Handout 1
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”).
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One might be asking, “O.K. Why not just stick to the simpler Aristotelian way of expressing
these things?” But Frege’s 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 Aristotle’s 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
4. Bsf
2,  elimation
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 Frege’s
semantic theory for this language.
Mathematics constitutively involves relations that cannot be expressed in Aristotelian logic, but
which can be expressed in Frege’s 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.”
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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.
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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 isn’t 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) that’s 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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