Predicate Logic

In Predicate Logic, the fundamental component of representation of propositions is the predicate. We'll use capital letters to stand in for predicates. The letter W, for example, might stand for the predicate of being wise. With this symbolization, the proposition "Socrates is wise" would be represented as Ws. In general, we will uses the lower case letters of the alphabet, with the notable exceptions of x, y and z, to stand in for individuals. They are called individual constants.

The Conventions of Predicate Logic
The conventions of predicate logic use all the conventions found in propositional logic. For example, the proposition "If Socrates is wise, then Plato is wise" can be represented as Ws ⊃ Wp. And we use the variable "x", symbolized in this case, as "Wx" as a place marker for any actual individual constant (like Socrates or Plato, usually proper names of individuals) so that a singular proposition might result.

The various singular propositions Wa, Wb, Wc, have a bivalent truth function: either true or false, but "Wx", as a propositional function, has an undetermined truth value until an actual constant is substituted. A propositional function is defined as an expression that contains an individual variable and becomes a statement when an individual constant (a, b, c...), is substituted for the individual variable. Such a situation can be called a substitution instance, and these substitutions are either true or false. We can call such proposition functions simple predicates: a proposition function having some true and some false substitution instances, each of which is an affirmative singular proposition.

To review the symbolization of singular sentences

1. Individual constants: small letters a,b,c,...,w stand for a particular subject (named  individual). Ex. j=John, m=Mary, r=Rover 2. Predicate letters: capital letters A,B,C,... stand for predicates. Ex. W=wise, I=Intelligent. 3. Individual variables: small letters x,y,z stand for any small letter at all, any individual. 4) Propositional functions: Ax, By, Pz stand for all sentences with a given predicate. Ex: Ax: x is in Australia; By: y is blue. 5) Singular sentences: Aj, Bm, Pr symbolize particular sentences. Example: Einstein and Newton were geniuses. e = Einstein; n = Newton; G= Genius: Ge, Gn.

Quantification
As you may recall from earlier sections, George Boole uncovered the 'existential error" found within the Traditional Square, which reduce many of the previous mediate references that the Traditional Square allowed. The logician Frege created the concept of quantification to allow us to restore many of the traditions of the classical logicians - we'll see that the Traditional Square of Opposition will be nearly completely restored (only subalterns and superalterns will remain lost to us.) This process will allow us to create particular affirmative and negative propositions, and universal affirmative and negative propositions, just like in categorical logic.

http://candleinthedark.com/frege.jpg

Universal Quantifier
The expressions "all", "some", "no" and "none", are called quantifiers. Predicate logic contains symbols for two quantifiers, and they are nearly identical to the quantities in classical syllogistic logic. However, in Frege's theory, we can hold make a universal claim, without an existential error, by using the phrase "Given any X, X, has the characteristic of A" We can symbolize this thusly:

(x) Ax

Where "x" is the universal quantifier, and "A" is the specified characteristic. (Always place the variable in parentheses in front of the propositional function.)

Existential Quantifier
Recall from our discussion of the Traditional Square of Opposition that that propositions with a particular quantity have existential import - again, they assert that it is the case that at least one "x" exists. This is why the particular quantifier is referred to as the existential quantifier.

Propositions with a particular quantity can be stated thusly: "There is at least one 'x', such that it has the character of A" We represent such propositions thusly:

(∃ x)Ax

Where "∃ " is the existential quantifier, and "A" again, is the specified characteristic.

From what we have learned so far, propositions may be formed from propositional functions either by instantiation (substituting an individual constant for the individual variable) or by generalization (by placing a universal or existential quantifier before it.) It should be clear that the universal quantification is true only if all of its substitution instances are true, whereas the existential quantifier is true as long as there is at least one true substitution instance.

Quality
What about quality: affirmative and negative propositions? Well, in quantification theory, all we need do is add the "~' symbol before any predicate, to create universal negatives and particular negatives.

Let's now take a look at some relationships between universal and existential quantification. The universal proposition of "Everyone is mortal" is denied by the existential proposition "Someone is not mortal". These statements are symbolized as "(x)Mx" and "($x) ~Mx", respectively. Since one is the denial of the other, we can negate one of them, and create the following true biconditionals:

[~(x)Mx] [(∃ x)~Mx] I.e: "It is not the case that everyone is mortal = There exists at least one person who is not mortal"

Or:

[(x)Mx]≡ [(~∃x)~Mx] i.e.: "Everyone is mortal = there isn't a case of someone who isn't mortal"

If we use the capital letter "A" here as a variable, to represent any simple predicate, all the relationships between universal and existential quantification can be symbolized here:

1) [(x)Ax]≡ [~(∃x)~Ax] 2) [(∃x)Ax] ≡ [~(x)~Ax] 3) [(x)~Ax]≡ [~(∃ x)Ax] 4) [(∃x)~Ax] ≡ [~(x)Ax]

We can express the above statements in words, thusly:

1) "Given any x, x has the characteristic of A" is equivalent to "There doesn't exist a case where an "x" doesn't have the characteristic of A" 2) "There's at least one x with the characteristic of A" is equivalent to: "It's not the case that  given any x, you'll find a case without the characteristic of A. 3) "Given any x, none of them will have the character A" is equal to "There isn't a case of x with  the characteristic of A" 4) "It is the case that no X has the characteristic of A" is equal to "It is not the case that given any x, they will have the characteristic of A"

Clever readers will realize that we are moving towards restoring the Traditional Square of Opposition.

The New Square of Opposition
Using Universal and Existential quantifiers, we now have, in a sense, a new Square of opposition, sans only super and sub alterns.

http://candleinthedark.com/newsquare.jpg

Continuing to assume the existence of at least one individual, we can say that:

The two top propositions are contraries, they might both be false, but cannot both be true The two bottom propositions are subcontraries, they might both be true, but both can't be false. Propositions that are on the opposite end of the diaganols are contradictories: one must be true, and the other must be false. Finally, on each side of the square, the truth of the lower proposition is implied by the truth directly above it. But we don't use the terms super and sub alterns.

We can even recreate the traditional subject-predicate propositions of the Traditional Square, A, E, I and O:

Consider four permutations on the phrase "All humans are mortal"

A (x)(Hx ⊃ Mx) - All humans are mortal E (x)(Hx ⊃ ~Mx) - No humans are mortal I (∃x)(Hx & Mx) - Some humans are mortal O (∃x)(Hx & ~Mx) - Some humans are not mortal

Those taking the Course in Logic 101 should proceed to the next section on Inductive Logic