When translating in the prepositional calculus we abbreviated sentences with capital letters. When translating in the predicate calculus we shall also want to abbreviate predicates and designators. And we shall need to specify a domain. Again what results form this process will be formulae, and the process is formalisation.
Suppose we want to formalise
Let us take as our domain everything (i.e the universal set).
Then as a first step we will get
Then let us use the following abbreviations:
Cx: x is a cow
Lxy: x likes y
Then we arrive at the formula
The key, together with the specification of the domain, is an interpretation.
capital letters to abbreviate sentences:
e.g. A, B, P, Q, T (These are sentence letters.)
capital letters followed by the appropriate number of different individual variables (lower case letters from the end of the alphabet) for predicates:
e.g. Cx, Gx, Hxy, Rxyz
lower case letters from the beginning of the alphabet to abbreviate designators:
e.g. a, b, g, h (These are individual constants, or names)
Note: the rules for the language of the predicate calculus dictate that, if you use individual constants, they must stand for something. So you should not, for instance, use "a" to abbreviate, say, "the largest prime number", because there is no such thing. It follows that, if you use individual constants, your domain must not be empty. (It must, of course, contain at least the things that your individual constants stand for.)
We will use the following interpretation to do some more formalisations.
Our interpretation is:
Cx: x is a cow
Px: x is a person
Lxy: x likes y
Only cows like Buttercup
There are various ways of saying this:
"x[¬Cx®¬Lxa] (i.e. "Anything which isn't a cow doesn't like Buttercup")
"x[Lxa®Cx] (i.e. "Anything which likes Buttercup is a cow")
¬$x[LxaÙ¬Cx] (i.e. "there is nothing which likes Buttercup but isn't a cow")
These all come to the same thing. ("[¬j®¬y]" is equivalent to "[y®j]"; "¬$x" is equivalent to ""x¬"; "¬[jÙ¬y]" is equivalent to "[j®y]".)
Notice, however, that these formalisations will all be true if nothing likes Buttercup. Maybe the original sentence implies that some things (namely some cows) like Buttercup. If that is what one thinks one will prefer one of the following formalisations:
These two formalisations come to the same thing, since if something likes Buttercup and anything that likes Buttercup is a cow, it follows that some cow likes Buttercup.
Buttercup likes all cows that like themselves.
Advice: when one is formalising things which are at all complicated it is a good idea to do it in a series of steps,
First we get:
"x[x is a cow which likes itself ® Buttercup likes x]
Then we formalise "Buttercup likes x" as "Lax", and "x is a cow which likes itself" as "[CxÙLxx]". So the whole thing becomes:
If anyone that likes a cow likes Buttercup, then someone likes Buttercup.
[Anyone that likes a cow likes Buttercup ® someone likes Buttercup]
["x[x is a person who likes a cow ® Lxa]®$x[PxÙLxa]]
["x[PxÙ x likes a cow]® Lxa]®$x[PxÙLxa]]
["x[PxÙ $y[CyÙLxy]]® Lxa]®$x[PxÙLxa]]