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Logical Calculus, Part 1

 

by Vern Crisler

 

Copyright 2000; 2005

 

1.  Introduction:

 

In this essay, I want to stipulate that the logical calculus I am developing here differs from our previous logical algebra primarily in the choice of symbols used to represent logical operations.  In this logical calculus, we are using the symbols of the logic of compounds to represent both the logic of terms and the logic of propositions.  The third column represents the traditional symbols from the propositional calculus, and we use the disjunction form in the fourth column for the universal propositions.  (Note that the negation sign “-” is equivalent to the tilde “~”, which is also used for negation.  We also use either the ® sign or the É for implication.)

 

 

English

Algebraic

Compound

Arithmetical

1. All A is B

AaB

A ® B

-A + B

2. No A is B

AeB

A ® ~B

-A + -B

3. Some A is B

AiB

A & B

A & B

4. Some A is not B

AoB

A & ~B

A & -B

           

 

The reader should remember that compound logical symbols used in logic textbooks are almost always used to symbolize propositions rather than terms.  Hence, such a logic is usually called a propositional logic or a propositional calculus.  It is often the case that a logic of terms is expressed using variables such as A, B, C, X, Y, or Z, etc., while a logic of propositions is expressed with such letters as p, q, r, and so on.  Nevertheless, in my opinion, since the same logical laws govern both systems of logic—term and compound—it is legitimate to refer to both using the expression “logical calculus” rather than the more restrictive expression “propositional calculus.”  Indeed, Fred Sommers spoke of a “structural isomorphism” between categorical and hypothetical logic, and I think this is correct.  In terms of “deep structure” both types of logic appear to be virtually the very same thing.  The only substantive difference between them is that compound logic (i.e., the logic of propositions) has extended itself over categorical logic by one proposition, and this extra proposition can be derived in post-Boolean logic due to the concept of inclusive disjunction adopted since at least the days of Stanley Jevons.

 

Historical Note: Gottfried Leibniz was the first to make the attempt to develop a logical calculus, and his work anticipated much that was to follow, including representing logic by lines, by circles, and by numbers (cf., Frege, Venn, and Godel respectively). 

 

In logical algebra, we found that Sommers’ notion of “polarizing” the middle terms¾giving them “opposite distribution values”¾helped us cancel them so that we could derive a valid conclusion.  There are different ways of representing this with symbols, but it seems that the fractional and the arithmetical approaches are the clearest or easiest to use.  In order to represent a logical calculus using fractional symbols, we have to make a change to the traditional disjunction symbol (the “v” sign).  As the name implies, the fractional symbol for disjunction would be a division sign.  For the arithmetical approach, the traditional + sign will stand for the “v” sign.  Hence, A v B, (i.e., “A or B”) will be symbolized either as the fractional form:

 

 

Or as the arithmetical form  A + B.

 

Similarly, A ® B, (i.e., “If A then B” or “All A is B”) will be symbolized as:

 

 

or as  -A + B

 

 

This is because the translational equivalent of “If A, then B” is “Either not-A or B.”  It should be kept in mind that when we are using the letters A or B as terms, the traditional compound symbols need to be interpreted as categorical statements rather than as compound propositions.  Hence, in the above, the sentence “If A then B” will be equivalent in a calculus of terms to the sentence “All A is B.”  If we were talking about a calculus of propositions, a sentence such as “If p, then q” would refer to a compound of subject-predicate sentences, i.e., “If (S is P) then (M is R).”

 

Note: It appears that a certain Mr. Johnson (apparently W.E. Johnson) was one of the first to suggest using the fraction sign to represent alternation (as noted by John Keynes in his Formal Logic, p. 533).  He also suggested that the vertical line should represent conjunction.  Here are Keynes’ examples of the use of the signs:

 

AB or CD was represented by:

 

AB

CD

 

(A or C) and (B or D) was represented by:

 

A

 B

C

 D

 

Technically, there should be division signs between A and C and between B and D, but Keynes only uses the vertical line in order to emphasize the sign of conjunction.  As will become clear, these signs were developed in a different way by Johnson and Keynes from the way we are using them in logical calculus.

 

2.  Arguments with Three Terms:

 

Fractional Symbolism:  To use fractional symbolism for an argument such as:

 

All B is C:                                 B ® C

All A is B:                                 A ® B

Therefore, All A is C                A ® C

 

We would use the division sign in the following symbolic form:

 

 

As can be seen, the B’s are polarized, one positive, one negative, and after cancelling, it is easy to derive the conclusion.  Moreover, the fractional representation of the conclusion can always be converted back to the textbook representation by simple translation: i.e., ~A v C to A ® C, (“If A then C”).  It might be helpful to strike out the polarized terms by means of a slash sign or some other sign that will show the cancellation of the linking terms.  (I usually use a slash sign “/” when working arguments out on paper.)

 

 

The arithmetical way of symbolizing this is by using the following format, using the + sign in place of the “v” sign, as follows:

 

1  ~B  +  C

2.   B           +  ~A

      0  +  C  +  ~A

 

The conclusion is 0 + C + ~A, or using the vel sign, 0 v C v ~A.  The 0 can be dropped and the remaining expression is C v ~A, or ~A v C, or A É C. 

 

In a traditional Boolean-based logic one can find the following simplification rules:

 

0 v P = P

1 v P = 1

(1) · P = P

(0) · P = 0.

 

In our “neo-Boolean” approach, the first simplification is the only one that is relevant.  Here are some comparisons with textbook ways of symbolizing the same logical processes.  As can be seen, the neo-Boolean symbolism is virtually the same as the traditional symbolisms.  What’s more, it will allow a straightforward cancellation procedure for deriving conclusions:

 

 

Neo-Boolean

Set Theory

Predicate Calculus

Arithmetical P-Calculus

 

 

 

 

1.  ~B + C

1. ~B È C

1. (x) Bx ® Cx

1. (x) ~ Bx + Cx

2.  ~A + B

2. ~A È B

2. (x) Ax ® Bx

2. (x)    Bx +       + ~Ax

3.  ~A + C

3. ~A È C

3. (x) Ax ® Cx

3. (x)    0x  + Cx + ~Ax

 

 

Note: The above term “Arithmetical P-Calculus” refers to the standard predicate calculus presented in arithmetical format.  It should also be remembered that the neo-Boolean symbolism above is a symbolism for terms or concepts.  We would use p, q, or r if we were symbolizing propositions.

 

The following is an example of an invalid argument, where S = subject term; M=middle term; and P=predicate term:

 

If P, then M                              P É M

If S, then M                              S É M

Therefore, if S, then P   S É P

 

In terms of our new symbolism, this would be expressed as:

 

 

It’s fairly obvious that the middle (or linking) terms are unpolarized.  In a fractional expression, this is a sign of invalidity, and it basically means that the middle terms are not distributed.  Therefore, the expression ~S v P is an invalid conclusion.  Here is the expression in arithmetical form, showing the lack of polarization:

 

1.  ~P  + M

2.        + M + ~S

     no conclusion 

 

Both M’s are positive, thus undistributed.  Without distribution there can be no conclusion.   Now, let us see how a fractional expression can be combined with a simple conjunctive expression as in the following:

 

M É P

S · M

Therefore, S · P

 

In the symbolism of logical calculus, this would be:

 

 

The conclusion, SP or (S · P) is valid because the middle terms are polarized.  A fractional combined with a conjunction can only result in a conjunctive conclusion.  In arithmetical form, we have:

 

1.  ~M + P

2.  +M  & S    

3.   0 + S & P

 

The presence of a & sign in one of the premises requires that the conclusion will be particular.  Also, in arithmetical logic, the 0 in the conclusion should always be preceded or followed by a + sign so as not to read the conclusion as (0) · SP, which would result in a conclusion of 0.  A fractional combined with a fractional gives a universal conclusion but only if the middle terms are polarized.  As we have seen, if the middle terms are unpolarized, the argument is invalid.

 

The following is a valid argument:

 

M ® P

M ® S

Therefore, S · P

 

But when we try to symbolize it by either fractional or arithmetical notation (or even check it by a truth table), we find that the argument appears to be invalid.  Why is this so?  This is due to the fact that the second premiss must be converted to its per accidens form, SM, before a valid conclusion of SP can be derived.   So, the argument should not be in this form:

 

 

But should instead be represented as:

 

 

In arithmetical symbolism, the form of the argument would be:

 

1.  ~M + P

2.  +M & S  

3.   0 + S & P

 

Conversio per accidens can only be used when both premises are universal in form, and as we shall see, when both premises are not negative.

 

Is the following argument valid?

 

M ® ~P

M ® ~S

Therefore, S · P

 

 

Or,

 

1.  ~M + ~P

2.  ~M        + ~S

 

Here, the middle terms are not polarized and they cannot be polarized by converting one of the universal sentences to its particular form.  The first and second premises can be converted per accidens (though not at the same time) but that still leaves the M’s with the same sign.  So conversion by limitation won’t work in this case.  By logical algebra, we can see what is wrong:

 

MeP + MeS

MeP + SeM = ?

 

The problem is that we have two original negative premises, meaning unpolarized middles, and so the conlusion SiP is invalid.  Therefore, any combination of fractions in which both original “denominators” contain negative terms will be invalid, since such a combination will always involve two negative premises (and hence undistributed middles).  Take care to note that we are talking here about original premises.  Once the original premises have been translated into fractional form, then the premises can be moved top to bottom, or bottom to top, since alternation is freely convertible.

 

3.  Basic Rules for Logical Calculus:

 

In this logical calculus, the basic rules for determining validity or invalidity are summarized as follows: 

 

1)  A fractional expression combined with a fractional expression leads to a fractional conclusion.  (The middle terms must be polarized.)

 

2)  A fractional expression combined with a conjunctive expression leads to a conjunctive conclusion.  (The middle terms must be polarized.)

 

3)  A conjunction combined with a conjunction results in an invalid conclusion, since two particular statements fail to distribute the middle terms.

 

4)  A fractional expression combined with a fractional expression, with like middle terms, may result in a valid conclusion but only if conversion per accidens is able to polarize the middles; otherwise, the argument is invalid.  Conversion per accidents, of course, cannot be used when both premises are negative.

 

The arithmetical rules are pretty much the same.  The middles must be polarized; and if the premises contain a particular proposition, the conclusion must be particular (i.e., conjunctive).  Moreover, two particulars lead to no conclusion; and conversion per accidens may be used to reduce a universal expression to particular form so as to polarize the middles; otherwise, the argument is invalid. 

 

4.  Sommers & the Calculus of Terms

 

The arithmetical format used in this logical calculus was inspired by Fred Sommers’ arithmetical term logic, but is relevantly different.  The arithmetical logic given in this essay is more of a middle ground approach between Boole and Sommers.  It still largely retains a Boolean-type structure, whereas Sommers has moved farther away from Boole.  In my opinion, Sommers’ +’s and –’s seem to be bearing too much symbolic weight.  Clifton McIntosh says,

 

“TFL [traditional formal logic] contains the full logic of propositions, all of syllogistic reasoning, a portion of the logic of identity, and some of the logic of relations.  All of this is represented with terms and the two algebraic signs ‘+’ and ‘-’.  This requires ‘+’ and ‘-’ to do many jobs.” (“Appendix F,” in Fred Sommers, The Logic of Natural Languages, p. 390.)

 

Too many jobs perhaps.  In another context, Sommers says,

 

“The logic [of terms] will make use of standard arithmetic operations to test for validity.  As with Boole, arithmetic is accepted as given.  There are no rules of inference.  Validity is arithmetical….” (“The Calculus of Terms,” in ed. G. Engelbretsen, The New Syllogistic, p. 14.)

 

We would point out, however, that logic is not really arithmetic, not really mathematics, that reasoning is not really calculating.  Isn’t this the basic mistake of modern mathematical logicians, thinking that logic could be turned into mathematics?  Contrary to Sommers, however, the rules of inference are still operative in Boolean logic, even if “behind the scenes.”  A purely arithmetical approach to logic is impossible.

 

In the following, we present Sommers’ logical symbolism using the traditional names as given by Aldrich and Carroll.  As can be seen, Sommers is able to polarize the M’s in many of the “equations” and arrive at the correct conclusion, but cannot do so for the ones in brackets:

 

Figure 1

1.  Barbara                   - M + P - S + M =  - S + P

2.  Celarent                  - M - P - S + M =  - S - P

3.  Darii                        - M + P + S + M =  + S + P

4.  Ferio                       - M - P + S + M =  + S - P

 

Figure 2

5.  Cesare                    - P - M - S + M =  - S - P

6.  Camestres               - P + M - S - M =  - S - P

7.  Festino                    - P - M + S + M =  + S - P

8.  Baroko                    - P + M + S - M =  + S - P

 

Figure 3

9.  Darapti                    [ - M + P - M + S = ? ]  

10.  Disamis                 + M + P - M + S =  + S + P

11.  Datisi                     - M + P + M + S =  + S + P

12.  Felapton                [ - M - P - M + S = ? ] 

13.  Bokardo                + M - P - M + S = + S - P

14.  Ferison                  - M - P + M + S = + S - P

 

Figure 4

15.  Bramantip              [ - P + M - M + S = ? ]

16.  Camenes               - P + M - M - S =  - S - P

17.  Dimaris                  + P + M - M + S = + S + P

18.  Fesapo                  [ - P - M - M + S = ? ]

19.  Fresison                - P - M + M + S =  + S - P

 

Sommers’ rule for determining validity is that:

 

“An inference is valid if and only if (i) the sum of the premises equals the conclusion, and (ii) the number of particular conclusions equals the number of particular premises.” (quoted in G. Englebretsen, Something to Reckon With: The Logic of Terms, p. 114.)

 

If a purely arithmetical procedure is used with the above stated rules, four of the arguments of traditional logic will not work.  Also, as we shall see, Sommers needs more than the above rules in order for his procedure to work.  First of all, (9) Darapti, (12) Felapton, and (18) Fesapo have unpolarized middles under the above symbolism, and the valid conclusions for these arguments cannot be derived.  Respectively, the conclusions should have been,

 

(9)      + S + P

(12)    + S - P 

(18)    + S - P

 

But the unpolarized middles prevent summation (or cancellation of the middles), thus violating rule (i), even though these are valid arguments in traditional logic.  The suggestion has been made that for 9, 12, and 18, we add a tautological premiss, for instance, m + m for the following:

 

1. -m + p

2. -m + s

3.  + m + m

4. + s + p

 

This conclusion “+ s + p” is the valid conclusion of Darapti, but consider the following:

 

1.  -m + -p

2.  -m + -s

3.   + m + m

4.  -s + - p

 

Here we have two universal premises, now with the addition of the tautological premiss.  We succeed in cancelling all the “m’s”, but the conclusion is invalid because there were two negative premises in the original argument.  Apparently, the addition of the extra premiss can only work for positive rather than negative premises, so a rule about two negative premises has to be adopted in addition to Sommers’ first rule.

 

Why can’t we add a new premiss to the following to derive a conclusion?

 

1.  + m + p

2.  + m + s

3.  -m  -m

4.  +s + p

 

Obviously, while the additional premiss polarizes the “m’s, the new premiss is actually a contradiction, i.e., “No m is m.”  Moreover, in accordance with Sommers’ rule 2, the conclusion is invalid because we have two particular premises in the original argument.  Thus, in addition to a purely arithmetical procedure, rules have to be adopted that, first, a new tautological premiss can only be positive and particular in form, and secondly, that two particular premises do not result in a valid conclusion.  So much for the lack of any rules for inference.

 

There is another problem with Sommers’ method.  The argument (15) Bramantip has polarized middles under Sommers’ rule 1, but this does not really help, nor do the other rules of preserving validity help.  Arithmetically, we derive the conclusion as:

 

(15)      + S - P

 

But the correct conclusion is

 

(15)*    + S + P.

 

The claim that we can add a tautological premiss does not work in the case of Bramantip, since the problem is not with the middle terms but with a negative term operator on one of the extreme terms.  Consider:

 

1.  - p  +  m

2.  - m + s

3.   + m + m

4.  + s  - p

 

The “m’s” do not cancel out, and the valid conclusion should have been + s + p.  What rule should be adopted for this difficulty?  Here is the solution of Bramantip using our own symbolism:

 

1.  -p + m

2.         -m + s

3.  -p + 0  + s

 

Or -p + s, which is “All p is s.”  We can then use conversio per accidens to derive s & p, or “Some s is p” since we want to state conclusions in subject-predicate form.  Can this be done with Sommers’ notation?  Consider again Bramantip using Sommer’s arithmetical procedure:

 

1. -p + m

2. -m + s

3.  -p + s

 

Here we should note that the position of the terms in the conclusion is just as important as any rule about “the sum of the premises.”  If “s” and “-p” were reversed, i.e., (+ s - p), we would have “Some s is not p” rather than the correct “All p is s.”  In a Boolean-type symbolism, the order of the signs does not matter.  For instance, the form -p + s is equivalent to s + -p.   However, in Sommers’ notation, the position of the terms is crucial for the conclusions in Ferio, Festino, Baroko, Felapton, Bokardo, Ferison, Bramantip, Fesapo, and Fresison.  If we were to reverse the + S - P conclusions of these argument to -P + S, we would not have the same conclusions.  (This problem was noticed by John Venn in another connection, cf. Symbolic Logic, p. 103.)

 

So, in addition to the rules already mentioned, we would need another rule stating that the S P form should be maintained.  However, the only way this could be done for Bramantip is by conversio per accidens, i.e., reducing -p + s to sp, and by this preserving the S P form of the conclusion.

 

Thus, we have four perfectly good syllogisms that do not work in Sommers’ calculus of terms, and for two different reasons, one the lack of polarization in 9, 12, and 18, and the other simply because the “-” before the predicate cannot be changed to a “+” and brought over to the conclusion in 15.  Therefore, in order for them to work, Sommers needs to add more rules to his arithmetical procedure, and his claim that the rules of inference are not operative in his calculus of terms was certainly premature.

 

The differences between Sommers’ symbolism and Boolean-type symbolism can be seen in the following comparative chart:

 

 

English

Sommers

Neo-Boolean

1.  All S is P

- (S) + (P)

-(S) + (P)

2.  No S is P

- (S) – (P)

-(S) + -(P)

3.  Some S is P

+ (S) + (P)

(S) & (P)

4.  Some S is not P

+ (S) – (P)

(S) & -(P)

 

 

In neo-Boolean symbolism, the “+” sign represents logical addition, that is, alternation.  It is merely replacing the vel or “v” sign in many representations of logic found in textbooks.  It is the same “+” sign found in books on digital or Boolean logic and means the very same thing, that is “or.”  In the “Neo-Boolean” column, the “+” or “&” standing in between the S and P (i.e., subject and predicate) represent logical connectives (for disjunction and conjunction), not term operators.  Notice that in 2 and 4, we have the “+” and “&” signs coming in the middle but not standing for term negation.  That is what the “-” is next to the parentheses, a term operator.  Sommers has his “+” and “-” doing double duty, standing for “is” in 1 and 3, and for “un” in 2 and 4.  Moreover, they are not reversible, as we have seen.

 

Sommers gives the following schedule in The New Syllogistic, p. 23:

 

A.  All S (is) P              - S + P

E.  All S (is) un-P                     - S - P

I.  Some S (is) P                       + S + P

O.  Some S (is) un-P    + S - P

 

The arithmetic symbols above appear to represent both copulas and terms.  Moreover, the failure to distinguish the conjunctive forms with a different symbol—such as “&”—will make it difficult for Sommers to use a logic grid involving particular statements (see more under section 12).

 

5.  The 19 Valid Syllogisms

 

It will help the reader to memorize the rules of logical calculus by working through the 19 valid syllogisms of traditional logic, using both the fractional and arithmetical symbolism.  We will state all conclusions of fractional calculus in terms of subject and predicate form, i.e., with S coming before P.

 

1.  Barbara:

 

M É P

S É M

\S É P

 

 

1.  ~M + P

2.  +M       + ~S

3.   0 +   P + ~S 

 

 

2.  Celarent

 

M É ~P

S É M

\S É ~P

 

 

1.  ~M + ~P

2.  +M         + ~S

3.   0 +   ~P + ~S

 

 

3.  Darii

 

M É P

S · M

\S · P

 

 

1.  ~M + P

2.  +M & S    

3.   0 + S & P

 

 

Note that rule 2 is satisified.

 

4.  Ferio

 

M É ~P

S · M

\S · ~P

 

 

1.  ~M + ~P

2.  +M & S    

3.   0 + S & ~P

 

 

Rule 2 satisfied.

 

 

5.  Cesare

 

P É ~M

S É M

\S É ~P

 

 

1.  ~P + ~M

2.          +M + ~S

3.  ~P +  0   + ~S

 

 

In terms of rule 1, the middle terms are polarized.

 

 

6.  Camestres

 

P É M

S É ~M

\S É ~P

 

 

1.  ~P + M

2.        ~M + ~S

3.  ~P + 0  + ~S

 

 

7.  Festino

 

P É ~M

S · M

\S · ~P

 

 

1.  ~P +    ~M

2.        S  & M

3.  ~P & S + 0

 

It is assumed in all these arguments that M standing by itself is a positive M, so that a + sign is implicit.  Up to now, I’ve been including the plus sign before a positive variable just to show how the middles are polarized and reduce to zero.  Nevertheless, when we have an expression such as in premiss 2, it is best to leave the plus sign as implicit, since to express it between the conjunction of S and M might lead to mistaking it for the + of alternation.

 

8.  Baroko

 

P É M

S · ~M

\S · ~P

 

 

1.  ~P        + M

2.       S  & ~M

3.  ~P& S + 0

 

9.  Darapti

 

M É P

M É S

\S · P

 

 

1.  ~M + P

2.  +M & S  

3.  0 +  S & P

 

In keeping with rule 4, we were able to polarize the middles by means of conversion per accidens.

 

 

10.  Disamis

 

M · P

M É S

\S · P

 

 

1.    M & P

2.  ~M + S

3.  0 +  P & S

 

11.  Datisi

 

M É P

M · S

\S · P

 

 

1.  ~M + P

2.  +M  & S  

3.   0 +  S & P

 

12.  Felapton

 

M É ~P

M É S

\S · ~P

 

 

1.  ~M + ~P

2.  +M & S   

3.   0 + S & ~P

 

Note that Rule 4 is satisfied.

 

 

13.  Bokardo

 

M · ~P

M É S

\S · ~P

 

 

1.    M  & ~P

2.  ~M   + S

3.   0 + ~P & S

 

14.  Ferison

 

M É ~P

M · S

\S · ~P

 

 

1.  ~M + ~P

2.  +M & S    

3.   0 + S & ~P

 

15.  Bramantip

 

P É M

M É S

\S · P

 

 

1.  ~P + M

2.        ~M + S

3.  ~P + 0  + S

 

The conclusion ~P v S is P É S, or  SP, by limitation (since we want the subject of our conclusion to be S).

 

16.  Camenes

 

P É M

M É ~S

\S É ~P

 

 

1.  ~P + M

2.        ~M + ~S

3.  ~P + 0  + ~S

 

17. Dimaris

 

P · M

M É S

\S · P

 

 

1.     P & M

2.   S + ~M

3.   S  &  P

 

18.  Fesapo

 

P É ~M

M É S

\S · ~P

 

 

1.  ~P + ~M

2.        S & M  

3.  ~P    & S

 

See, rule 4.  Also, the plus sign on the M of the second premiss is implicit rather than explicit in order to avoid confusing conjunction with alternation.

 

19.  Fresison

 

P É ~M

M · S

\S · ~P

 

 

1.  ~P +   ~M

2.        S & M

3.  ~P & S

 

 

End of Part 1