Sentential Logic（SL） 1.Syntax: The language of SL / Symbolize 2.Semantic: a sentence / compare two sentences / compare a set of sentences 3.DDerivation.

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Sentential Logic（SL） 1.Syntax: The language of SL / Symbolize 2.Semantic: a sentence / compare two sentences / compare a set of sentences 3.DDerivation

Syntax: The Language SL Vocabularies: A, B, C, …,Y, Z, A1... Logical connection: &, ~, , ,  Punctuation marks: ( ) Sentences

A recursive definition of sentences of SL 1.Every sentence letter is a sentence. 2.If P is a sentence, then ~ P is a sentence. 3.If P and Q are sentences, then ( P & Q ) is a sentence. 4.If P and Q are sentences, then ( P  Q ) is a sentence. 5.If P and Q are sentences, then ( P  Q ) is a sentence. 6.If P and Q are sentences, then ( P  Q ) is a sentence. 7.Nothing is a sentence unless it can be formed by repeated application of clauses 1-6.

Syntax: Symbolize

Conjunction (&) P Q P & Q T T F F T F T F T T F F T F T F T F F F

Negation (~) P ~ P T T F F T T F F F F T T

Disjunction (  ) P Q P  Q T T F F T F T F T T F F T F T F T T T F

Conditional (  ) P Q P  Q T T F F T F T F T T F F T F T F T F T T

Biconditional (  ) P Q P  Q T T F F T F T F T T F F T F T F T F F T

Semantic: a sentence Truth-functional truth: A sentence P of SL is truth-functionally true if and only if P is true on every truth-value assignment. T ruth-functional false : A sentence P of SL is truth-functionally false if only if P is false on every truth value assignment. Truth-functionally indeterminate: A sentence P of S L is truth-functionally indeterminate if and only if P is neither truth-functionally true or truth- functionally false.

A B (A  B)  (~A  B) T T F F T F T F T T F F F F T T T F F T Truth-functionally truth T T F F T F T T T F T F T F T T T T T T

A B ~ ( (~A  B)  ~ (A  B) ) T T F F T F T F Truth-functionally false T T F F F F T T T F T F T F T T T T F F T F T F T F T T F T F F T T T T F F F F

Truth-functional indeterminate A B ( A  B )  A T T F F T F T F T T F F T F T F T F T T T T F F T T F F

Semantic: compare two sentences T ruth-functionally equivalent : Sentences P and Q of S L are truth-functionally equivalent if and only if there is no truth-value assignment on which P and Q are different truth-values. T ruth-functionally contradictory : Sentences P and Q of S L are truth-functionally contradictory if and only if there is no truth-value assignment on which P and Q are the same truth-values. T ruth-functionally independent : Sentences P and Q of S L are truth-functionally independent if they are neither truth-functionally equivalent nor truth-functionally contradictory.

A B A & B / ~ (A  ~B) T T F F T F T F Truth-functionally equivalent T F F F T T F F T F T T T T F F T F T F F T T T T F F F F T F T

A B A  B / ~ ((~A  B) & (~B  A)) T T F F T F T F Truth-functionally contradictory T T F F T F T F T F F T T T F F F F T T T F T F T F T T T F T F F T F T T T F F T F F T T F F T F T T F

A B A & B / A  B T T F F T F T F Truth-functionally independent T F F F T T F F T F T T T T F F T F T F T T T F

Semantic: compare a set of sentences Truth-functionally consistent: A set of sentences of SL is truth-functionally if and only if there is at least one truth- functionally assignment on which all the numbers of the set are true. Truth-functionally inconsistent: A set of sentences of SL is truth-functionally inconsistent if and only if it is not truth-functionally consistent.

A B H A / B  H / B T T T T F F F F T T F F T T F F Truth-functionally consistent T F T F T F T F T T T T F F F F T T F F T T F F T F T F T F T F T F T T T F T T T T F F T T F F i n v a l i d

J H (J  J)  H / ~ J / ~ H T T F F T F T F Truth-functionally inconsistent T T F F T T F F T T T T T F T F T F T T T T F F F F T T T F T F F T F T v a l i d

Derivation (1) Derive: ~N 1 H  ~N Assumption 2 ( H  G) & ~M Assumption 3 ~N  ( G  B ) Assumption 4 H  G 2 &E 5 H Assumption 6 ~ N 1, 5  E 7 G Assumption 8 G  B 7  I 9 ~ N 3, 8  E 8 ~N 4, 5-6, 7-9  E

(2) Derive: L & ~K 1 (~L  K)  A Assumption 2 A  ~A Assumption 3 ~L Assumption 4 ~L  K 3  I 5 A 1, 4  E 6 ~A 2, 5  E 7 L Assumption 8 K Assumption 9 ~L  K 8  I 10 A 1, 9  E 11 ~A 2, 10  E 12 ~K 8-11 ~I 13 L & ~K 7,12 &I

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