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## PowerPoint Slideshow about ' Elementary Logic' - italia

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Presentation Transcript

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’ truthand 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

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

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

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

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 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 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 semantics

Exclusive disjunction semantics (truth table)

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

T TTFFT

T F TTTF

F T TTTF

F FFFTF

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

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 F F T

F T TF

F F T T

Logical semantics

Biconditional semantics (truth table)

P Q (P → Q) & (Q → P)

T TTTT

T F FFT

F T T F F

F FTTT

(“…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) → ~ (Q v P)

T TTTTT

T F T F F T

F T F T T F

F FFFFF

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 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 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 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 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 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

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

P P & ~P

T T F

F F T

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 formulas

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

P P & ~P

T TFF

F FFT

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

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 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. 68-77

- identical in purpose to tables

- more efficient

- but no time = no need

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