F T F F T

F T F T T F

F FFFFF

Combining truth tables

P Q (~P & Q) → ~ (Q v P)

T T F T T TT

T F FT F F T

F T TF T T F

F F T FFFF

Combining truth tables

P Q (~P & Q) → ~ (Q v P)

T T F T F T TT

T F FT F F F T

F T TF T T T F

F F T FFFFF

Combining truth tables

P Q (~P & Q) → ~ (Q v P)

T TFT F TTTT

T F FT F FF T T

F T TF T TTT F

F FTFFFFFF

Combining truth tables

P Q (~P & Q) → ~ (Q v P)

T TFT F T F TTT

T F FT F FFF T T

F T TF T T F TTF

F FTFFF T FFF

Combining truth tables

P Q (~P & Q) → ~ (Q v P)

T TFT F TT F TTT

T F FT F F T F FTT

F T TF T T F F TTF

F FTFFF T T FFF

Combining truth tables

P Q (~P & Q) → ~ (Q v P)

T TFTFTTFTTT

T F FTFFTFFTT

F T TFTTFFTTF

F FTFFFTTFFF

Tautologies – true in all cases

P P v ~P

T T F

F F T

Tautologies – true in all cases

P P v ~P

T TTF

F FT T

Tautologies – true in all cases

P P v ~P

T TTF

F FTT

Contradictory (or truth-functionally inconsistent) – false in all cases

P P & ~P

T T F

F F T

Contradictory (or truth-functionally inconsistent) – false in all cases

P P & ~P

T T F F

F FFT

Contradictory (or truth-functionally inconsistent) – false in all cases

P P & ~P

T TFF

F FFT

Contingent – can be both true and false

Z R Z & R

T TT

T F F

F T F

F FF

Either Peter or Saul went to the bar.

Peter did not go.

Therefore, Saul went.

1) P v S

2) ~P

3) ∴ S

1) P v S

2) ~P

3) ∴ S

What’s this argument’s form?

Disjunctive syllogism.

1) P v S

2) ~P

3) ∴ S

[(P v S) & ~P] → S

P S [(P v S) & ~P] → S

T TTTTT

T F T F T F

F T F T F T

F FFFFF

P S [(P v S) & ~P] → S

T TTTFTT

T F T F FT F

F T F T TF T

F FFFTFF

P S [(P v S) & ~P] → S

T TTTT F TT

T F TTFFT F

F T FTTTF T

F FFFF T FF

P S [(P v S) & ~P] → S

T TTTTFFTT

T F TTFFFT F

F T FTTTTF T

F FFFFFTFF

P S [(P v S) & ~P] → S

T TTTTFFTTT

T F TTFFFTTF

F T FTTTTFTT

F FFFFFTFTF

This argument is valid; there is no line where the premises are all true and the conclusion is false.

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.

- Truth / refutation trees, S. pp. 68-77

- identical in purpose to tables

- more efficient

- but no time = no need