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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.

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**SL: Basic syntax -alphabet**

SL alphabet: - A, B, C, ... - ~, &, , , - (, )

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**SL: Basic syntax - propositions**

A, B, C, ... propositions. have value T or F

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**SL: Basic syntax - negation**

~A is read: “not A” A

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**SL: Basic syntax - conjunction**

A&B is read: ”A and B” A is left conjunct, B is right conjunct AB, AB

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**SL: Basic syntax - disjunction**

AB is read: “A or B” A is left disjunct, B is right disjunct

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**SL: Basic syntax - implication**

material implication AB is read “A implies B”, or “If A, then B” A is the antecedent B is the consequent AB, AB

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**SL: Basic syntax - biconditional**

material biconditional or equivalence AB is read: “A is equivalent to B” AB, AB

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**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 AB is read: “A or B” A is left disjunct, B is right disjunct material implication AB is read “A implies B”, or “If A, then B” A is the antecedent B is the consequent material biconditional or equivalence AB is read: “A is equivalent to B”

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**SL: Basic syntax - formulae**

Atomic formulae – have no connectives

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**SL: Basic syntax - formulae**

Atomic formulae – have no connectives If A is a formula, so is ~A

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**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, AB, AB, AB

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**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, AB, AB, AB nothing else is a formula

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**SL: Basic syntax – main connective**

For more complex formulae brackets tell as what is the main connective:

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**SL: Basic syntax – main connective**

For more complex formulae brackets tell as what is the main connective: ~( ) is negation

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**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.

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**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.)

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**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.

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**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.

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**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)

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

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**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

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**A is left disjunct, B is right disjunct**

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

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** material implication**

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

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** material biconditional or equivalence**

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

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**Truth table for (AB)(~AB)**

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**Truth table for (AB)(~AB)**

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**Truth table for (AB)(~AB)**

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**Truth table for (AB)(~AB)**

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**Truth table for (AB)(~AB)**

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**Truth table for (AB)(~AB)**

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

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Peirce’s law ((A B) A) A T F

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Peirce’s law ((A B) A) A T T F F

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Peirce’s law ((A B) A) A T T F F

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Peirce’s law ((A B) A) A T T F F

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Peirce’s law ((A B) A) A T T F F

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Peirce’s law ((A B) A) A T T F F

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Peirce’s law ((A B) A) A T T F F

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

1 ( (A B) C ) T F

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

2 (B C)) 1 ( (A B) C ) T F

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**Truth table for ( (AC) (BC) ) ( (AB)C )**

3 C) 2 (B C)) 1 ( (A B) C ) T F 4 5 6 7 8

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**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)

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**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.

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**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.

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**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.

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**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.

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**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.

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**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.

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