Exam 5: Full First-Order Logic

(∀x)[Px ⊃ (∃y)Qxy] ⊃ [(∃x)Px ⊃ (∃x)(∃y)Qxy] -Consider assuming '(∀x)[Px ⊃ (∃y)Qxy]' for a conditional proof of the above logical truth. Which of the Following propositions is a legitimate second step in that proof?

select the best translation into predicate logic, using the following translation key: b: Britain c: Charles e: Elizabeth t: Tescos Px: x is a person Qx: x is a Queen of England Wx: x is a woman Exy: x is more exalted than y Ixy: x is in y Pxy: x shops at y Sxy: x is a son of y -Any son of Elizabeth or Charles is more exalted than anyone who is not Charles, Elizabeth, or a son of either.

1. (∀x)[Ax ⊃ (∃y)(By • Cxy)] 2. (∃x)(Ax • Dx) 3. (∀x)(Bx ⊃ Ex) -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

Translate each sentence into predicate logic, using the given translation keys. use: h: Hume l: Locke Px: x is a philosopher Rx: x is a rationalist Ixy: x influenced y Sxy: x is more skeptical than y -Hume is not more skeptical than some philosophers.

derive the conclusions of each of the following arguments using the rules of inference for F. -1. (?x)(?y)Axy ? (?x)(?y)Bxy 2. (?x)(?y)?Bxy / (?x)(?y)?Axy

1. (∃x)(∃y){Gx • Gy • x≠y • (∀z)[Gz ⊃ (z=x $\lor$ z=y)]} 2. Ga • Gb • Gc -Which of the following propositions is derivable from the given premises in F?

translate each sentence of predicate logic into natural, English sentences using the following translation key: t: two Ox: x is odd Ex: x is even Nx: x is a number Gxy: x is greater than y -(∀x)[Nx ⊃ (Ox ⊃ ∼Ex)]

select the best translation into predicate logic, using the following translation key: Cx: x is a cheetah Lx: x is a lion Tx: x is a tiger Fxy: x is faster than y Lxy: x is larger than y -No tiger is faster than all cheetahs.

select the best English interpretation of the given statements of predicate logic, using the following translation key: t: two Ox: x is odd Ex: x is even Nx: x is a number Gxy: x is greater than y -~(∃x)[(Nx • ∼Ex) • Gxt]

use the following translation key to write the sentences below in FF. a: one b: two c: three f(x): the successor of x g(x,y): the sum of x and y Nx: x is a number Dxy: x is divisible by y Gxy: x is greater than y -If a number is divisible by three, then the sum of its successor with the successor of one is also divisible by three.

translate the given paragraphs into arguments written in F, using identity and the given translation key. Then, derive their conclusions using the rules of inference for F, including the rules for identity. -Rey received the highest grade on Test #1. Spencer, who is not Rey, received a grade on Test #1. So, Rey's grade is higher than Spencer's. (r: Rey; s: Spencer; t: Test #1; Gxy: x is a grade on y; Hxy: x is higher than y; Rxy: x received y)

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} a = 1 e = 21 b = 2 f = 23 c = 4 g = 27 d = 19 h = 29 Ex = {2, 4, 6, ..., 28, 30} Ox = {1, 3, 5, ..., 27, 29} Px = (2, 3, 5, 7, 11, 13, 17, 19, 23, 29} Sxyz = The set of all triples such that the first is the sum of the second and third {<2, 1, 1>, <3, 1, 2>, <3, 2, 1>, <4, 1, 3>, <4, 2, 2>, <4, 3, 1>, <5, 1, 4>, ... } -Construct a theory of at least two sentences which uses at least two constants.

derive the conclusions of each of the following arguments using the rules of inference for F, including the rules for identity. -1. (∀x)(Ecx ⊃ x=d) 2. (∀x){(Fx • Gx) ⊃ (∀y)[(Fy • Gy) ⊃ y=x]} 3. (∃x)(Fx • Gx • Ecx) 4. Fa • Ga / a=d

use: b: Britain c: Charles e: Elizabeth t: Tescos Px: x is a person Qx: x is a Queen of England Wx: x is a woman Exy: x is more exalted than y Ixy: x is in y Pxy: x shops at y Sxy: x is a son of y -Elizabeth has exactly three sons.

translate the given paragraphs into arguments written in F, using identity and the given translation key. Then, derive their conclusions using the rules of inference for F, including the rules for identity. -The virtue ethicist who teaches Aristotle does not publish on moral theory. Smith is a virtue ethicist who teaches Aristotle. Either one publishes on moral theory or one publishes on applied ethics. So, Smith publishes on applied ethics. (s: Smith; Ax: x publishes on applied ethics; Mx: x publishes on moral theory; Tx: is teaches Aristotle; Vx: x is a virtue ethicist)

select the best translation into predicate logic, using the following translation key: b: Britain c: Charles e: Elizabeth t: Tescos Px: x is a person Qx: x is a Queen of England Wx: x is a woman Exy: x is more exalted than y Ixy: x is in y Pxy: x shops at y Sxy: x is a son of y -All people in Britain shop at Tescos, except Elizabeth.

1. (∀x)(∀y)(∀z)[(Ix • Jx • Iy • Jy • Iz • Jz) ⊃ (x=y $\lor$ y=z $\lor$ x=z)] 2. Ia • Ja • Ka • Ib • Jb • Kb • a≠b -Which of the following propositions is an immediate (one-step) consequence in F of the given premises?

translate the given paragraphs into arguments written in F, using the given translation key. Then, derive their conclusions using the rules of inference for -All philosophers respect each other. Some philosopher doesn't study some philosopher. Anything which respects something without studying it is open-minded, if ignorant. So something is open-minded and ignorant. (Ix: x is ignorant; Ox: x is open-minded; Px: x is a philosopher; Rxy: x respects y; Sxy: x studies y)

Translate each sentence into predicate logic, using the given translation keys. use: h: Hume l: Locke Px: x is a philosopher Rx: x is a rationalist Ixy: x influenced y Sxy: x is more skeptical than y -All philosophers are more skeptical than some rationalists.

construct a derivation of each logical truth of identity theory. -(?x)[Px • (?y)(Qy • y=x)] ? (?x)(Px • Qx)