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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

Elementary Logic

PHIL 105-302

Intersession 2013

MTWHF 10:00 – 12:00

ASA0118C

Steven A. Miller

Day 4

formalizing review
Formalizing review

Symbolization chart:

It is not the case = ~

And = &

Or = v

If … then = →

If and only if = ↔

Therefore = ∴

logical semantics
Logical semantics

Our interpretations are concerned with statements’ truthand falsity.

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

logical semantics1
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 semantics2
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 semantics3
Logical semantics

Negation semantics (truth table)

P ~P

T F

F T

logical semantics4
Logical semantics

Conjunction semantics

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

is true when

“My name is Steven Miller.”

logical semantics5
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 semantics6
Logical semantics

Conjunction semantics (truth table)

P Q P & Q

T TT

T F F

F T F

F FF

logical semantics7
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 semantics8
Logical semantics

Disjunction semantics

“…or both”:

“Soup or salad?”

logical semantics9
Logical semantics

Disjunction semantics

Inclusive disjunction:

this, or that, or both

Exclusive disjunction:

this, or that, but not both

logical semantics10
Logical semantics

Disjunction semantics

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

“or” means:

this, or that, or both

logical semantics11
Logical semantics

Disjunction semantics (truth table)

P Q P v Q

T TT

T F T

F T T

F FF

logical semantics12
Logical semantics

Disjunction semantics

Exclusive disjunction symbolization:

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

logical semantics13
Logical semantics

Exclusive disjunction semantics (truth table)

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

T TTT

T F T F

F T T F

F FFF

logical semantics14
Logical semantics

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

logical semantics15
Logical semantics

Exclusive disjunction semantics (truth table)

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

T TT F FT

T F TTTF

F T TTTF

F FFFTF

logical semantics16
Logical semantics

Exclusive disjunction semantics (truth table)

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

T TTFFT

T F TTTF

F T TTTF

F FFFTF

logical semantics17
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 semantics18
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 semantics19
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 semantics20
Logical semantics

Material conditional semantics (truth table)

P Q P → Q

T TT

T F F

F T T

F FT

logical semantics21
Logical semantics

Biconditional semantics

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

P ↔ Q = (P → Q) & (Q → P)

logical semantics22
Logical semantics

Biconditional semantics (truth table)

P Q (P → Q) & (Q → P)

T TTT

T F F T

F T TF

F F T T

logical semantics23
Logical semantics

Biconditional semantics (truth table)

P Q (P → Q) & (Q → P)

T TTTT

T F FFT

F T T F F

F FTTT

logical semantics24
Logical semantics

Biconditional semantics (truth table)

P Q (P ↔ Q)

T TT

T F F

F T F

F F T

slide28

Seventh Inning Stretch

(“…Buy Me Some Peanuts …”)

logical semantics25
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 semantics26
Logical semantics

Combining truth tables

P Q ~~ (P & Q)

T TT

T F F

F T F

F FF

logical semantics27
Logical semantics

Combining truth tables

P Q ~~ (P & Q)

T T F T

T F T F

F T TF

F F T F

logical semantics28
Logical semantics

Combining truth tables

P Q ~~ (P & Q)

T TTFT

T F FTF

F T F TF

F FFTF

logical semantics29
Logical semantics

Combining truth tables

P Q ~~ (P & Q)

T TTFT

T F FTF

F T FTF

F FFTF

logical semantics30
Logical semantics

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

logical semantics31
Logical semantics

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

logical semantics32
Logical semantics

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

logical semantics33
Logical semantics

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

logical semantics34
Logical semantics

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

logical semantics35
Logical semantics

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

logical semantics36
Logical semantics

Combining truth tables

P Q (~P & Q) → ~ (Q v P)

T TFTFTTFTTT

T F FTFFTFFTT

F T TFTTFFTTF

F FTFFFTTFFF

three kinds of formulas
Three kinds of formulas

Tautologies – true in all cases

P P v ~P

T T F

F F T

three kinds of formulas1
Three kinds of formulas

Tautologies – true in all cases

P P v ~P

T TTF

F FT T

three kinds of formulas2
Three kinds of formulas

Tautologies – true in all cases

P P v ~P

T TTF

F FTT

three kinds of formulas3
Three kinds of formulas

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

P P & ~P

T T F

F F T

three kinds of formulas4
Three kinds of formulas

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

P P & ~P

T T F F

F FFT

three kinds of formulas5
Three kinds of formulas

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

P P & ~P

T TFF

F FFT

three kinds of formulas6
Three kinds of formulas

Contingent – can be both true and false

Z R Z & R

T TT

T F F

F T F

F FF

putting it all together
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 together1
Putting it all together

1) P v S

2) ~P

3) ∴ S

What’s this argument’s form?

Disjunctive syllogism.

putting it all together2
Putting it all together

1) P v S

2) ~P

3) ∴ S

[(P v S) & ~P] → S

putting it all together3
Putting it all together

P S [(P v S) & ~P] → S

T TTTTT

T F T F T F

F T F T F T

F FFFFF

putting it all together4
Putting it all together

P S [(P v S) & ~P] → S

T TTTFTT

T F T F FT F

F T F T TF T

F FFFTFF

putting it all together5
Putting it all together

P S [(P v S) & ~P] → S

T TTTT F TT

T F TTFFT F

F T FTTTF T

F FFFF T FF

putting it all together6
Putting it all together

P S [(P v S) & ~P] → S

T TTTTFFTT

T F TTFFFT F

F T FTTTTF T

F FFFFFTFF

putting it all together7
Putting it all together

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.

putting it all together8
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
Things we’re skipping

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

- identical in purpose to tables

- more efficient

- but no time = no need

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