Exam 4: Monadic Predicate Logic

use: t: Tortuga Bx: x creates bricks Cx: x is a city Nx: x is nicely placed Px: x is productive Sx: x is a settlement Tx: x has a trading port Wx: x is on the water -Either only cities are nicely placed or some settlements are not productive.

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. -Everything is material, or ideal or transcendental. All atoms are not ideal and not transcendental. And nothing is material. So there are no atoms. (Ax: x is an atom; Ix: x is ideal; Mx: x is material; Tx: x is transcendental)

1. (∃x)(Kx $\lor$ Lx) 2. (∀x)(Jx ⊃ ~Lx) -Which of the following propositions is derivable from the given premises in M?

construct a model for each of the given theories in the following domain by assigning members of the domain to the constants used in the theory and sets of members of the domain to the predicates used in the theory. Domain = {Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune} -(∃x)[Ax • (Bx • ∼Cx)] (∃x)[Ax • (Cx • ∼Bx)] (∀x)(Bx ≡ ∼Cx)

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 three sentences, at least one of which uses an existential quantifier and at least one of which uses a universal quantifer.

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 which uses at least two constants.

1. (∀x)(Px ⊃ Qx) 2. ∼(∃x)[(Px • Rx) • Qx] 3. (∃x)Rx / ∼(∀x)Px -Which of the following propositions isnot a likely last line of the indented sequence for an indirect proof of the given argument?

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)Ex • (∃x)∼Ex] ⊃ (∀x)(Ex $\lor$ ∼Ex)

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

Some materialist empiricists are either libertarians or hard determinists. But no empiricist is a hard determinist. So some materialists are libertarians. -Which of the following propositions is an immediate (one-step) consequence in M of the given premises?

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 websites have open comments which are not anonymous. Any website is either anonymous or requires a login. So something with open comments requires a login. (Ax: x is anonymous; Ox: x has open comments; Rx: x requires a login; Wx: x is a website)

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 play professional sports if, and only if, they have determination.

refer to the following formula: (∀x)[(Ex $\lor$ Fx) ⊃ (Gx • Hd)] -Is the formula open or closed?

For each of the following sentences, select the best translation into predicate logic, using the given constants and predicates. -All mammals feed their young.

1. (∀x)(Jx ⊃ Kx) 2. ∼(∀x)Kx -Which of the following propositions is derivable from the given premises in M?

select the best translation into predicate logic. -If Tortuga is a city, then some settlements are not nicely placed.

select a counterexample for the given invalid argument. -1. (∀x)(Mx ⊃ Nx) 2) (∃x)(Mx • Ox) 3) Oa / Oa • Na

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 formula is a logical truth of M or not. If it is not a logical truth, select a false valuation. -(∀x)(Ax ⊃ ∼Bx) $\lor$ (∃x)(Ax • Bx)

show that the given formula is a logical truth of M, using the rules of inference including conditional or indirect proof. -[(∃x)(Px • Qx)] ⊃ [(∃x)Px • (∃x)Qx]