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Translating First Order Logic Into Natural Language
Contemporary standard logical notation for first order logic translates into English in the following manner.
((x) = for all x, ((x) = there exists an x such that, (meaning, there exists at least one x such that)((A) = It is not the case that, (A) (A ( B) = (If A, then B) (A ( B) = (A, or B) (A ( B) = (A, and B) (Rx) = x has the property R, (Rxy) = x is in the relation R to y (Rxyz) = w, x, y, and z are in the relation R
(Rxyzz1) = x, y, z, and z1 are in the relation R
Etc. (one can do this for any finite number of variables or constants)
[Note that we have not included translations for sentences involving the identity sign (e.g. x = y) or function symbols (e.g. f(x) = y), which are usually used when formalizing mathematics. However, it is provable that the above kind of language is just as expressively powerful as languages with identity and function symbols, so nothing other than convenience is lost.]
Here is how one expresses the Aristotelian syllogistic figures A, I, O, and E in this language.
((x)(Sx ( Px) =
for all x, (if x has the property S, then x has the property P) (another way to say Every S is a P),
((x)(Sx ( Px) =
there exists an x such that, (x has the property S, and x has the property P) (another way to say Some Ss are Ps),
((x)(Sx ( (Px) = there exists an x such that, (x has the property S, and it is not the case that (x has the property P)) (another way to say Some S is not a P),
(((x)(Sx ( Px) = it is not the case that, (x has the property S, and x has the property P),
((x)(Sx ( (Px) = for all x, (if x has the property S, then it is not the case that x has the property P)
((x)((Sx ( Px) = for all x, it is not the case that, (x has the property S, and x has the property P) (three ways to say No S is a P).
One might be asking, O.K. Why not just stick to the simpler Aristotelian way of expressing these things? But Freges notation allowed a fundamental advance that had bedeviled logicians for two thousand years. It allowed for the expression of relations. Consider the inference,
Sam is bigger than Bill.
Everyone who is bigger than Bill is bigger than Fred.
Sam is bigger than Fred.
This is intuitively logically valid in that it is not logically possible for the premises to be true and the conclusion false. But if we are stuck with Aristotles notation, there is no way to express these sentences such that the logical properties that render the inference valid are exposed. In a Fregean language such as the above we simply denote them in the following manner.
Bsb
((x)(Bxb ( Bxf)
Bsf
Thus we can express the key premises in the argument. A system of deductive logic gives us logical rules such that proofs of inferences such as the above can be checked by a machine. An example of a proof in such a system is the following.
1. Bsb
2. ((x)(Bxb ( Bxf)
3. Bsb ( Bsf 2, ( elimation
4. Bsf 1,3 ( elimation
In PHIL 2010 we learn such a system for first order logic. It is astoundingly powerful, allowing us to express most proofs in mathematics such that the correctness of those proofs can be checked by a machine. In this class we are presently concerned just with Freges semantic theory for this language.
Mathematics constitutively involves relations that cannot be expressed in Aristotelian logic, but which can be expressed in Freges language. For example, when we say that three is the sum of two and one, we are asserting a relationship between three numbers; when we say that three is greater than two we are asserting a relationship between two numbers.
One last thing. Note that in using logic to translate natural language, philosophers and linguists diverge in one small way. Philosophers will usually let the letters such as P and S stand for natural language words via a translation manual, while linguists will often use natural language words themselves. For example, if a philosopher wanted to represent the argument, Every person is a mortal. Socrates is a person. Therefore, Socrates is a mortal, she would first translate man, mortal and Socrates into symbols of the predicate logic in this manner:
Let person translate into P, Let mortal translate into M, and Let Socrates translate into s.
Then the argument can be translated into this.
((x)(Px ( Mx) Ps Ms
In Linguistics literature you are much more likely to see something like this
((x)(person'(x) ( mortal'(x)) person'(s) mortal'(s).
Here Socrates has been translated to avoid confusion, but person and mortal are only translated by putting the dash after them.
In reading these, one can slavishly follow the kind of translation given at the beginning of this handout, for example, reading ((x)(person'(x) --> mortal'(x)) as for all x, (if x has the property person' then x has the property mortal'). This is easy to do, as you only need to cut and paste from the above translation suggestions. For example, (x((y(Oxy ( (Yyx) can be translated step by step.
1 For all x, ((y(Oxy ( (Yyx)
2 For all x, it is not the case that (y(Oxy ( (Yyx)
3 For all x, it is not the case that there exists a y such that (Oxy ( (Yyx)
4 For all x, it is not the case that there exists a y such that (x is in the relation O to y ( (Yyx)
5 For all x, it is not the case that there exists a y such that (x is in the relation O to y and (Yyx)
6 For all x, it is not the case that there exists a y such that (x is in the relation O to y and it is not the case that Yyx)
7 For all x, it is not the case that there exists a y such that (x is in the relation O to y and it is not the case that y is in the relation Y to x)
(1) Note that, the method of translation requires us to keep the parentheses in step 4! This is as is dictated by the translation manual that starts this handout. Always leave in the parentheses so that your translation does not become logically ambiguous! (2) Note how we are underlining the property term. This is a good thing to do because a mechanical translation according to the above will usually lead to forced English grammar, and the underlining lets you know that you are referring to the property denoted by the word. For example, assume that Oxy = x is older than y, Yx,y = x is younger than y, then we can continue the translation.
8 For all x, it is not the case that there exists a y such that (x is in the relation is older than to y and it is not the case that y is in the relation is younger than to x)
From this we can use common sense to finally get the following.
9 For all x, it is not the case that there exists a y such that (x is older than y and it is not the case that y is younger than x)
In this manner, when you get more comfortable with the language it is easy to thus shorten has the property in the above to something closer to natural language. Our earlier sentence for all x, (if x has the property person' then x has the property mortal') becomes for all x, (if x is a person, then x is a mortal). This isnt mechanical though, since logic predicates can be translates of many nouns, adjectives, and verbs. A quick rules of thumb- if the predicate is a mass noun, change has the property to is, if the predicate is a singular count noun, change has the property to is a, if the predicate is an intransitive verb then simply delete has the property. More than one place predicates get more complicated.
Exercises
Where c = Charlie (and by = here we mean translates into), m = Mary, d = Dan, Mx = x is male, Fx = x is female, Hx = x is human, Cx = x is canine, Txy = x is taller than y, Oxy = x is older than y, Yx,y = x is younger than y, Sxy= x is the same age as y, translate the following sentences of first order logic into English. If you want to stop with a sentence that mentions the properties (as in line 8 in the above derivation), that is fine. If you want to go further (as in line 9) thats fine too.
1 Mc
2 Fm
3 (Fc
4 Hc ( Mm
5 Hd ( Hm
6 (x(Hx ( (Cx)
7 (x((Hx ( (Cx)
8 (x((Hx ( Cx)
9 ((x(Hx ( Cx)
10 Omc ( Odc
11 (x(y(Oxy ( Yyx)
12 (x(y(Oxy ( (Oyx)
13 (x(y((Oxy ( (Oyx) ( Sxy))
14 (x(y((Oxy ( Yxy) ( Sxy))
15 (x((Hx ( (Cx)
16 (x((Hx ( Cx)
17 ((x(Hx ( Cx)
18 ((x(Txc)
19 (x(Tcx ( Sxc)
20 (x(y(Sxy)
21 (x(y(Yxy ( Sxy)
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