Predicate Calculus: Revised Version
Part I:
(Rough Draft)
by Vern Crisler
(First Published: April, 2000)
Introduction:
I think one of the reasons that the views of Frege,
Russell, and others within the tradition of mathematical logic have dominated
logic during the last century is primarily because of their adaptation of the functional notation of
mathematics. This notation was
especially effective in symbolizing relations of the sort one encounters in
mathematics. A relation involves an
ordered pair, or an ordered set of coordinates. You may remember the ordered pair (x, y) from algebra, which pair
was used to plot graphs (x being the horizontal part of the graph, and y being
the vertical part of the graph).
Despite your best efforts to forget mathematics, you may still recall an
equation such as the following:
1. f(x) = x^{2}+5x+10
In
the above equation “x” is called a variable, since it stands for any number you
want. You could replace the x in (1) by
any number, and solve the equation on the right of the equal sign. For instance, if x in the function f(x) just so happens to be 2, then you would
replace “x” in the right sided equation
(wherever the x may occur) with the number 2; hence,
2. If x=(2), then f(x) = (2)^{2}+5(2)+10,
i.e., 4+10+10 = 24
Now
a relation would involve more than one variable. A relation involves an ordered pair, or an ordered triple, or an
ordered ntuple. The expression ntuple,
when you get right down to it, basically means that the writer doesn’t know or
can’t remember the Greek or Latin endings for any relation going over maybe
four or five variables, and hence, it is a convenient etcetera. Moreover, since you may want to go a lot
higher in terms of variables in a relation, the n is a placeholder for
whatever number you want the relation to go up to. (In part 2 of this revised edition of predicate calculus, I’ll
show how it can be used to symbolize relations.)
There
is an analogy between functional notation in mathematics of the sort mentioned
above and logical notation, but only up to a point. I for one have no trouble at all in “spoiling” the mathematicians
of their symbols, but that does not mean the logician should forsake pure logic
for the temptations of
mathematics. We must be careful
not to confuse the purely formal or intentional relations of
logic with the real or material relations of mathematics; that is, one must not
make the mistake of having the mathematical tail wag the logical dog, for logic
and mathematics are not the same, nor does one dominate the other. (For instance, the subjectpredicate
relation of formal logic is not the same as the material or real numerical
relations of mathematics, nor the same as the argumentfunction relation of
mathematical logic.)
The Square of Opposition
The following four propositions represent
the traditional square of opposition in this revised version of predicate calculus. Note that the absolute value sign is the
same as the traditional omnisoperator in formal logic, i.e.,
“All”:
All S is P 
Ps 
Some S is P 
Ps 
No S is P 
~Ps 
Some S is not P 
~Ps 
In
the above, “No S is P” (SeP) is equivalent to “All S is nonP” (Sa~P), and in predicate
calculus symbolism, we have used the latter form, i.e., ~Ps, which
means “All S is nonP,” or more idiomatically, “No S is P.” The particular negative, “Some S is not
P” (SoP) is equivalent to “Some S is
nonP” (Si~P), and we’ve also used the latter form in this case as well, i.e.,
~Ps means “Some S is ~P.” In this
version of predicate calculus, the lack of the absolute value sign always means
the subject is particular rather than universal.
The
above symbolism is in part a simplification (or what I take to be a simplification)
of F. Sommers’ more elaborate symbolism in his seminal essay, “On a Fregean
Dogma,” in Imre Lakatos’s book, Problems in the Philosophy of Mathematics
(1965, 67), pp. 4781. In my judgment,
Sommers’ notation is not perspicuous, nor is it easy to use, and he appears
also to have abandoned it later in favor of a different symbolism discussed in
his book, The Logic of Natural Languages. Nevertheless, I believe Sommers was on the right track in using
this symbolism, and with a few modifications, it can even be connected up with
the predicate calculus of Frege and Russell.
Here
is how Sommers formulated the square of opposition using his functional
symbolism:
A

Ps 
All S are P 
E 
_{} 
No S are P 
I 
_{} 
Some S are P 
O 
P’s 
Some S are not P 
Now
the A and E propositions involve term negation. In the universal affirmative proposition A, Sommers’
symbolism Ps says the “All S are P” or “S’s are P’s.” The E expression says that “No S is P,” or
“S’s are unP’s.” In both these cases,
a predicate is being affirmed or denied of a subject. On the other hand, the I and O propositions leave off simple term
negation, and involve also sentence negations.
The I proposition says that “Some S are P” or “S’s aren’t unP’s” and
the O proposition says that “Some S are not P” or S’s aren’t P’s.” Here is how it would look in logical
algebra, and you can see the transition from predicate affirmation or denial to
sentence affirmation or denial in Sommers’ interpretation of his symbolism:
A

Ps 
S’s are P’s 
SaP 
E 
_{} 
S’s are unP’s 
Sa~P 
I 
_{} 
S’s aren’t unP’s 
~(Sa~P) 
O 
P’s 
S’s aren’t P’s 
~(SaP) 
Refer
to the last two propositions in parentheses.
The negative sentence operator standing before both sentences means that
both sentences are being denied, not the terms. In this revised version of the predicate
calculus, the use of the absolute value sign for universal quantity, as well as
the lack of an absolute value sign for particularity should help us avoid this
transition, and we will therefore be able to keep all predication as term
predication.
Moving
on from our basic functional symbolism to some other operations in logic, let
us represent the process of simple
conversion as follows:
All S is P to Some P is S 
Ps
= Sp 
Some S is P to Some P is S 
Ps
= Sp 
No S is P to No P is S 
~Ps
= ~Sp 
Some S is not P 
~Ps = not converted 
The
expression “Some S is not P” cannot be simply converted to “Some P is not S.”) This corresponds to the rule in logical
algebra that SoP cannot be converted simply to PoS.
One
other form of conversion is also necessary in this revised version of predicate
calculus, and this form is basically the transposition rule in propositional
calculus. If you will recall, we can
transpose A É
B by negating B and then negating A, hence, ~B É
~A. In logical algebra, this is the
operation, AaB, Ae~B, ~BeA, to ~Ba~A.
In predicate calculus, the same operation is represented by going from:
Ba
to
~A~b
i.e.,
“All nonB is nonA,” which is the same as ~Ba~A, or ~B É ~A. This contraposition method is basically what
rule 2 of the decision procedure in logical algebra is talking about, and we
will need it for the same reasons we needed it in logical algebra or
propositional calculus. (Examples given
later.)
Solving
Logical Equations in Functional Notation
One
of the major differences between this revised version of predicate calculus and
both logical algebra and propositional calculus is that the middle term is no
longer polarized or charged. On the
contrary, in this revised version of predicate calculus, the signs of the
middle term must be equivalent, or the argument will not arrive at a valid
conclusion. In short, both middle terms
must be either positive or negative in this version of predicate calculus.
In
solving the argument, All humans are mortal; Adam is human; therefore, Adam is
mortal, we set up the functional notation as follows:
Mh · Ha
É Ma
In
the above argument the absolute value sign symbolizes universal quantity, i.e.,
“All” and the absolute value sign around Adam is due to the fact that
individuals have what Sommers calls “wild” quantity, meaning they can take on
either universal or particular value, e.g., “All Adam” or “Some Adam” or
better, “All individuals identical to Adam are…” or “Some individuals identical
to Adam are….” and this is because there is only one Adam in either case. The absolute value sign can help us give
universal quantity to individuals, but for the most part, it will be assumed
that individuals have universal quantity, whether they appear as subjects or
predicates, or whether they have the absolute value sign or not.
Here
are some basic rules for using this functional calculus to solve syllogisms:
1. Since in the conclusion, the predicate term
is capital “P” and the subject term is small “s”, the premises must also
reflect this form. For instance,
the above argument can be generalized to the form, Pm · Ms
É Ps, where P equals the predicate term
mortal; m equals the middle term men; and s equals the subject term Adam. Note how the P is capital and the s is
lowercase in the conclusion, so this means it must be the same way in the
premises as well. Now if the premises
are not in this form (capital P, lowercase s), they can be changed by
means of conversion or inversion. For
instance, consider the argument All M is P; Some M is S; therefore, Some S is
P. The equation:
Pm · Sm = ?
can
only be solved by converting Sm to Ms by simple conversion:
Pm · Ms É Ps.
This
will follow the rule that the subject term must be in lowercase to derive a
valid conclusion. Here is the equation
again, though this time I’ve highlighted the subject and predicate term so that
you can see what I mean:
Pm · Ms É Ps.
The
conclusion, Some S is P, is a valid conclusion.
2. The middle terms must have like signs,
whether positive or negative. In
the equation,
Pm · Ms
É Ps
both
m’s are positive, so it meets the requirement of rule 2. In other arguments, both m’s can be
negative, but in no case can one m be positive and the other negative,
and still result in a valid conclusion.
(The opposite principle is used in logical algebra and logical
calculus.)
3. At least one of the middle terms must have
the absolute value sign. In the
above example, m means that that the subject is universally quantified. This is our familiar rule from logical
algebra that at least one of the middle terms must be connected to “All” or
“No.” The absolute value sign around a
middle term means that the term has been distributed. If the sign is missing from the equation, then the middle terms
have not been distributed, and the resulting conclusion will be invalid. But sometimes an equation will lack an
absolute value sign on first reading.
Is such an equation always invalid?
The answer is no. Sometimes, the
equation has to be converted to proper form, and this will result in a
distribution of the middle term. (We
met with this in rule 2 of logical algebra.)
Take the argument, All P is M; No S is M; therefore, No S is P. In equation form it would be:
Mp · ~Ms
= ?
Since
the P is not capital, and at least one of the middle terms is negative, we have
to change the premises by means of conversion, hence:
~P~m · ~Ms
É ~Ps
As
we said above, Mp can be inverted to ~P~m, i.e., MaP can be obverted
to Me~P, then converted to ~PeM, then obverted again to ~Pa~M, which you can
see is similar to ~P~m in form. Note
also that the middle terms have the same sign, and at least one of them is now
universally qauntified with an absolute value sign. The above argument is therefore valid.
4. The subject term in the conclusion takes
on the quantity value of the subject term in the premises. As you can see in the above example, because
s is universally quantified in the premises, it must also be universally
quantified in the conclusion as s.
If the subject term of the premises lacks the absolute value sign, it is
particular, and the conclusion must also lack the absolute value sign, and be a
particular conclusion of the form, “Some S is/is not….”
Here
are the 19 valid arguments in traditional logic expressed in function notation:
1. Barbara
All
M is P
All
S is M
Therefore,
All S is P.
2. Celarent
No
M is P
All
S is M
Therefore,
No S is P
~Pm · Ms
É ~Ps
3. Darii
All
M is P
Some
S is M
Therefore,
Some S is P
Pm · Ms É Ps
4. Ferio
No
M is P
Some
S is M
Therefore,
Some S is not P
~Pm · Ms É ~Ps
5. Cesare
No
P is M
All
S is M
Therefore,
No S is P
~Mp
· Ms;
~Pm · Ms
É ~Ps
6. Camestres
All
P is M
No
S is M
Therefore,
No S is P
Mp · ~Ms
= ?
~P~m · ~Ms
É ~Ps
7. Festino
No
P is M
Some
S is M
Therefore,
Some S is not P
~Mp · Ms = ?
~Pm · Ms É ~Ps
8. Baroko
All
P is M
Some
S is not M
Therefore,
Some S is not P
Mp · ~Ms = ?
~P~m · ~Ms
É ~Ps
9. Darapti
All
M is P
All
M is S
Therefore,
Some S is P
Pm · Sm = ?
Pm · Ms É Ps
10. Disamis
Some
M is P
All
M is S
Therefore,
Some S is P
Pm · Sm = ?
Pm · Ms É Ps
11. Datisi
All
M is P
Some
M is S
Therefore,
Some S is P
Pm · Sm = ?
Pm · Ms É Ps
12. Felapton
No
M is P
All
M is S
Therefore,
Some S is not P
~Pm · Sm = ?
~Pm · Ms É ~Ps
13. Bokardo
Some
M is not P
All
M is S
Therefore,
Some S is not P
~Pm · Sm = ?
~Pm · Ms É ~Ps
14. Ferison
No
M is P
Some
M is S
Therefore,
Some S is not P
~Pm · Sm = ?
~Pm · Ms É ~Ps
15. Bramantip
All
P is M
All
M is S
Therefore,
Some S is P
Mp · Sm = ?
Pm · Ms É Ps
16. Camenes
All
P is M
No
M is S
Therefore,
No S is P
Mp · ~Sm
= ?
~P~m · ~Ms
É ~Ps
17. Dimaris
Some
P is M
All
M is S
Therefore,
Some S is P
Mp · Sm
= ?
Pm · Ms É Ps
18. Fesapo
No
P is M
All
M is S
Therefore,
Some S is not P
~Mp · Sm
= ?
~Pm · Ms É ~Ps
19. Fresison
No
P is M
Some
M is S
Therefore,
Some S is not P
~Mp · Sm = ?
~Pm · Ms É ~Ps
Transposed
Syllogisms
1. Barbara
All
S is M
All
M is P
Therefore,
All S is P.
2. Celarent
All
S is M
No
M is P
Therefore,
No S is P
Ms
· ~Pm É ~Ps
3. Darii
Some
S is M
All
M is P
Therefore,
Some S is P
Ms · Pm
É Ps
4. Ferio
Some
S is M
No
M is P
Therefore,
Some S is not P
Ms · ~Pm
É ~Ps
5. Cesare
All
S is M
No
P is M
Therefore,
No S is P
Ms ·
~Mp = ?
Ms
· ~Pm
É ~Ps
6. Camestres
No
S is M
All
P is M
Therefore,
No S is P
~Ms · Mp = ?
~Ms ·
~P~m É ~Ps
7. Festino
Some
S is M
No
P is M
Thereofre,
Some S is not P
Ms · ~Mp
= ?
Ms ·
~Pm É ~Ps
8. Baroko
Some
S is not M
All
P is M
There,
Some S is not P
~Ms · Mp
= ?
~Ms ·
~P~m É ~Ps
9. Darapti
All
M is S
All
M is P
Therefore,
Some S is P
Sm · Pm
= ?
Ms · Pm
É Ps
10. Disamis
All
M is S
Some
M is P
Therefore,
Some S is P
Sm ·
Pm = ?
Ms · Pm É Ps
11. Datisi
Some
M is S
All
M is P
Therefore,
Some S is P
Sm ·
Pm = ?
Ms ·
Pm É Ps
12. Felapton
All
M is S
No
M is P
Therefore,
Some S is not P
Sm ·
~Pm = ?
Ms ·
~Pm É ~Ps
13. Bokardo
All
M is S
Some
M is not P
Therefore,
Some S is not P
Sm · ~Pm = ?
Ms ·
~Pm É ~Ps
14. Ferison
Some
M is S
No
M is P
Therefore,
Some S is not P
Sm
· ~Pm = ?
Ms
· ~Pm É ~Ps
15. Bramantip
All
M is S
All
P is M
Therefore,
Some S is P
Sm
· Mp = ?
Ms · Pm
É Ps
16. Camenes
No
M is S
All
P is M
Therefore,
No S is P
~Sm
· Mp = ?
~Ms · ~P~m
É ~Ps
17. Dimaris
All
M is S
Some
P is M
Therefore,
Some S is P
Sm
· Mp = ?
Ms
· Pm
É Ps
18. Fesapo
All
M is S
No
P is M
Therefore,
Some S is not P
Sm
· ~Mp = ?
Ms
· ~Pm É ~Ps
19. Fresison
Some
M is S
No
P is M
Therefore,
Some S is not P
Sm ·
~Mp = ?
Ms ·
~Pm É ~Ps
These
four rules have been followed in determining the validity of the arguments in
all four figures. For a more indepth
explanation, see the discussion near the beginning of this essay:
1. Since in the conclusion, the predicate term
is capital “P” and the subject term is small “s”, the premises must also
reflect this form.
2. The middle terms must have like signs,
whether positive or negative.
3. At least one of the middle terms must have
the absolute value sign.
4. The subject term in the conclusion takes
on the quantity value of the subject term in the premises.
First
Figure:
MP
\SP
AA1: Pm
· Ms
É Ps = valid
AE1: Pm
· ~Ms
É = invalid, already in 1^{st} figure;
middle terms not same.
AI1: Pm
· Ms É Ps = valid
AO1: Pm · ~Ms É = invalid; middles
not same.
EA1:
~Pm · Ms
É ~Ps = valid
EE1: ~Pm
· ~Ms
É
= invalid; middles not same.
EI1: ~Pm
· Ms É ~Ps = valid
IA1: Pm
· Ms
É = invalid; at least one m must have
absolute value sign.
II1: Pm
· Ms É = invalid; no
distribution.
IO1: Pm
·
~Ms
É
= invalid; middles not same; no distribution.
OA1: ~Pm
· Ms
É = invalid; middle undistributed.
OE1: ~Pm
· ~Ms
É = invalid; middles not same; no
distribution.
OI1: ~Pm
· Ms É = invalid; middle undistributed.
OO1: ~Pm
· ~Ms É = invalid; middles not same; undistributed.
Second
Figure:
PM
SM
\SP
AA2: Mp
· Ms
É ?
Pm · Ms
É = invalid; middles undistributed.
AE2: Mp
· ~Ms
É ?
~P~m · ~Ms
· ~Ps
= valid
AI2: Mp
· Ms
= ?
Pm · Ms É = invalid; middles
undistributed
AO2: Mp · ~Ms = ?
~P~m · ~Ms É ~Ps
= valid
~Pm · Ms
É ~Ps
= valid
~Pm · ~Ms
É invalid; unlike middles
EI2: ~Mp
· Ms =
?
~Pm · Ms É ~Ps
= valid
~Pm · ~Ms É invalid; unlike middles
IA2: Mp
· Ms
= ?
Pm · Ms
É invalid; middles undistributed
IE2: Mp
· ~Ms
= ?
Pm · ~Ms
É invalid; unlike middles
II2: Mp
· Ms =
?
Pm · Ms É invalid; middles undistributed
IO2: Mp
· ~Ms
= ?
Pm · ~Ms É invalid; unlike middles
OA2: ~Mp
· Ms
= invalid; O not convertible
OE2: ~Mp
· ~Ms
= invalid; O not convertible
OI2: ~Mp
· Ms =
invalid; O not convertible
OO2: ~Mp
· ~Ms
= invalid; O not convertible
MP
MS
\SP
AA3 Pm
· Sm
= ?
Pm · Ms É Ps = valid
Pm · ~Ms
É invalid; unlike middles
AI3 Pm
· Sm
= ?
Pm · Ms É Ps = valid
Pm · ~Ms É invalid; unlike middles
EA3 ~Pm
· Sm
= ?
~Pm · Ms É ~Ps
= valid
~Pm · ~Ms
É invalid; unlike middles
EI3 ~Pm
· Sm
= ?
~Pm · Ms É ~Ps = valid
~Pm · ~Ms É invalid; unlike middles
IA3 Pm
· Sm
= ?
Pm · Ms É Ps = valid; middles distributed in
original equation
Pm · ~Ms
É invalid; unlike middles
II3 Pm
· Sm
= ?
Pm · Ms É invalid; middles undistributed
IO3 Pm
· ~Sm
= ?
Pm · ~Ms É invalid; unlike middles
OA3 ~Pm
· Sm
= ?
~Pm · Ms É ~Ps
= valid; middles distributed in original equation
~Pm · ~Ms
É invalid; unlike middles
OI3 ~Pm
· Sm = ?
~Pm · Ms É invalid; undistributed middle
OO3 ~Pm
· ~Sm
= ?
~Pm · ~Ms É invalid; unlike middles; also undistributed
Fourth
Figure
PM
MS
\SP
AA4 Mp
· Sm
= ?
Pm · Ms É Ps =
valid; middle distributed in original equation
AE4 Mp
· ~Sm
= ?
~P~m · ~Ms
É ~Ps
= valid
AO4 Mp
· ~Sm = ?
Pm · ~Ms É invalid; unlike middles
EA4 ~Mp
· Sm
= ?
~Pm · Ms É ~Ps
= valid
~Pm · ~Ms
É invalid; unlike middles
EI4 ~Mp
· Sm = ?
~Pm · Ms É ~Ps = valid; middle distributed in
second equation
~Pm · ~Ms É invalid; unlike middles
IA4 Mp
· Sm
= ?
Pm · Ms É Ps = valid; middle distributed in
original equation
Pm · ~Ms
É invalid; unlike middles
II4 Mp
· Sm = ?
Pm · Ms É invalid; middles undistributed
IO4 Mp
· ~Sm = ?
Pm · ~Ms É invalid; unlike middles
OA4 ~Mp
· Sm
= invalid; O not convertible
OI4 ~Mp
· Sm = invalid; O not convertible
OO4 ~Mp
· ~Sm = invalid; O not convertible