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

Elementary Logic

PHIL 105-302Intersession 2013

MTWHF 10:00 – 12:00ASA0118C

Steven A. MillerDay 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 ~PT FF T

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 & QT T TT F FF T FF F F

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


“…or both”:

“Soup or salad?”

Logical semantics


Inclusive disjunction:this, or that, or both

Exclusive disjunction:this, or that, but not both

Logical 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 QT T TT F TF T TF F F

Logical 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 TT F T FF T T FF F F F

Logical semantics


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 semantics


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 semantics


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 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


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


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

Antecedent = falseConsequent = false

Logical semantics

Material conditional semantics (truth table)

P Q P → QT T TT F FF T TF F T

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 T T TT F F TF T T FF F T T

Logical semantics


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

Logical semantics


P Q (P ↔ Q)T T TT F FF T FF 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


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

Logical semantics


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

Logical semantics


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

Logical semantics


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

Logical semantics


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 semantics


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 semantics


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 semantics


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 semantics


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 semantics


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 semantics


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 formulas

Tautologies – true in all cases

P P v ~PT T FF F T



P P v ~PT T T FF F T T



P P v ~PT T T FF F T T


Contradictory (or truth-functionally inconsistent) – false in all casesP P & ~PT T FF F T


Contradictory (or truth-functionally inconsistent) – false in all casesP P & ~PT T F FF F F T


Contradictory (or truth-functionally inconsistent) – false in all casesP P & ~PT T F FF F F T


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 S2) ~P3) ∴ S


1) P v S2) ~P3) ∴ S

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


1) P v S2) ~P3) ∴ S

[(P v S) & ~P] → S


P S [(P v S) & ~P] → ST T T T T T

T F T F T F F T F T F T F F F F F F


P S [(P v S) & ~P] → ST 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


P S [(P v S) & ~P] → ST 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


P S [(P v S) & ~P] → ST 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


P S [(P v S) & ~P] → ST 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.


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

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

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