Exam 4: Monadic Predicate Logic

select the best translation into predicate logic. -Either only cities are nicely placed or some settlements are not productive.

use: b: Berkeley h: Hume Ax: x is an apriorist Cx: x is consistent Ex: x is an empiricist Ix: x is an idealist Px: x is a person Rx: x is a rationalist Sx: x is a skeptic Tx: x is a theist -If some idealists are not skeptics then not all theists are rationalists.

1. (∀x)(Cx ⊃ Dx) 2. (∀x)(Ex ⊃ ∼Dx) -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

refer to the following formula: (∃x)[Mx • (∼Nc $\lor$ ∼Ox)] ≡ (Py • Pb) -Which is the main operator of the formula?

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample. -1. (∃x)[Ax • (Bx $\lor$ Cx)] 2) (∀x)(Bx ⊃ ∼Cx) 3) (∃x)Bx 4) Ca / (∃x)(Ax • Cx)

provide a conterexample in a finite domain to each given invalid argument. -1. (∃x)(∼Ax ≡ Cx) 2. (∃x)(Ax • Cx) 3. (∀x)(Bx ⊃ Ax) / (∀x)(Cx ⊃ Bx)

1. (∀x)(Hx ⊃ ∼Jx) 2. (∀x)(Ix ⊃ Jx) 3. Ha • Ib -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

use: b: Berkeley h: Hume Ax: x is an apriorist Cx: x is consistent Ex: x is an empiricist Ix: x is an idealist Px: x is a person Rx: x is a rationalist Sx: x is a skeptic Tx: x is a theist -No apriorist rationalists are skeptics, but Hume is.

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)(Gx ⊃ Hx) 2. (∃x)(Gx • Ix) 3. (∃x)(Gx • Jx) 4. (∀x)(Jx ⊃ Hx) / (∃x)(Ix • Jx)

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

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 -Some athletes who don't work hard receive scholarships, if, and only if, no athletes who play professional sports don't have determination.

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 • ∼Ix) 2. (∀x)(Hx ⊃ Jx) 3. ∼(∃x)(Jx • Kx) / (∃x)∼(Ix $\lor$ Kx)

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

use: b: Berkeley h: Hume Ax: x is an apriorist Cx: x is consistent Ex: x is an empiricist Ix: x is an idealist Px: x is a person Rx: x is a rationalist Sx: x is a skeptic Tx: x is a theist -Some apriorist is a skeptic if, and only if, s/he is an inconsistent empiricist.

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

1. (∃x)Qx ⊃ (∀x)(Rx ⊃ Sx) 2. (∀x)∼Qx ⊃ (∃x)Sx 3. (∀x)Rx / (∃x)Sx -Which of the following propositions is an appropriate assumption for an indirect proof of the conclusion of the given argument?

refer to the following formula: ∼(∀x){(Ix • Jx) ⊃ [Kx ≡ (La • Lb)]} -List all of the subformulas in the scope of '(∀x)'.

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 a universal quantifier.

refer to the following formula: (∃x)[(Ax • ∼Bx) • ∼(Cx $\lor$ Dx)] -What is the main operator of the given formula?

determine whether the given formula is a logical truth of M or not. If it is not a logical truth, select a false valuation. -[(∃x)Ex • (∃x)∼Ex)] ⊃ (∀x)(Ex $\lor$ ∼Ex)