elementary logic

Elementary LogicPHIL 105-302

Intersession 2013MTWHF 10:00 – 12:00

ASA0118CSteven A. Miller

Day 4

Formalizing reviewSymbolization 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 semanticsNegation 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 semanticsNegation 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 semanticsNegation semantics (truth table)

P ~PT FF T

Logical semanticsConjunction semantics

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

“My name is Steven Miller.”

Logical semanticsConjunction semantics

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

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

Logical semanticsConjunction semantics (truth table)

P Q P & QT T TT F FF T FF F F

Logical semanticsDisjunction semantics

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

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

Logical semanticsDisjunction semantics

“…or both”:

“Soup or salad?”

Logical semanticsDisjunction semantics

Inclusive disjunction:this, or that, or both

Exclusive disjunction:this, or that, but not both

Logical semanticsDisjunction semantics

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

“or” means:this, or that, or both

Logical semanticsDisjunction semantics (truth table)

P Q P v QT T TT F TF T TF F F

Logical semanticsDisjunction semantics

Exclusive disjunction symbolization:

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

Logical semanticsExclusive disjunction semantics (truth

table)

P Q (P v Q) & ~ (P & Q)T T T TT F T FF T T FF F F F

Logical semanticsExclusive disjunction semantics (truth

table)

P Q (P v Q) & ~ (P & Q)T T T F TT F T T FF T T T FF F F T F

Logical semanticsExclusive disjunction semantics (truth

table)

P Q (P v Q) & ~ (P & Q)T T T F F TT F T T T FF T T T T FF F F F T F

Logical semanticsExclusive disjunction semantics (truth

table)

P Q (P v Q) & ~ (P & Q)T T T F F TT F T T T FF T T T T FF F F F T F

Logical semanticsMaterial 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 semanticsMaterial 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 semanticsMaterial conditional semantics

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

Antecedent = falseConsequent = false

Logical semanticsMaterial conditional semantics

(truth table)

P Q P → QT T TT F FF T TF F T

Logical semanticsBiconditional semantics

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

P ↔ Q = (P → Q) & (Q → P)

Logical semanticsBiconditional semantics (truth table)

P Q (P → Q) & (Q → P)T T T TT F F TF T T FF F T T

Logical semanticsBiconditional semantics (truth table)

P Q (P → Q) & (Q → P)T T T T TT F F F TF T T F FF F T T T

Logical semanticsBiconditional semantics (truth table)

P Q (P ↔ Q)T T TT F FF T FF F T

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

Logical semanticsCombining 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 semanticsCombining truth tables

P Q ~~ (P & Q)T T TT F FF T FF F F

Logical semanticsCombining truth tables

P Q ~~ (P & Q)T T F TT F T FF T T FF F T F

Logical semanticsCombining truth tables

P Q ~~ (P & Q)T T T F TT F F T FF T F T FF F F T F

Logical semanticsCombining truth tables

P Q ~~ (P & Q)T T T F TT F F T FF T F T FF F F T F

Logical semanticsCombining truth tables

P Q (~P & Q) → ~ (Q v P)T T T T T TT F T F F TF T F T T FF F F F F F

Logical semanticsCombining truth tables

P Q (~P & Q) → ~ (Q v P)T T F T T T TT F F T F F TF T T F T T FF F T F F F F

Logical semanticsCombining truth tables

P Q (~P & Q) → ~ (Q v P)T T F T F T T TT F F T F F F TF T T F T T T FF F T F F F F F

Logical semanticsCombining truth tables

P Q (~P & Q) → ~ (Q v P)T T F T F T T T TT F F T F F F T TF T T F T T T T FF F T F F F F F F

Logical semanticsCombining truth tables

P Q (~P & Q) → ~ (Q v P)T T F T F T F T T TT F F T F F F F T TF T T F T T F T T FF F T F F F T F F F

Logical semanticsCombining truth tables

P Q (~P & Q) → ~ (Q v P)T T F T F T T F T T TT F F T F F T F F T TF T T F T T F F T T FF F T F F F T T F F F

Logical semanticsCombining truth tables

P Q (~P & Q) → ~ (Q v P)T T F T F T T F T T TT F F T F F T F F T TF T T F T T F F T T FF F T F F F T T F F F

Three kinds of formulasTautologies – true in all cases

P P v ~PT T FF F T

Three kinds of formulasTautologies – true in all cases

P P v ~PT T T FF F T T

Three kinds of formulasTautologies – true in all cases

P P v ~PT T T FF F T T

Three kinds of formulasContradictory (or truth-functionally

inconsistent) – false in all casesP P & ~PT T FF F T

Three kinds of formulasContradictory (or truth-functionally

inconsistent) – false in all casesP P & ~PT T F FF F F T

Three kinds of formulasContradictory (or truth-functionally

inconsistent) – false in all casesP P & ~PT T F FF F F T

Three kinds of formulasContingent – 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 togetherEither Peter or Saul went to the bar.Peter did not go.Therefore, Saul went.

1) P v S2) ~P3) ∴ S

Putting it all together1) P v S2) ~P3) ∴ S

What’s this argument’s form?Disjunctive syllogism.

Putting it all together1) P v S2) ~P3) ∴ S

[(P v S) & ~P] → S

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

T T T 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 togetherA 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