TRUTH TABLES TYPES OF COMPOUND STATEMENTS Gut idea (1) Certain compound statements are truth-functional. A compound statement is truth-functional if its truth value is completely determined by the truth value of the atomic statements. (2) In that case, we can systematically define the truth value of certain compound statements by using truth tables. (2) Given this systematic definition, we will be able to tell whether many arguments are valid or invalid in a mechanical way.

Negations p T F

~p F T

~p: opposite truth-value Illustration: God does not exist. (G: God exists) G T F

~G F T

Conjunctions (components called ‘conjuncts’) p T T F F

q T F T F

p●q T F F F

p ● q: always false except when both conjuncts are true Illustration: God exists and Moses gave the Ten Commandments to Israel. (G: God exists; M: Moses gave the Ten Commandments to Israel) G T T F F

M T F T F

G●M T F F F

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Disjunctions (components called ‘disjuncts’) Note: this is a truth table for inclusive disjunction p T T F F

q T F T F

pvq T T T F

p v q: always true except when both disjuncts are false Illustration: Either God exists or Barack Obama exists. (G: God exists; B: Barack Obama exists) G T T F F

B T F T F

GvB T T T F

Material Conditionals (components called ‘antecedent’ and ‘consequent’) Conditional are complicated. But there’s one very important fact about all conditionals: if the antecedent is true and the consequent is false, then the conditional as a whole is false. This fact is so important, logicians have defined one type of conditional as being false only when the antecedent is true and the consequent is false: the material conditional. p T T F F

q T F T F

pÆq T F T T

p Æ q: always true except when the antecedent is true and the consequent is false •

Illustrations of conditionals that conform to the truth table for material conditionals: (1) True conditional with true antecedent and true consequent: If Seattle is in Washington, then Seattle in the US. (True) T T 2

(2) False conditional with true antecedent and false consequent: If Seattle is in Washington, then Seattle is in Europe. (False) T F (3) True conditional with false antecedent and true consequent: If Seattle is in Kentucky, then Seattle is in the US. (True) F T (4) True conditional with false antecedent and false consequent: If Seattle is in Alabama, then Seattle is in a southern state. (True) F F •

Note: not every English conditional conforms to the truth table for material conditionals. p T T F F

q T F T F

pÆq T F T T

To illustrate: If Seattle is in Germany, then it’s in the US. F T This conditional is false (Seattle would be in Europe), but the truth table says it is true. See row 3. To illustrate: If Seattle is in Germany, then it’s on the moon. F F This conditional is false (Seattle would be in Europe), but the truth table says it is true. See row 4. Moral: Not every conditional conforms to the truth table that defines the material conditional. That is, not every conditional is a material conditional. So, the system must be amended in order to capture, for example, all conditionals with false antecedents, otherwise known as counterfactual conditionals. (We do the rudiments of counterfactual logic in another class: Philosophy 330, Metaphysics.)

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Material Biconditionals p T T F F

q T F T F

p↔q T F F T

p ↔ q: always true except when its two constituent statements have different truth values Illustration: God exists if and only if our world is the best possible world. G: God exists B: Our world is the best possible world G T T F F

B T F T F

G↔B T F F T

Conjunction of material conditionals G T T F F

B T F T F

GÆB T F T T

BÆG T T F T

(G Æ B) ● (B Æ G) T F F T

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Overview: Truth Tables for Compounds Negation, ~p: opposite truth-value p T F

~p F T

Conjunction, p ● q: always false except when both conjuncts are true p T T F F

q T F T F

p●q T F F F

Disjunctions, p v q: always true except when both disjuncts are false p T T F F

q T F T F

pvq T T T F

Material conditionals, p Æ q: always true except when the antecedent is true and the consequent is false p T T F F

q T F T F

pÆq T F T T

Material biconditionals, p ↔ q: always true except when its two constituent statements have different truth values p T T F F

q T F T F

p↔q T F F T

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