Phil 120 week 1 About this course. Introducing the language SL. Basic syntax. Semantic definitions of the connectives in terms of their truth conditions.

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Presentation transcript:

Phil 120 week 1 About this course. Introducing the language SL. Basic syntax. Semantic definitions of the connectives in terms of their truth conditions using truth tables.

SL: Basic syntax -alphabet SL alphabet: - A, B, C, ... - ~, &, , ,  - (, )

SL: Basic syntax - propositions A, B, C, ... propositions. have value T or F

SL: Basic syntax - negation ~A is read: “not A” A

SL: Basic syntax - conjunction A&B is read: ”A and B” A is left conjunct, B is right conjunct AB, AB

SL: Basic syntax - disjunction AB is read: “A or B” A is left disjunct, B is right disjunct

SL: Basic syntax - implication  material implication AB is read “A implies B”, or “If A, then B” A is the antecedent B is the consequent AB, AB

SL: Basic syntax - biconditional  material biconditional or equivalence AB is read: “A is equivalent to B” AB, AB

SL: Basic syntax - ~& A, B, C, ... propositions. have value T or F ~ negation ~A is read: “not A” & conjunction A&B is read: ”A and B” A is left conjunct, B is right conjunct  disjunction AB is read: “A or B” A is left disjunct, B is right disjunct  material implication AB is read “A implies B”, or “If A, then B” A is the antecedent B is the consequent  material biconditional or equivalence AB is read: “A is equivalent to B”

SL: Basic syntax - formulae Atomic formulae – have no connectives

SL: Basic syntax - formulae Atomic formulae – have no connectives If A is a formula, so is ~A

SL: Basic syntax - formulae Atomic formulae – have no connectives If A is a formula, so is ~A If A and B are formulae, so are A&B, AB, AB, AB

SL: Basic syntax - formulae Atomic formulae – have no connectives If A is a formula, so is ~A If A and B are formulae, so are A&B, AB, AB, AB nothing else is a formula

SL: Basic syntax – main connective For more complex formulae brackets tell as what is the main connective:

SL: Basic syntax – main connective For more complex formulae brackets tell as what is the main connective: ~(-------) is negation

SL: Basic syntax – main connective For more complex formulae brackets tell as what is the main connective: ~(-------) is negation (---------) & (---------) is a conjunction (---------)  (----------) is an implication, etc.

SL: Basic syntax – main connective For more complex formulae brackets tell as what is the main connective: ~(-------) is negation (---------) & (---------) is a conjunction (---------)  (----------) is an implication, etc. Formula (((A&B)  ~C)  D)  ~B is an implication. (If A and B or not C is equivalent to D, then not B.)

SL: Basic syntax – main connective For more complex formulae brackets tell as what is the main connective: ~(-------) is negation (---------) & (---------) is a conjunction (---------)  (----------) is an implication, etc. Formula (((A&B)  ~C)  D)  ~B is an implication. (If A and B or not C is equivalent to D, then not B.) The antecedent is a biconditional.

SL: Basic syntax – main connective For more complex formulae brackets tell as what is the main connective: ~(-------) is negation (---------) & (---------) is a conjunction (---------)  (----------) is an implication, etc. Formula (((A&B)  ~C)  D)  ~B is an implication. (If A and B or not C is equivalent to D, then not B.) The antecedent is a biconditional. The consequent is a negation.

SL: Basic syntax – main connective What is the main connective in the formulae below? ((C ∨ B) ⊃ A) ⊃ (H & ∼ L) (F ∨ (G ∨ D)) & (∼ (F ∨ G) ∨ (∼ (F ∨ D) ∨ ∼ (G ∨ D))) (C & N) ∨ ((C & T) ∨ (N & T)) (∼ A ∨ (H ⊃ J)) ⊃ (A ∨ J) ∼ (A ∨ B) ⊃ (∼ A ∨ ∼B)

SL: Truth table for ~ ~ negation ~A is read: “not A” A ~A T F

A is left conjunct, B is right conjunct SL: Truth table for & & conjunction A&B is read: ”A and B” A is left conjunct, B is right conjunct A B A&B T F

A is left disjunct, B is right disjunct SL: Truth table for   disjunction AB is read: “A or B” A is left disjunct, B is right disjunct A B A  B T F

 material implication SL: Truth table for   material implication AB is read “A implies B”, or “If A, then B” A is the antecedent. B is the consequent. A B A  B T F

 material biconditional or equivalence SL: Truth table for   material biconditional or equivalence AB is read: “A is equivalent to B” A B A  B T F

Truth table for (AB)(~AB)

Truth table for (AB)(~AB)

Truth table for (AB)(~AB)

Truth table for (AB)(~AB)

Truth table for (AB)(~AB)

Truth table for (AB)(~AB)

SL: Truth table for & A B A&B T F A & B T F

Peirce’s law ((A  B) A) A T F

Peirce’s law ((A  B) A) A T T F F

Peirce’s law ((A  B) A) A T T F F

Peirce’s law ((A  B) A) A T T F F

Peirce’s law ((A  B) A) A T T F F

Peirce’s law ((A  B) A) A T T F F

Peirce’s law ((A  B) A) A T T F F

Truth table for ( (AC)  (BC) )  ( (AB)C )

Truth table for ( (AC)  (BC) )  ( (AB)C )

Truth table for ( (AC)  (BC) )  ( (AB)C )

Truth table for ( (AC)  (BC) )  ( (AB)C )

Truth table for ( (AC)  (BC) )  ( (AB)C ) 1 ( (A  B) C ) T F

Truth table for ( (AC)  (BC) )  ( (AB)C ) 2 (B C)) 1 ( (A  B) C ) T F

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Truth table for ( (AC)  (BC) )  ( (AB)C ) 3 C)  2 (B C)) 1 ( (A  B) C ) T F 4 5 6 7 8

Construct truth tables for: ~A  (A  ~B) ~(B~B) ~B  (A  ~A) Practice Construct truth tables for: ~A  (A  ~B) ~(B~B) ~B  (A  ~A)

TF truth, falsity, and indeterminacy P is truth-functionally true iff it has the value T for any truth-value assignment. P is truth-functionally false iff it has the value F for any truth-value assignment. P is truth-functionally indeterminate iff it has the value T for some truth-value assignments, and the value F for some other truth-value assignments.

TF truth, falsity, and indeterminacy P is truth-functionally true iff it has the value T for any truth-value assignment. P is truth-functionally false iff it has the value F for any truth-value assignment. P is tf-false iff ~P is tf-true P is truth-functionally indeterminate iff it has the value T for some truth-value assignments, and the value F for some other truth-value assignments.

TF truth, falsity, and indeterminacy P is truth-functionally true iff it has the value T for any truth-value assignment. P is truth-functionally false iff it has the value F for any truth-value assignment. P is tf-false iff ~P is tf-true P is truth-functionally indeterminate iff it has the value T for some truth-value assignments, and the value F for some other truth-value assignments. P is tf-indeterminate iff it is neither tf-true nor th-false.

TF equivalence and consistency P and Q are truth-functionally equivalent iff P and Q do not have different truth-values for any truth-value assignment.

TF equivalence and consistency P and Q are truth-functionally equivalent iff P and Q do not have different truth-values for any truth-value assignment. A set of sentences is truth-functionally consistent iff there is a truth-value assignment that on which all the members of the set have the value T.

TF equivalence and consistency P and Q are truth-functionally equivalent iff P and Q do not have different truth-values for any truth-value assignment. A set of sentences is truth-functionally consistent iff there is a truth-value assignment that on which all the members of the set have the value T. A set of sentences is truth-functionally inconsistent iff it is not tf-consistent.