Quantification Theory
In this final lesson on symbolic logic, we'll take a very brief look at modern
methods of representing the internal structure of propositions in first-order predicate
calculus (or quantification theory). Incorporating all of the propositional calculus
along with a few new symbols and rules of inference, the predicate calculus provides
another way of handling the same logical forms we examined in our study of
categorical logic.
Individuals, Predicates, and Quantifiers
In addition to the familiar symbols of the propositional calculus, quantification
theory also employs special symbols of four special sorts:


individual constants ( a, b, c, etc. ) represent particular individual
things—Allison, Bill, or this car, for example.
predicate constants ( F, G, H, etc. ) represent specific predicates—free,
greedy, or heavy, for example.
Writing an individual symbol directly after a predicate symbol signifies that this thing
has that property. Thus, using the above examples, Fa, Gb, and Hc would
signify "Allison is free," "Bill is greedy," and "This car is heavy" respectively. Since
these are complete statements, they can be combined using the statement connectives
to form
such
compound
statement
individual
predicate
statements
a, b, c...
F, G, H...
constants A, B, C...
as Fa 
...x, y, z
variables p, q, r...
Gb, Hg 
Gb, and
operators
~, &, v,
(x)
, 
(x)
Hc 
~Hc .


individual variables ( x, y, z, etc. ) are employed to refer to any
predicate generally.
quantifiers are logical operators that signify the range of individuals to which
individual variables apply, either all
(x)
or some
(x) .
Put together, these symbols can be used to represent each of the four major forms of
categorical proposition:

(x)(Sx  Px)

(x)(Sx  ~Px)
corresponds to the E proposition, "No S are P."

(x)(Sx  Px)
corresponds to the I proposition, "Some S are P."

(x)(Sx  ~Px)
corresponds to the O proposition, "Some S are not
corresponds to the A proposition, "All S are P."
P."
Any of the rules of replacement from the propositional calculus may be applied to
these statements of quantification theory. Thus, for example, we can easily
demonstrate the converse of an E proposition:
1. (x)(Sx  ~Px)
premise
2. (x)(~~Px  ~Sx)
1 Trans.
3. (x)(Px  ~Sx)
2 D.N.
Quantification Rules
In order to prove the validity of syllogisms, however, we first need to strip the
quantifiers from each statement, apply the appropriate rules of inference, and then
restore quantifiers to each statement. The four quantification rules dictate the
conditions under which it is permissible to delete or add a quantifier:



Using Øx to represent any expression in which the individual variable "x"
appears, Universal Instantiation (UI) can be stated as an argument of the form:
(x)( Øx )
_______


Øu
The "u" in this case can be any arbitrarily chosen individual constant or
variable. In the context of a proof, for example, the truth of "(x)[Fx (Gx 
Hx)]" could be used to justify that of "Fb  (Gb  Hb)." If the statement
holds for all x, then it certainly must hold for b.




In similar fashion, Universal Generalization (UG) has the form:
Øy
_______

(x)( Øx )
In this case, however, it is crucial that the "y" is an arbitrarily chosen
individual—that is, an individual that was introduced into the proof by an
application of UI. Only then can we be sure that what holds of it is not some
special feature but something that would hold equally well of all " x."

Removing an existential quantifier requires application of Existential
Instantiation (EI), and argument of the form:

(x)( Øx )

_______


Øu
In this case, the individual constant "u" must be one which has never been
used in any earlier line of the proof; otherwise, we might mistakenly associate
two things which have nothing in common. Thus, it is usually best to employ
EI as soon as possible (certainly, before any application of UI) in a proof.



Finally, Existential Generalization (EG) is an argument of the form:
Øu
_______


(x)( Øx )
Here, as in UI, "u" may be any individual constant or variable.
Using these quantification rules together with the inference and replacement rules of
the propositional calculus, it is possible to prove the validity of any of the valid
categorical syllogisms. For EIO-1, for example:
1. (x)(Mx  ~Px)
premise
2. (x)(Sx  Mx)
premise
3. Sd  Md
2 EI
4. Md  ~Pd
1 UI
5. Sd
3 Simp.
6. Md  Sd
3 Comm.
7. Md
8. ~Pd
6 Simp.
4, 7 M.P.
9. Sd  ~Pd
10. (x)(Sx  ~Px)
5, 8 Conj.
9 EG
Asyllogistic Inferences
Quantification theory makes it possible to prove the validity of many arguments
that could not easily be expressed in categorical logic at all. In fact, Russell and
Whitehead showed less than a hundred years ago that a higher-order predicate
calculus (one that fills in the chart above by including variables and quantifiers for
predicates as well as for individuals) is sufficient to provide logical demonstrations of
simple arithmetical truths. A few decades later, however, Gödel showed that any such
powerful system must contain at least one proposition whose truth or falsity cannot be
proven.