Exam 5: Full First-Order Logic

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 -All lions and tigers are larger than some cheetahs, but not faster than all cheetahs.

1. (∀x)(∀y)[f(x)=y ⊃ (Pax • Qay)] 2. ∼Pab -Which of the following propositions is an immediate (one-step) consequence in FF of the given premises?

determine whether the given argument is valid or invalid. If it is invalid, select a counterexample. -1. (?x)(?y)(Pxy • ?Pyx) 2) (?x)[(?y)Pxy ? (?y)Qxy] / (?x)(?y)(Qxy • ?Pyx)

1. (∀x)(∀y){Pf(x,y) ⊃ [Pf(x,x) • Pf(y,y)]} 2. a=f(d,b) 3. Pa -Which of the following propositions is an immediate (one-step) consequence in FF of the given premises?

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 its successor is not.

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 -No number is greater than the sum of that number and two.

1. (?x)(?y)[f(x)=y ? (Pax • Qay)] 2. ?Pab / ?(?x)f(b)=x

select the best translation into predicate logic, using the following translation key: b: Bhavin c: Chloe m: Megha n: Nietzsche p: Plato Ax: x is an altruist Jx: x is joyful Px: x is a philosopher Rx: x is Russian Tx: x is thoughtful Bxy: x is a brother of y Mxy: x mocks y Rxy: x is richer than y Sxy: x is smarter than y -The richest philosopher is smarter than any of Chloe's brothers.

select the best translation into predicate logic, using the following translation key: b: Bhavin c: Chloe m: Megha n: Nietzsche p: Plato Ax: x is an altruist Jx: x is joyful Px: x is a philosopher Rx: x is Russian Tx: x is thoughtful Bxy: x is a brother of y Mxy: x mocks y Rxy: x is richer than y Sxy: x is smarter than y -Some Russian philosopher other than Chloe is an altruist.

determine whether the given argument is valid or invalid. If it is valid, derive the conclusion using our rules of inference and equivalence. If it is invalid, provide a counterexample. -1. (?x)[Tx ? (?y)(Sy • Wxy)] 2. (?x)(Sx • Vx) 3. (?x)(Tx • Rx) / (?x)[(Rx • Vx) • (?y)Wxy]

derive the conclusions of each of the following arguments using the rules of inference for F. -1. (?x)[Ex ? (?y)(Fy • Gxy)] 2. (?x)(Ex • Hxb) / (?x)(?y)(Gxy • Hxy)

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){(Ex • Nx) ⊃ (∀y)[(Oy • Ny) ⊃ ∼Gxy]}

1. (?x)(?y)(Bxy ? Dyx) 2. (?x)Bxf(x) / (?x)Df(x)x

select the best translation into predicate logic, using the following translation key: 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 -Some rationalists influenced some philosophers who were more skeptical than them.

select the best translation into predicate logic, using the following translation key: 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 -The sum of one and two is three.

1. Fab • (∀x)(Fax ⊃ x=b) 2. ∼Fac -Which of the following propositions is derivable from the given premises in F?

1. (∀x)(∀y)(Bxy ≡ Dyx) 2. (∃x)Bxf(x) -Which of the following propositions is derivable from the given premises in FF?

provide a conterexample in a finite domain to each given invalid argument. -1. Ad ⊃ (∀x)Fdx / (∃x)Fdx

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

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 three sentences which uses all three predicates and at least three different constants.