Lecture Notes 8Lecture Notes 8

CS1502CS1502

Example ProofExample Proof

A A (B (B C) C)

(A (A B) B) (A (A C) C)

Valid ArgumentValid Argument

PP11

P P22

… … P Pnn

Q Q

Q is a tautological (logical) consequence of PQ is a tautological (logical) consequence of P11, P, P22, …, P, …, Pnn

(P(P1 1 P P2 2 … … P Pnn) ) Q is a tautology (logical necessity). Q is a tautology (logical necessity). NEW IDEANEW IDEA

Valid Argument

ExampleExample

Show Show P is a tautological consequence of P is a tautological consequence of (P (P Q). Q).

Methods of attack:Methods of attack:BooleBoole

– Show Show P is a tautological consequence of P is a tautological consequence of (P (P Q). Q).

– Show Show (P (P Q) Q) P is a tautology.P is a tautology.

FitchFitch– Show Show (P (P Q) is a valid argument Q) is a valid argument

P

Tautological ConsequenceTautological Consequence

TautologyTautology

Using FitchUsing Fitch

ExampleExample

Show Show P is not a tautological consequence of P is not a tautological consequence of (P (P Q). Q).

Method of attack:Method of attack:BooleBoole

– Show Show P is not a tautological consequence of P is not a tautological consequence of (P (P Q). Q).– Show Show (P (P Q) Q) P is not a tautology.P is not a tautology.

Build a worldBuild a world– Show Show (P (P Q) is an invalid argument Q) is an invalid argument

P

Not a Tautological Not a Tautological ConsequenceConsequence

Not a TautologyNot a Tautology

Build a WorldBuild a World

Let P be assigned true and Q false.Let P be assigned true and Q false. (P (P Q) is true while Q) is true while P is false.P is false.

premises conclusion

ExampleExample

Show the following argument is valid.Show the following argument is valid.

Cube(b) Cube(b) (Cube(c) (Cube(c) Cube(b)) Cube(b)) Cube(c) Cube(c)

Logical ConsequenceLogical Consequence

Cube(b) Cube(c) Cube(b) ~(Cube(b) ̂Cube(c )) ~Cube(c) is spurious?T T T F F noT F T T T noF T F T F noF F F T T no

Logical NecessityLogical Necessity

Cube(b) Cube(c) Cube(b) ~(Cube(b) ̂Cube(c) ~Cube(c) (Cube(b) ̂~(Cube(b) ̂Cube(c))) -> ~Cube(c)T T T F F TT F T T T TF T F T F TF F F T T T

Every non-spurious row is true! In fact, every row is true, so a Tautology!!

FitchFitch

Non-consequenceNon-consequence

Show the following argument is invalid.Show the following argument is invalid.

Cube(a) Cube(a) Cube(b) Cube(b) (Cube(c) (Cube(c) Cube(b)) Cube(b)) Cube(c)Cube(c)

CounterexampleCounterexample

Inference PatternsInference Patterns

Modus PonensModus Ponens P P Q Q P P Q Q

Tautological ConsequenceTautological Consequence

TautologyTautology

EliminationElimination

P P Q Q … …

P P … … Q Q Elim

IntroductionIntroduction

PP … …

Q Q P P Q Q Intro

EliminationElimination

P P Q Q … …

P P … … Q Q Elim

IntroductionIntroduction

PP … … Q Q

Q Q … … P P P P Q Q

Intro

Inference PatternsInference Patterns

Modus TollensModus Tollens P P Q Q QQ PP

Modus TollensModus Tollens