Exam 2: Propositional Logic: Syntax and Semantic

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -{X $\lor$ [C • (Y ⊃ B)]} ⊃ {Z ⊃ [Z ⊃ (Z ⊃ Z)]}

For each of the following questions, determine whether the given formulas is a wff or not. If it is a wff, indicate its main operator. -I • E ⊃ ∼F

construct a complete truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. -F • (∼G $\lor$ H) and F ≡ (∼H • G)

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, choose an option which presents a counterexample. (There may be other counterexamples as well.) -(G $\lor$ ∼H) $\lor$ ∼∼I / G $\lor$ ∼H

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction. Justify your answers by appeal to the meanings of those terms. -(G ? H) ? ? [(G • H) $\lor$ (?G • ?H)]

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, specify a counterexample. -S ⊃ (T $\lor$ U) ∼T T $\lor$ S / U • S

construct a complete truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. -L $\lor$ (M ⊃ N) and ∼L • (M • ∼N)

Instructions: For questions 11-20, construct complete a truth table for each of the following pairs of propositions. Then, using the truth table, determine whether the statements are logically equivalent or contradictory. If neither, determine whether they are consistent or inconsistent. Justify your answers. -G ≡ ∼H and (H • ∼G) $\lor$ (G • ∼H)

Construct a complete truth table for each of the following arguments. Then, using the truth table, determine whether the argument is valid or invalid. If the argument is invalid, specify a counterexample. -∼A ≡ B (∼A $\lor$ B) ⊃ C / C

For each of the following questions, determine whether the given formulas is a wff or not. If it is a wff, indicate its main operator. -[A $\lor$ B • C] ⊃ (A $\lor$ C)

use indirect truth tables to determine whether each of the following arguments is valid. If the argument is invalid, specify a counterexample. -(M • N) ? (O • P) (M • O) ? (Q $\lor$ ?P) R ? (Q ? N) / (?M • ?N) $\lor$ (O • P)

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction. -(A • ∼B) • (B $\lor$ ∼A)

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator. -[(W ⊃ X) • (Y $\lor$ ∼X)] ≡ [∼(Z $\lor$ Y) ⊃ ∼W]

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -∼(Q ⊃ C) $\lor$ (Z • ∼X)

use the following key to translate each of the given arguments into symbols of PL. B: Brouwer is an intuitionist. F: Frege is a logicist. G: Gödel is a platonist. H: Hilbert is a formalist. -If Frege is a logicist, then Brouwer is an intuitionist. If Brouwer is an intuitionist, then Gödel is a platonist only if Hilbert is a formalist. Gödel is a platonist. Frege is a logicist. So, Hilbert is a formalist.

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -∼Y ⊃ [A ≡ (Y • B)]

For each of the following questions, determine whether the given formula is a wff or not. If it is a wff, indicate its main operator. -[(P ≡ Q) • ∼Q] ⊃ (P ⊃ R)

Assume A, B, C are true; X, Y, Z are false; and P and Q are unknown. Evaluate the truth value of each complex expression. -(P ⊃ ∼Q) $\lor$ ∼P

Note: the solutions to most of the multiple choice questions in these sections use what call the standard assignment of truth values to atomic propositions. The standard assignment of truth values assigns the values given here to the variables in the wffs in the exercises, when read left to right. So, the first variable in the formula read left to right gets the α assignment; the second variable in the formula read left to right (if any) gets the β assignment; the third variable in the formula read left to right (if any) gets the γ assignment; and the fourth variable in the formula read left to right (if any) gets the δ assignment. For exercises with only one propositional variable, the standard assignment is: \alpha 1 0 For exercises with two propositional variables, the standard assignment is: \alpha 1 1 1 0 0 1 0 0 For exercises with three propositional variables, the standard assignment is: \alpha \beta \gamma 1 1 1 1 1 0 1 0 1 1 0 0e 0 1 1 0 1 0 0 0 1 0 0 0 For exercises with four propositional variables, the standard assignment is: \alpha \gamma \delta\ 1 1 1 1 1 1 1 0 1 1 0 1 1 1 0 0 1 0 1 1 1 0 1 0 1 0 0 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 0 1 0 1 0 0 0 0 1 1 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 1 For each of the given propositions, determine which of the given sequences properly describes the column under the main operator, given the standard assignment of truth values to atomic propositions. -{[X ? ?(Y $\lor$ Z)] • [Y ? ?(X $\lor$ Z)]} ? [(X • Z) $\lor$ ?(X • ?Z)]

construct a complete truth table for each of the following propositions. Then, using the truth table, classify each proposition as a tautology, a contingency, or a contradiction. -(G ≡ H) ⊃ ∼[(G • H) $\lor$ (∼G • ∼H)]