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Predicate Calculus: Revised Version

Part I:

 

(Rough Draft)

 

by Vern Crisler

 

Copyright 2000

 

(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) = x2+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 n-tuple. The expression n-tuple, when you get right down to it, basically means that the writer doesnt know or cant 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, Ill 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 subject-predicate relation of formal logic is not the same as the material or real numerical relations of mathematics, nor the same as the argument-function 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 omnis-operator in formal logic, i.e., All:

 

 

All S is P

P|s|

Some S is P

Ps

No S is P

~P|s|

Some S is not P

~Ps

 

 

In the above, No S is P (SeP) is equivalent to All S is non-P (Sa~P), and in predicate calculus symbolism, we have used the latter form, i.e., ~P|s|, which means All S is non-P, or more idiomatically, No S is P. The particular negative, Some S is not P (SoP) is equivalent to Some S is non-P (Si~P), and weve 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 Lakatoss book, Problems in the Philosophy of Mathematics (1965, 67), pp. 47-81. 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

Ps

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 Ss are Ps. The E expression says that No S is P, or Ss are un-Ps. 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 Ss arent un-Ps and the O proposition says that Some S are not P or Ss arent Ps. 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

Ss are Ps

SaP

E

Ss are un-Ps

Sa~P

I

Ss arent un-Ps

~(Sa~P)

O

Ps

Ss arent Ps

~(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

P|s| = Sp

Some S is P to Some P is S

Ps = Sp

No S is P to No P is S

~P|s| = ~S|p|

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 B by negating B and then negating A, hence, ~B ~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:

 

B|a|

 

to ~A~|b|

 

i.e., All nonB is nonA, which is the same as ~Ba~A, or ~B ~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:

 

M|h| H|a| M|a|

 

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, P|m| M|s| P|s|, 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 lower-case 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, lower-case 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:

 

P|m| Sm = ?

 

can only be solved by converting Sm to Ms by simple conversion:

 

P|m| Ms Ps.

 

This will follow the rule that the subject term must be in lower-case to derive a valid conclusion. Here is the equation again, though this time Ive highlighted the subject and predicate term so that you can see what I mean:

 

P|m| Ms 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,

 

P|m| M|s| P|s|

 

both ms are positive, so it meets the requirement of rule 2. In other arguments, both ms 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:

 

M|p| ~M|s| = ?

 

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| ~M|s| ~P|s|

 

As we said above, M|p| 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.

 

Valid Arguments

 

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.

 

P|m| M|s| P|s|

 

 

2. Celarent

 

No M is P

All S is M

Therefore, No S is P

 

~P|m| M|s| ~P|s|

 

 

3. Darii

 

All M is P

Some S is M

Therefore, Some S is P

 

P|m| Ms Ps

 

 

4. Ferio

 

No M is P

Some S is M

Therefore, Some S is not P

 

~P|m| Ms ~Ps

 

 

5. Cesare

 

No P is M

All S is M

Therefore, No S is P

 

~M|p| M|s|;

~P|m| M|s| ~P|s|

 

 

6. Camestres

 

All P is M

No S is M

Therefore, No S is P

 

M|p| ~M|s| = ?

~P~|m| ~M|s| ~P|s|

 

 

7. Festino

 

No P is M

Some S is M

Therefore, Some S is not P

 

~M|p| Ms = ?

~P|m| Ms ~Ps

 

 

8. Baroko

 

All P is M

Some S is not M

Therefore, Some S is not P

 

M|p| ~Ms = ?

~P~|m| ~Ms ~Ps

 

 

9. Darapti

 

All M is P

All M is S

Therefore, Some S is P

 

P|m| S|m| = ?

P|m| Ms Ps

 

 

10. Disamis

 

Some M is P

All M is S

Therefore, Some S is P

 

Pm S|m| = ?

Pm Ms Ps

 

 

11. Datisi

 

All M is P

Some M is S

Therefore, Some S is P

 

P|m| Sm = ?

P|m| Ms Ps

 

 

12. Felapton

 

No M is P

All M is S

Therefore, Some S is not P

 

~P|m| S|m| = ?

~P|m| Ms ~Ps

 

 

13. Bokardo

 

Some M is not P

All M is S

Therefore, Some S is not P

 

~Pm S|m| = ?

~Pm Ms ~Ps

 

 

14. Ferison

 

No M is P

Some M is S

Therefore, Some S is not P

 

~P|m| Sm = ?

~P|m| Ms ~Ps

 

 

15. Bramantip

 

All P is M

All M is S

Therefore, Some S is P

 

M|p| S|m| = ?

Pm Ms Ps

 

 

16. Camenes

 

All P is M

No M is S

Therefore, No S is P

 

M|p| ~S|m| = ?

~P~|m| ~M|s| ~P|s|

 

 

17. Dimaris

 

Some P is M

All M is S

Therefore, Some S is P

 

Mp S|m| = ?

Pm Ms Ps

 

 

18. Fesapo

 

No P is M

All M is S

Therefore, Some S is not P

 

~M|p| S|m| = ?

~P|m| Ms ~Ps

 

 

19. Fresison

 

No P is M

Some M is S

Therefore, Some S is not P

 

~M|p| Sm = ?

~P|m| Ms ~Ps

 

 

 

Transposed Syllogisms

 

1. Barbara

 

All S is M

All M is P

Therefore, All S is P.

 

M|s| P|m| P|s|

 

 

2. Celarent

 

All S is M

No M is P

Therefore, No S is P

 

M|s| ~P|m| ~P|s|

 

 

3. Darii

 

Some S is M

All M is P

Therefore, Some S is P

 

Ms P|m| Ps

 

 

4. Ferio

 

Some S is M

No M is P

Therefore, Some S is not P

 

Ms ~P|m| ~Ps

 

 

5. Cesare

 

All S is M

No P is M

Therefore, No S is P

 

M|s| ~M|p| = ?

M|s| ~P|m| ~P|s|

 

 

6. Camestres

 

No S is M

All P is M

Therefore, No S is P

 

~M|s| M|p| = ?

~M|s| ~P~|m| ~P|s|

 

 

7. Festino

 

Some S is M

No P is M

Thereofre, Some S is not P

 

Ms ~M|p| = ?

Ms ~P|m| ~Ps

 

 

8. Baroko

 

Some S is not M

All P is M

There, Some S is not P

 

~Ms M|p| = ?

~Ms ~P~|m| ~Ps

 

 

9. Darapti

 

All M is S

All M is P

Therefore, Some S is P

 

S|m| P|m| = ?

Ms P|m| Ps

 

 

10. Disamis

 

All M is S

Some M is P

Therefore, Some S is P

 

S|m| Pm = ?

Ms Pm Ps

 

 

11. Datisi

 

Some M is S

All M is P

Therefore, Some S is P

 

Sm P|m| = ?

Ms P|m| Ps

 

 

12. Felapton

 

All M is S

No M is P

Therefore, Some S is not P

 

S|m| ~P|m| = ?

Ms ~P|m| ~Ps

 

 

13. Bokardo

 

All M is S

Some M is not P

Therefore, Some S is not P

 

S|m| ~Pm = ?

Ms ~Pm ~Ps

 

 

14. Ferison

 

Some M is S

No M is P

Therefore, Some S is not P

 

Sm ~P|m| = ?

Ms ~P|m| ~Ps

 

 

15. Bramantip

 

All M is S

All P is M

Therefore, Some S is P

 

S|m| M|p| = ?

Ms Pm Ps

 

 

16. Camenes

 

No M is S

All P is M

Therefore, No S is P

 

~S|m| M|p| = ?

~M|s| ~P~|m| ~P|s|

 

 

17. Dimaris

 

All M is S

Some P is M

Therefore, Some S is P

 

S|m| Mp = ?

Ms Pm Ps

 

 

18. Fesapo

 

All M is S

No P is M

Therefore, Some S is not P

 

S|m| ~M|p| = ?

Ms ~P|m| ~Ps

 

 

19. Fresison

 

Some M is S

No P is M

Therefore, Some S is not P

 

Sm ~M|p| = ?

Ms ~P|m| ~Ps

 

 

Functional Calculus in Four Figures

 

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

SM

\SP

 

 

AA-1: P|m| M|s| P|s| = valid

AE-1: P|m| ~M|s| = invalid, already in 1st figure; middle terms not same.

AI-1: P|m| Ms Ps = valid

AO-1: P|m| ~Ms = invalid; middles not same.

EA-1: ~P|m| M|s| ~P|s| = valid

EE-1: ~P|m| ~M|s| = invalid; middles not same.

EI-1: ~P|m| Ms ~Ps = valid

EO-1: ~P|m| ~Ms = invalid; middles not same.

IA-1: Pm M|s| = invalid; at least one m must have absolute value sign.

IE-1: Pm ~M|s| = invalid; middles not same.

II-1: Pm Ms = invalid; no distribution.

IO-1: Pm ~Ms = invalid; middles not same; no distribution.

OA-1: ~Pm M|s| = invalid; middle undistributed.

OE-1: ~Pm ~M|s| = invalid; middles not same; no distribution.

OI-1: ~Pm Ms = invalid; middle undistributed.

OO-1: ~Pm ~Ms = invalid; middles not same; undistributed.

 

 

Second Figure:

 

PM

SM

\SP

 

AA-2: M|p| M|s| ?

Pm M|s| = invalid; middles undistributed.

AE-2: M|p| ~M|s| ?

~P~|m| ~M|s| ~P|s| = valid

AI-2: M|p| Ms = ?

Pm Ms = invalid; middles undistributed

AO-2: M|p| ~Ms = ?

~P~|m| ~Ms ~Ps = valid

EA-2: ~M|p| M|s| = ?

~P|m| M|s| ~P|s| = valid

EE-2: ~M|p| ~M|s| = ?

~P|m| ~M|s| invalid; unlike middles

EI-2: ~M|p| Ms = ?

~P|m| Ms ~Ps = valid

EO-2: ~M|p| ~Ms = ?

~P|m| ~Ms invalid; unlike middles

IA-2: Mp M|s| = ?

Pm M|s| invalid; middles undistributed

IE-2: Mp ~M|s| = ?

Pm ~M|s| invalid; unlike middles

II-2: Mp Ms = ?

Pm Ms invalid; middles undistributed

IO-2: Mp ~Ms = ?

Pm ~Ms invalid; unlike middles

OA-2: ~Mp M|s| = invalid; O not convertible

OE-2: ~Mp ~M|s| = invalid; O not convertible

OI-2: ~Mp Ms = invalid; O not convertible

OO-2: ~Mp ~Ms = invalid; O not convertible

 

 

 

Third Figure

 

MP

MS

\SP

 

 

AA-3 P|m| S|m| = ?

P|m| Ms Ps = valid

AE-3 P|m| ~S|m| = ?

P|m| ~M|s| invalid; unlike middles

AI-3 P|m| Sm = ?

P|m| Ms Ps = valid

AO-3 P|m| ~Sm = ?

P|m| ~Ms invalid; unlike middles

EA-3 ~P|m| S|m| = ?

~P|m| Ms ~Ps = valid

EE-3 ~P|m| ~S|m| = ?

~P|m| ~M|s| invalid; unlike middles

EI-3 ~P|m| Sm = ?

~P|m| Ms ~Ps = valid

EO-3 ~P|m| ~Sm = ?

~P|m| ~Ms invalid; unlike middles

IA-3 Pm S|m| = ?

Pm Ms Ps = valid; middles distributed in original equation

IE-3 Pm ~S|m| = ?

Pm ~M|s| invalid; unlike middles

II-3 Pm Sm = ?

Pm Ms invalid; middles undistributed

IO-3 Pm ~Sm = ?

Pm ~Ms invalid; unlike middles

OA-3 ~Pm S|m| = ?

~Pm Ms ~Ps = valid; middles distributed in original equation

OE-3 ~Pm ~S|m| = ?

~Pm ~M|s| invalid; unlike middles

OI-3 ~Pm Sm = ?

~Pm Ms invalid; undistributed middle

OO-3 ~Pm ~Sm = ?

~Pm ~Ms invalid; unlike middles; also undistributed

 

 

 

Fourth Figure

 

PM

MS

\SP

 

 

AA-4 M|p| S|m| = ?

Pm Ms Ps = valid; middle distributed in original equation

AE-4 M|p| ~S|m| = ?

~P~|m| ~M|s| ~P|s| = valid

AI-4 M|p| Sm = ?

Pm Ms invalid; middles undistributed

AO-4 M|p| ~Sm = ?

Pm ~Ms invalid; unlike middles

EA-4 ~M|p| S|m| = ?

~P|m| Ms ~Ps = valid

EE-4 ~M|p| ~S|m| = ?

~P|m| ~M|s| invalid; unlike middles

EI-4 ~M|p| Sm = ?

~P|m| Ms ~Ps = valid; middle distributed in second equation

EO-4 ~M|p| ~Sm = ?

~P|m| ~Ms invalid; unlike middles

IA-4 Mp S|m| = ?

Pm Ms Ps = valid; middle distributed in original equation

IE-4 Mp ~S|m| = ?

Pm ~M|s| invalid; unlike middles

II-4 Mp Sm = ?

Pm Ms invalid; middles undistributed

IO-4 Mp ~Sm = ?

Pm ~Ms invalid; unlike middles

OA-4 ~Mp S|m| = invalid; O not convertible

OE-4 ~Mp ~S|m| = invalid; O not convertible

OI-4 ~Mp Sm = invalid; O not convertible

OO-4 ~Mp ~Sm = invalid; O not convertible

 

Continued: Part 2 of Predicate Calculus: Revised Version