x(Dx->Mx)| SENTENCE: | All dogs are mammals |
| FIRST REWRITE AS: | All dogs=x: (x is a mammal) |
| REWRITE AS: | Every x: (if x is a dog then x is a mammal) |
| TRANSLATION SCHEME: | Dx=x is a dog, Mx=x is a mammal |
| TRANSLATION: | x(Dx->Mx) |
| SENTENCE: | Some sharks are ovoviviparous |
| FIRST REWRITE AS: | Some sharks=x: (x is ovoviviparous) |
| REWRITE AS: | Some x: (x is a shark and x is ovoviviparous) |
| TRANSLATION SCHEME: | Sx=x is a shark, Ox=x is ovoviviparous |
| TRANSLATION: |  x(Sx&Ox) |
| SENTENCE: | No fishes are endothermic |
| FIRST REWRITE AS: | No fishes=x: (x is endothermic) |
| REWRITE AS: | Every fish=x: not (x is endothermic) |
| REWRITE AS: | Every x: (if x is a fish then not (x is endothermic)) |
| TRANSLATION SCHEME: | Fx=x is a fish, Ex=x is endothermic |
| TRANSLATION: | x(Fx->~Ex) |
In this sentence, note that the sentential connective 'not' has (as
usual) wider scope than the 'all': so, we analyze it first.
| EXERCISE 4: | Not all fishes are pelagic |
| Sentential connective first: | Not (all fishes are pelagic) |
| REWRITE AS: | Not (all fishes=x: (x is pelagic) |
| REWRITE AS: | Not (Every x: (if x is a fish then x is pelagic)) |
| TRANSLATION SCHEME: | Fx=x is a fish, Px=x is pelagic |
| TRANSLATION: | ~x(Fx->Px) |
This sentence is sententially compound. Once we spell out the two conjuncts,we
have a second problem: how do we interpret 'Reptiles are not endothermic'?
The text translates it as equivalent to 'No reptiles are endothermic';
this is one possibility.
| SENTENCE: | Reptiles and amphibians are not endothermic |
| Sentential connective first: | Reptiles are not endothermic and amphibians are not endothermic |
| REWRITE (TWO AT ONCE): | (Reptiles=x: (x is not endothermic)) and (amphibians=y: (y is not endothermic)) |
| REWRITE (taking plurals as universals): | (Every x: (if x is a reptile then x is not endothermic)) and (every y: (if y is an amphibian then y is not endothermic)) |
| TRANSLATION SCHEME: | Rx=x is a reptile, Ax=x is an amphibian, Ex=x is endothermic |
| TRANSLATION: | x(Rx->~Ex)&y(Ry->~Ey) |
6. Some primates and rodents are arboreal
This involves a sentential connective as well, but here the quantifier
is explicit:
| SENTENCE: | Some primates and rodents are arboreal |
| Sentential connective first: | (Some primates are arboreal) and (some rodents are arboreal) |
| REWRITE (TWO AT ONCE): | (Some primates=x: (x is arboreal)) and (some rodents=y: (y is arboreal)) |
| REWRITE: | ((Some x: (x is a primate and (x is arboreal))) and (some y: (y is a rodent and (y is arboreal))) |
| TRANSLATION SCHEME: | Px=x is a primate, Rx=x is a rodent, Ax=x is arboreal |
| TRANSLATION: | x(Px&Ax)&x(Rx&Ax) |
The next example involves a new English expression: 'only'. We need
a new recipe for handling this. To get one, notice that 'Only Fs are Gs'
can be rewritten as something like 'It's a G only if it's an F'; we can
then translate the 'only if' as '->'. In more detail, and with this example:
| SENTENCE: | Only lagomorphs gnaw |
| FIRST REWRITE AS: | Only lagomorphs=x: (x gnaws) |
| REWRITE AS: | Every x: (x gnaws only if x is a lagomorph) |
| Notice how this works: we turn 'only' into 'every' and then put 'only if' + (the first predicate) at the end of the translation. | |
| TRANSLATION SCHEME: | Gx=x gnaws, Lx=x is a lagomorph |
| TRANSLATION: | x(Gx->Lx) |
The next example adds two separate complications to the last one. We could translate 'Only tarantulas are poisonous' in the same way as the last example. 'Only tarantulas and black widows are poisonous', however, is more difficult. You might think this is sententially compound, that is:
Only tarantulas are poisonous and only black widows are poisonousBut that's obviously not what the sentence means. So, you might think that the 'and' simply forms part of the first predicate expression in this manner:
It's poisonous only if it's a tarantula and it's a black widowUnfortunately, that can't be right, since it would mean that the only poisonous things are those things which are simultaneously tarantulas and black widows, and nothing is (tarantulas aren't black widows, and black widows aren't tarantulas). So what does it mean?
The answer, alas, is that in this case we have to rewrite 'and' as 'or', in the following way:
It's poisonous only if it's a tarantula or it's a black widowSo, rewriting this to make the quantifier explicit, our sentence is:
Every x: (x is poisonous only if x is a tarantula or x is a black widow)Taking care to put the parentheses in the right place, this becomes:
But that's still not the original sentence: number 8 begins with the phrase 'among spiders'. How do we handle this? Take a simpler case:
Among spiders, only tarantulas eat gerbils(For simplicity's sake, let's treat '___ eats gerbils' as a one-place predicate here.) The phrase 'among spiders' here has the effect of saying 'we're talking about (all) spiders'. We can express that by treating it just like 'every spider'. We can do that in two stages, first analyzing 'only;
Among spiders, (every x: (x eats gerbils only if x is a tarantula))Then we translate 'among spiders' to 'if x is a spider then' and put this after the quantifier:
Every x: (if x is a spider then (x eats gerbils only if x is a tarantula))Now let's try to combine these recipes for dealing with the original sentence:
Among spiders, only tarantulas and black widows are poisonous
Among spiders, (only (tarantulas or black widows)=x: (x is poisonous))
Among spiders, (every x: (x is poisonous only if (x is a tarantula
or black widow)))
Among spiders, (every x: (x is poisonous only if (x is a tarantula
or x is a black widow)))
Every x: (if x is a spider then (x is poisonous only if (x is a tarantula
or x is a black widow)))
x(Sx->(Px->(Tx
v Bx)))
(By the way, it's not true: tarantulas actually make nice pets, and anyone who's been bitten by a brown recluse knows there are other poisonous spiders.)
9. All and only marsupials have pouches
This actually can be handled using the recipes we've seen so far. The 'and' is a sentential connective, so this is
All marsupials have pouches and only marsupials have pouchesSo, doing the translations left and right, we have
(Every x: (if x is a marsupial then x has a pouch)) and (every x: (x has a pouch only if x is a marsupial))And so:
Some of you are already complaining that this violates the rule proposed earlier of using different variables for different quantifiers. However, here we have a point to make. If we were to combine the open forms 'Mx->Px' and 'Px->Mx' with '&', the end result would be equivalent to 'Mx<->Px'. It follows from that (and a little more logic) that our last translation is equivalent to:
So, we could also use this as a recipe for translating 'all and only'.
10. No fish have wings unless they belong to the family Exocoetidae
The first thing to notice about this is that it is not sententially compound: if you divide it at the connective 'unless', then the right half,
they belong to the family Exocoetidae,isn't actually a sentence because it contains 'they' (with no indication, in the sentence, as to who or what 'they' are). This is an extension of the test we used in sentential logic to determine what the main connective is. Note that 'they' is obviously connected to the quantifiying expression 'no fish'. When we're checking to see whether a sentence is sententially compound, we cannot separate a pronoun from the quantifying expression it's connected with (the result would be the English equivalent of an open formula).
Unfortunately, there's another problem in this sentence: 'unless'. We might try to translate the sentence this way:
No fish=x: (x has wings unless x belong to the family Exocoetidae)When we apply the recipe for 'no' to this, you might think we would get this:
Every fish=x: not (x has wings unless x belong to the family Exocoetidae)And that isn't what we want (if you finish the translation, you'll see that this comes out as equivalent to 'No fish has wings and no fish belongs to the family Exocoetidae'). But a small change will make it work: the 'not' only attaches to the part of the sentence before the connective 'unless':
No fish=x: (x has wings) unless x belongs to the family Exocoetidae11. Some organisms are chordates and some organisms are molluscs, but nothing is both a chordate and a mollusc
Every fish=x: (not (x has wings) unless x belongs to the family Exocoetidae)
Every x: (if x is a fish then (not (x has wings) unless x belongs to the family Exocoetidae)))
This is long but straightforward. The only mildly novel point is the compound predicate expression '__ is both a chordate and a mollusc', which we disassemble into two predicate expressions and 'and'. First, the sentential-logic structure:
((Some organisms are chordates) and (some organisms are molluscs,)) but (nothing is both a chordate and a mollusc)The two on the left are easy, so let's get them out of the way in one step:
((Some x: (x is an organism and x is a chordate)) and (some y: (y is an organism and y is a mollusc))) but (nothing is both a chordate and a mollusc)The right hand side, then, is analyzed as:
Nothing=x: (x is both a chordate and a mollusc)But the complex predicate becomes
Nothing=x: (both x is a chordate and x is a mollusc)And so this is:
Every x: not (both x is a chordate and x is a mollusc)Combining the two, we have:
((Some x: (x is an organism and x is a chordate)) and (some y: (y is an organism and y is a mollusc))) but (every z: not (both z is a chordate and z is a mollusc))12. None but phylogenists are intelligent(
Treat 'none but' just like 'only':
None but phylogenists=x: (x is intelligent)13. Animals behave normally if not watched
Only phylogenists=x: (x is intelligent)
Every x: (x is intelligent only if x is a phylogenist)
Notice that this is in effect an abbreviation: it's equivalent to
Animals behave normally if they are not watchedIf we think about the meaning of this sentence, it's clear that it is a universal quantification. We can rewrite it as:
Every animal behaves normally if it is not watchedThe 'it' is connected with 'every animal', so this is not sententially compound. We can translate by substituting x for both places in the original sentence:
Every animal=x: (x behaves normally if x is not watched)The rest of the translation is straightforward:
Every x: (if x is an animal then (x behaves normally if x is not watched))14. Animals behave normally only if not watched
Just as before:
Animals behave normally only if they are not watched15. Some sharks are pelagic fish, but not all pelagic fish are sharks
Every animal behaves normally only if it is not watched
Every animal=x: (x behaves normally only if x is not watched)
Every x: (if x is an animal then (x behaves normally only if x is not watched))
Note that '___ is a pelagic fish' = '___ is pelagic and ___ is a fish':
(Some sharks are pelagic fish,) but (not all pelagic fish are sharks)At this point, the right side still contains 'pelagic fish'. However, we just apply the recipes: first analyze the 'all', then analyze the compound predicate:
(Some shark=x: (x is pelagic and x is a fish,)) but (not (all pelagic fish=y: y is a shark))
every y: (if y is a pelagic fish then (y is a shark))So, putting this back into the rewrite of the whole sentence:
every y: (if y is pelagic and y is a fish) then (y is a shark))
(Some shark=x: (x is pelagic and x is a fish,)) but (not (every y: (if y is pelagic and y is a fish) then (y is a shark)))16. If Shamu is a whale and all whales are mammals, then Shamu is a mammal
Sentential structure first:
If (Shamu is a whale and all whales are mammals), then (Shamu is a mammal)With the scheme a = 'Shamu', Wx='x is a whale', Mx='x is a mammal', the three parts of this are:
Shamu is a whale: WaPutting these back into the original:
All whales are mammals:
Shamu is a mammal: Ma
If (Wa and17. No sparrow builds a nest unless it has a mate
(Wa &
Just like example 10:
No sparrow=x: (x builds a nest unless x has a mate)18. No organism that is edentulous is a predator
Every sparrow=x: (not (x builds a nest) unless x has a mate))
Every x: (if x is a sparrow then (not (x builds a nest) unless x has a mate))
To do this one, we'll need to use the recipe for relative clauses:
No organism that is edentulous=x: (x is a predator)19. All predators are not herbivorous
Every organism that is edentulous=x: not (x is a predator)
Every x: (if (x is an organism that is edentulous) then not (x is a predator))
Every x: (if (x is an organism and x is edentulous) then not (x is a predator))
This has two possible interpretations, according to whether you take the 'not' to have wider or narrower scope than the 'all'. On the first interpetation (which I think is more natural), it is:
Not (all predators are herbivorous)And that is:
~On the other interpretation, it could be rewritten thus:
All predators=x: (x is not herbivorous)That is,
20. Not all predators are carnivorous
This one isn't ambiguous at all:
Not (all predators are carnivorous
Not (all predators=x:(x is carnivorous))
Not (every x: (if x is a predator then x is carnivorous))
~x(Px->Hx)
21. A mammal with wings is a bat
22. A mammal with wings is flying
These two examples illustrate another source of ambiguity: predicates with nothing but an indefinite article attached can be either universal or existential. Number 21 is a universal case which we could rewrite as:
Every mammal with wings is a batNumber 22, by contrast, is really an existential quantification:
Every mammal with wings=x: (x is a bat)
Every x: (if x is a mammal with wings then x is a bat)'With wings' amounts to 'has wings', so we analyze the first predicate as compound:
Every x: (if (x is a mammal and x has wings) then x is a bat)
Some mammal with wings is flying
Some mammal with wings=x: (x is flying)
Some x: (x is a mammal with wings and (x is flying))
Some x: ((x is a mammal and x has wings) and (x is flying))