Exam 4: Monadic Predicate Logic

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates. -Efraim takes acting classes if, and only if, he gets time off from work.

select the best translation into predicate logic. -All tall athletes work hard.

construct theories for which the following interpretation is a model (i.e. construct at least two sentences which are true under the given interpretation). Domain = {1, 2, 3, ..., 28, 29, 30} E = {2, 4, 6, ..., 28, 30} O = {1, 3, 5, ..., 27, 29} P = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29} a = 1 d = 19 b = 2 e = 23 c = 3 f = 29 -Construct a theory of at least two sentences, at least one of which uses an existential quantifier.

consider the following domain, assignment of objects in the domain, and assignments sets to predicates. Domain = {1, 2, 3, ..., 28, 29, 30} N = {1, 2, 3, ..., 28, 29, 30} E = {2, 4, 6, ..., 28, 30} O = {1, 3, 5, ..., 27, 29} P = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29} a = 1 b = 2 c = 28 -Given the customary truth tables, which of the following theories is modeled by the above interpretation?

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)[(Dx • Ex) • Fx] 2. (∀x)[(Dx • Fx) ⊃ ∼Gx] / (∃x)(Ex • ∼Gx)

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. -Some materialist empiricists are either libertarians or hard determinists. But no empiricist is a hard determinist. So some materialists are libertarians. (Dx: x is a hard determinist; Ex: x is an empiricist; Lx: x is a libertarian; Mx: x is a materialist)

refer to the following formula: ∼(∀x){(Ix • Jx) ⊃ [Kx ≡ (La • Lb)]} -Are there any free variables? If so, which are they?

determine whether the given formula is a logical truth of M or not. If it is a logical truth, provide a proof of the formula. If it is not a logical truth, provide a counterexample in a finite domain. -(∀x)(Ax ⊃ ∼Bx) $\lor$ (∃x)(Ax • Bx)

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?

1. (∃x)(Gx • ∼Hx) ≡ (∃x)(Hx • ∼Ix) 2. (∀x)(Hx ⊃ Ix) -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

1. ∼(∃x)[Fx • (Gx • Hx)] 2. ∼(∃x)(Ix • ∼Fx) -Which of the following propositions is derivable from the given premises in M?

select the best translation into predicate logic. -There are no settlements on the water with trading ports.

select the best translation into predicate logic. -All cities and settlements are nicely placed and productive.

refer to the following formula: (∃x)[Mx • (∼Nc $\lor$ ∼Ox)] ≡ (Py • Pb) -List all of the subformulas in the scope of '(∃x)'.

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample. -1. (?x)(Ax ? Bx) ? (?x)Cx 2) (?x)(Ax • ?Bx) 3) (?x)(Dx ? Bx) / (?x)(Dx ? Cx)

refer to the following formula: (∀x)[(Ex $\lor$ Fx) ⊃ (Gx • Hd)] -Which is the main operator of the formula?

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)∼Ix ⊃ (∀x)(Jx $\lor$ Kx) 2. ∼(∀x)Ix • ∼Jb / Kb

refer to the following formula: (∃x)[(Ax • ∼Bx) • ∼(Cx $\lor$ Dx)] -Which variables are bound by the '(∃x)'?

Translate each sentence into predicate logic, using the given translation keys. Ax: x is an athlete Dx: x has determination Px: x plays professional sports Sx: x receives a scholarship Tx: x is tall Wx: x works hard -Only athletes who are tall and work hard play professional sports.

determine whether the given formula is a logical truth of M or not. If it is a logical truth, provide a proof of the formula. If it is not a logical truth, provide a counterexample in a finite domain. -(∃x)(Cx • Dx) $\lor$ (∃x)(Cx • ∼Dx)