Exam 4: Monadic Predicate Logic

select the best English interpretation of the given statements of predicate logic. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(Pf • Pg) • [(Rf • Rg) • (Sf • ∼Sg)]

1. (∀x)Ix ⊃ (∀x)Kx 2. (∀x)[Jx • (Ix $\lor$ Lx)] 3. (∀x)(Jx ⊃ ∼Lx) -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

select the best translation into predicate logic. -If some idealists are not skeptics then not all theists are rationalists.

translate the given paragraphs into arguments written in M, using the given translation key. Then, derive their conclusions using the rules of inference for M. -Things are pleasant if, and only if, they are not too crowded. Everything too crowded is noisy. So if something isn't noisy, then something is pleasant. (Cx: x is too crowded; Nx: x is noisy; Px: x is pleasant)

derive the conclusions of each of the following arguments using the rules of inference for M, including conditional or indirect proof. -1. (∃x)Qx ⊃ (∀x)(Rx ⊃ Sx) 2. (∀x)∼Qx ⊃ (∃x)Sx 3. (∀x)Rx / (∃x)Sx

consider the following domain, assignment of objects in the domain, and assignments sets to predicates. Domain = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune, Pluto} P = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune} I = {Mercury, Venus} O = {Mars, Jupiter, Saturn, Uranus, Neptune} a = Mercury b = Jupiter c = Saturn d = Pluto -Given the customary truth tables, which of the following theories is modeled by the above interpretation?

refer to the following formula: (∃x)[Mx • (∼Nc $\lor$ ∼Ox)] ≡ (Py • Pb) -Are there any free variables? If so, which are they?

refer to the following formula: ∼(∀x){(Ix • Jx) ⊃ [Kx ≡ (La • Lb)]} -What is the main operator of the given formula?

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample. -1. (∀x)[Fx ⊃ (Gx $\lor$ Hx)] 2) (∀x)(Gx ⊃ ∼Ix) 3) ∼(∃x)(Hx • Ix) 4) Fg / ∼Ig

Translate each of the following sentences into predicate logic, using the given constants and predicates. -Whatever Lola wants, Lola gets. (Gx: Lola gets x Wx: Lola wants x)

determine whether the given argument is valid or invalid. If it is valid, provide a derivation of the conclusion from the premises. If it is invalid, provide a counterexample. -1. (∀x)[Fx ⊃ (Gx $\lor$ Hx)] 2. (∀x)(Gx ⊃ ∼Ix) 3. ∼(∃x)(Hx • Ix) 4. Fg / ∼Ig

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates. -Whatever Lola wants, Lola gets.

select the best translation into predicate logic. -Only athletes who are tall and work hard play professional sports.

use the given interpretations to translate each sentence of predicate logic into natural, English sentences. f: Fifi g: Gigi Px: x is a poodle Qx: x is abused Rx: x is loved Sx: x will fetch balls Tx: x will fetch sticks. -(∃x)[Px • (∼Qx • Rx)] ⊃ (∀x)[(Px • Rx) ⊃ Tx)]

refer to the following formula: (∃x)[Mx • (∼Nc $\lor$ ∼Ox)] ≡ (Py • Pb) -Which of the following variables in the formula are free?

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample. -1. (∀x)(Ax ⊃ Bx) 2) (∃x)(Ax • Cx) / ∼(∀x)(Bx ⊃ ∼Cx)

1. (∀x)[Ax ⊃ (Bx ⊃ Cx)] 2. ∼(∀x)(Bx ⊃ Dx) -Consider assuming '(∀x)Ax' for conditional proof. Which of the following propositions is an immediate (one-step) consequence in M of the given premises with that further assumption for conditional proof?

determine whether the given argument is valid or invalid. If it is valid, provide a derivation of the conclusion from the premises. If it is invalid, provide a counterexample. -1. (∃x)[(Ax • Bx) • Cx] 2. (∀x)[(Ax • ∼Cx) ⊃ ∼Bx] / (∃x)(Ax • ∼Bx)

derive the conclusions of each of the following arguments using the rules of inference for M. Do not use conditional or indirect proof. -1. (∀x)(Hx ⊃ ∼Jx) 2. (∀x)(Ix ⊃ Jx) 3. Ha • Ib / ∼(Ia $\lor$ Hb)

Translate each of the following sentences into predicate logic, using the given constants and predicates. -Some yellow birds both chirp and sing. (Bx: x is a bird; Cx: x chirps; Sx: x sings; Yx: x is yellow)