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CS1502 Formal Methods in CS1502 Formal Methods in Computer ScienceComputer Science

Lecture Notes 4Lecture Notes 4Tautologies and Logical TruthTautologies and Logical Truth

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Constructing a Truth TableConstructing a Truth Table

Write down sentenceWrite down sentenceCreate the reference columnsCreate the reference columnsUntil you are done:Until you are done:– Pick the next connective to work onPick the next connective to work on– Identify the columns to considerIdentify the columns to consider– Fill in truth values in the columnFill in truth values in the column

EG: ~(A ^ (~A v (B ^ C))) v BEG: ~(A ^ (~A v (B ^ C))) v B(in Boole and on board)(in Boole and on board)

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TautologyTautology

A sentence S is a A sentence S is a tautologytautology if and only if if and only if every row of its truth table assigns true to every row of its truth table assigns true to S.S.

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ExampleExample

Is Is (A (A ( (A A (B (B C))) C))) B a tautology? B a tautology?

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ExampleExample

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Logical PossibilityLogical Possibility

A sentence S is A sentence S is logically possiblelogically possible if it could if it could be true (i.e., it is true in be true (i.e., it is true in somesome world) world)

It is It is TW-possibleTW-possible if it is true in if it is true in somesome world world that that can be built using the program can be built using the program

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ExamplesExamples

Cube(b) Cube(b) Large(b) Large(b)

(Tet(c) (Tet(c) Cube(c) Cube(c) Dodec(c)) Dodec(c))

e e e e

Logically possible TW-possible

Not TW-possibleLogically possible

Not Logically possible

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Spurious RowsSpurious Rows

A A spurious rowspurious row in a truth table is a row in a truth table is a row whose whose reference columnsreference columns describe a describe a situation or circumstance that is situation or circumstance that is impossibleimpossible to realize on logical grounds. to realize on logical grounds.

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ExampleExampleP Q R

Cube(a) a=b Cube(b)T T TT T FT F TT F FF T TF T FF F T

Spurious!

Spurious!

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Logical NecessityLogical Necessity

A sentence S is a A sentence S is a logical necessitylogical necessity ((logicallogical truthtruth)) if and only if S is true in every logical if and only if S is true in every logical circumstance.circumstance.A sentence S is a A sentence S is a logical necessitylogical necessity ((logicallogical truthtruth)) if and only if S is true in every non- if and only if S is true in every non-spurious row of its truth table.spurious row of its truth table.

Logical-Necessity

TW-Necessity

1111

ExampleExample

a=b b=b a=b v b=bT T TT F TF T TF F F

Logical Necessity TW-NecessityNot a tautology

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ExampleExample

Larger(a,b) Smaller(a,b) Larger(a,b) v Smaller(a,b)T T T spuriousT F TF T TF F F

Not a TW-NecessityNot a Logical NecessityNot a tautology

According to the book, the first row is spurious, because a cannot be both larger and smaller than b. Technically, though, “Larger” and “Smaller” might mean anyrelation between objects. So, the first row is really only TW-spurious. This issuewon’t come up with any exam questions based on this part of the book. (The bookrefines this later.)

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Sentence Hierarchy

Tautologies

Logical Necessities

TW Necessities

Tet(b) Tet(b)

a=a

Tet(b) Cube(b) Dodec(b)

Cube(a) Small(a)

Cube(a) v Cube(b)

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Tautological EquivalenceTautological Equivalence

Two sentences Two sentences SS and and SS’ are’ are tautologically tautologically equivalentequivalent if and only if every row of their if and only if every row of their joint truth table assigns the same values to joint truth table assigns the same values to SS and and S’S’..

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ExampleExample

S and S’ are Tautologically Equivalent

P Q P --> Q ~P v QT T T TT F F FF T T TF F T T

S S’

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Logical EquivalenceLogical Equivalence

Two sentences S and S’ are Two sentences S and S’ are logically logically equivalentequivalent if and only if every non-spurious if and only if every non-spurious row of their joint truth table assigns the row of their joint truth table assigns the same values to S and S’.same values to S and S’.

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ExampleExample

Not Tautologically equivalent Logically Equivalent

a=b Cube(a) Cube(b) a=b ̂Cube(a) a=b ̂Cube(b)T T T T TT T F T FT F T F TT F F F FF T T F FF T F F FF F T F FF F F F F

S S’

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Tautological ConsequenceTautological Consequence

Sentence Sentence QQ is a is a tautological consequencetautological consequence of of PP11, P, P22, …, P, …, Pnn if and only if every row that if and only if every row that assigns true to all of the premises also assigns true to all of the premises also assigns true to assigns true to QQ..Remind you of anything?Remind you of anything?P1,P2,…,PnP1,P2,…,Pn | | QQ is also a valid argument!is also a valid argument!A Con Rule:A Con Rule: Taut Tautological ological ConConsequencesequence

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Example Example

P Q P v Q ~Q PT T T F TT F T T TF T T F FF F F T F

Tautological consequence

premises conclusion

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Logical ConsequenceLogical Consequence

Sentence Sentence QQ is a is a logical consequencelogical consequence of of PP11, P, P22, …, P, …, Pnn if and only if if and only if every non-every non-spuriousspurious rowrow that assigns true to all of the that assigns true to all of the premises also assigns true to premises also assigns true to QQ..

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a=b b=c a=c a=b ̂b=c a=cT T T T TT T F T FT F T F TT F F F FF T T F TF T F F FF F T F TF F F F F

Not a tautological consequence

Is a logical consequence

premise conclusion

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SummarySummaryNecessaryNecessary

S is always trueS is always true

PossiblePossible

S could be trueS could be true

EquivalenceEquivalence

S and S’ always S and S’ always have the same have the same truth valuestruth values

ConsequenceConsequence

Whenever Whenever P1…Pn are P1…Pn are true, Q is also true, Q is also truetrue

TautologicalTautologicalAll rows in truth All rows in truth tabletable

S is a tautologyS is a tautology S is S is Tautologically Tautologically possiblepossible

S and S’ are S and S’ are Tautologically Tautologically equivalentequivalent

Q is a Q is a tautological tautological consequence of consequence of P1…PnP1…Pn

LogicalLogicalAll non-spurious All non-spurious rowsrows

S is logically S is logically necessary necessary (logical truth)(logical truth)

S is logically S is logically possiblepossible

S and S’ are S and S’ are logically logically equivalentequivalent

Q is a logical Q is a logical consequence of consequence of P1…PnP1…Pn

TWTWLogic + Tarski’s Logic + Tarski’s WorldWorld

S is TW S is TW necessarynecessary

S is TW S is TW possiblepossible

S and S’ are S and S’ are TW equivalentTW equivalent

Q is a TW-Q is a TW-consequence of consequence of P1…PnP1…Pn

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SummarySummary

Every tautological consequence of a set of premises is a Every tautological consequence of a set of premises is a logical consequence of these premises.logical consequence of these premises.Not every logical consequence of a set of premises is a Not every logical consequence of a set of premises is a tautological consequence of these premises.tautological consequence of these premises.

Tautological-Consequences of

P1…Pn

Logical-Consequences of

P1…Pn

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SummarySummary

Every tautological equivalence is a logical Every tautological equivalence is a logical equivalence.equivalence.Not every logical equivalence is a Not every logical equivalence is a tautological equivalence.tautological equivalence.

TautologicalEquivalences

Logical

Equivalences

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SummarySummary

Every tautology is a logical necessity.Every tautology is a logical necessity.Not every logical necessity is a tautology.Not every logical necessity is a tautology.

Tautologies

Logical

Necessities