Predicate formulae

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

            All cows like Buttercup

Let us take as our domain everything (i.e the universal set).

Then as a first step we will get

            "x[x is a cow ® x likes Buttercup]

Then let us use the following abbreviations:

            a:            Buttercup

            Cx:          x is a cow

            Lxy:         x likes y

Then we arrive at the formula

            "x[Cx®Lxa].

The key is an interpretation.

We use:

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 and domain to do some more formalisations.

Interpretation:

            a:            Buttercup

            Cx:          x is a cow

            Px:          x is a person

            Lxy:         x likes

Domain: everything

First example:

            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:

[$x[CxÙLxa]Ù"x[Lxa®Cx]] (i.e. "The is a cow which likes Buttercup and…")

 

[$xLxaÙ"x[Lxa®Cx]] (i.e. "Something likes Buttercup and…)

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.

Second example:

            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:

            "x[[CxÙLxx]®Lax].

Third example:

            If anyone that likes a cow likes Buttercup, then someone likes Buttercup.

First step:

            [Anyone that likes a cow likes Buttercup ® someone likes Buttercup]

Next step:

            ["x[x is a person who likes a cow ® Lxa]®$x[PxÙLxa]]

Next step:

            ["x[PxÙ x likes a cow]® Lxa]®$x[PxÙLxa]]

Finally:

            ["x[PxÙ $y[CyÙLxy]]® Lxa]®$x[PxÙLxa]]