3.2 Translation of English to Quantified Wffs









All dogs are mammals.
Some sharks are ovoviviparous.
No fishes are endothermic.
Not all fishes are pelagic.
Reptiles and amphibians are not endothermic.
Some primates and rodents are arboreal.
Only lagomorphs gnaw.
Among spiders, only tarantulas and black widows are poisonous.
All and only marsupials have pouches.
No fish have wings unless they belong to the family Exocoetidae
.
Some organisms are chordates and some organisms are molluscs, but
nothing is both a chordate and a mollusc.
None but phylogenists are intelligent.
Animals behave normally if not watched.
Animals behave normally only if not watched.
Some sharks are pelagic fish, but not all pelagic fish are sharks.
If Shamu is a whale and all whales are mammals, then Shamu is a ma
mmal.
No sparrow builds a nest unless it has a mate.
No organism that is edentulous is a predator.
All predators are not herbivorous.
Not all predators are carnivorous.
A mammal with wings is a bat.
A mammal with wings is flying.

Shamu can do every trick.
Shamu can do any trick.
Shamu cannot do every trick.
Shamu cannot do any trick.
If any whale can do a trick, Shamu can.
If every whale can do a trick, Shamu can.
If any whale can do a trick, any whale can do a trick.

Godzilla ate Bambi.
Something ate Bambi.
Godzilla ate something.
Bambi ate everything.
Everything ate Bambi.
Something ate something.
Something ate everything.
Everything ate something.
Everything ate everything.
Everything ate itself.
Something ate itself.
Nothing ate itself.
Something ate nothing.
Everyone said something to everyone.
Everyone said something to someone.
Everyone said nothing to someone.
No one said anything to anyone.
There is a reptile smaller than a cat but larger than a dog.
Some fishes swim slower than humans.
Some fishes are smaller than every mammal.
Some whales eat only fast-moving fishes.
Some whales do not eat any fast-moving fishes.
If anything eats fast-moving fishes, sharks do.
Jaguars' tails are longer than ocelots' tails.
If an organism is symbiotic with a clown fish then it is a sea ane
mone.
The phalanges of birds are homologous to the phalanges of humans w
hereas the eyes of octopi are analogous but not homologous to the eyes of mammals and birds.
Some whales eat more than all fishes.
There is a monkey who grooms all and only those monkeys who do not
 groom themselves.

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translation scheme	Definition. A translation scheme for the language of predicate logic consists of a pairing of predicate letters with English predicate phrases and of names of predicate logic with names in English. We also include metavariab les with the predicates and associated phrases to indicate the appropriate order for names and variables.


	Example. According to the translation scheme `Lab: a loves b` `a: Abigail,` the sentence `Everyone loves Abigail' is translated as `@xLxa.` Comment. It is possible to give several non-equivalent translation schemes for sen tences of English, depending on how many places are assigned to the predicates. Example In the translation scheme `Fa: a is the father of Mary` `a: John.` F is specified as a 2-place predicate with the first position (occupied by a) correspondin g to the subject of the verb `loves' and the second (occupied by b) corresponding to its object. Using this scheme, the sentence `John is Mary's father` is translated as `Fa.` In the translation scheme `Fab: a is the father of b a: John` `b: Mary,` F is specified as a 2-place predicate with the first position (occupied by a) correspondin g to the subject of the verb `loves' and the second (occupied by b) corresponding to its object. Using this scheme, the sentence `John is Mary's father` is translated as `Fab.` Comment. The choice of whether to represent English phrases with one-place or many -place predicates is de-pen-dent on the degree of structure that must be included in order for an argument to be analyzed adequately. In general, more detail is better than less detail, since arg uments may be labeled invalid erroneously if insufficient detail is represented. Comment. The logical forms of many English sentences can be captured with the quan tifiers introduced in section 3.1. The following is an incomplete list of some of the more common sentences.
universals	Variants whose logical form is `@xFx` include the following: `Everything is F.` `All things are F.` Variants whose logical form is `@x(Fx -> Gx)` include the following: `Every F is a G. All Fs are Gs. If it's an F, it's a G. Everything that is F is G. Anything that is an F is a G. Any F is G. If something is an F, it is a G.` `Only Gs are Fs.` There are several variants having the form `@x(Fx -> ~Gx)` including these: `No Fs are Gs. Not a single F is G. Fs are never Gs.` `Every F is not G.`
existentials	Variants with the form `$xFx` include the following: `Something is F. There exists an F.` `There is at least one F.` Variants having the form `$x(Fx & Gx)` include the following: `Some Fs are Gs. At least one F is G.` `There exists an F that is G.` Comment. Notice the difference between translating `Every F is G' and `Some Fs are G'. In the first case, an arrow is used in the scope of a universal quantifier. In the second, an ampersand is the appropriate connective in the scope of the existential quantifier. Comment. There are many subtleties in the translation of English quantifier phrase s into the language of predicate logic. Such phrases often introduce ambiguity into the expressio ns of English. The exercises below illus
Exercise 3.5	Give translation schemes and translate the following sentences of English into the language of predicate logic. If a sentence is ambiguous, give all the reason-able transl ations of it.
	(1-22: Translate using one-place predicates only.)
1	All dogs are mammals.
2	Some sharks are ovoviviparous.
3	No fishes are endothermic.
4	Not all fishes are pelagic.
5	Reptiles and amphibians are not endothermic.
6	Some primates and rodents are arboreal.
7	Only lagomorphs gnaw.
8	Among spiders, only tarantulas and black widows are poisonous.
9	All and only marsupials have pouches.
10	No fish have wings unless they belong to the family Exocoetidae .
11	Some organisms are chordates and some organisms are molluscs, but nothing is both a chordate and a mollusc.
12	None but phylogenists are intelligent.
13	Animals behave normally if not watched.
14	Animals behave normally only if not watched.
15	Some sharks are pelagic fish, but not all pelagic fish are sharks.
16	If Shamu is a whale and all whales are mammals, then Shamu is a ma mmal.
17	No sparrow builds a nest unless it has a mate.
18	No organism that is edentulous is a predator.
19	All predators are not herbivorous.
20	Not all predators are carnivorous.
21	A mammal with wings is a bat.
22	A mammal with wings is flying.
	(23-29: Try these first with one-place predicates, then with many-place predicates.)
23	Shamu can do every trick.
24	Shamu can do any trick.
25	Shamu cannot do every trick.
26	Shamu cannot do any trick.
27	If any whale can do a trick, Shamu can.
28	If every whale can do a trick, Shamu can.
29	If any whale can do a trick, any whale can do a trick.
	(30-57: Translations with many-place predicates.)
30	Godzilla ate Bambi.
31	Something ate Bambi.
32	Godzilla ate something.
33	Bambi ate everything.
34	Everything ate Bambi.
35	Something ate something.
36	Something ate everything.
37	Everything ate something.
38	Everything ate everything.
39	Everything ate itself.
40	Something ate itself.
41	Nothing ate itself.
42	Something ate nothing.
43	Everyone said something to everyone.
44	Everyone said something to someone.
45	Everyone said nothing to someone.
46	No one said anything to anyone.
47	There is a reptile smaller than a cat but larger than a dog.
48	Some fishes swim slower than humans.
49	Some fishes are smaller than every mammal.
50	Some whales eat only fast-moving fishes.
51	Some whales do not eat any fast-moving fishes.
52	If anything eats fast-moving fishes, sharks do.
53	Jaguars' tails are longer than ocelots' tails.
54	If an organism is symbiotic with a clown fish then it is a sea ane mone.
55	The phalanges of birds are homologous to the phalanges of humans w hereas the eyes of octopi are analogous but not homologous to the eyes of mammals and birds.
56	Some whales eat more than all fishes.
57	There is a monkey who grooms all and only those monkeys who do not groom themselves.