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 postBoolean 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 notA 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 subjectpredicate 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
Booleanbased logic one can find the following simplification rules:
0 v P = P
1 v P = 1
(1) ·
P = P
(0) · P = 0.
In our “neoBoolean” 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 neoBoolean symbolism is virtually the same as the traditional
symbolisms. What’s more, it will allow
a straightforward cancellation procedure for deriving conclusions:
NeoBoolean 
Set Theory 
Predicate Calculus 
Arithmetical
PCalculus 




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 PCalculus”
refers to the standard predicate calculus presented in arithmetical
format. It should also be remembered
that the neoBoolean 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
Booleantype 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 subjectpredicate 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
Booleantype 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
Booleantype symbolism can be seen in the following comparative chart:
English 
Sommers 
NeoBoolean 
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 neoBoolean 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
“NeoBoolean” 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) unP  S  P
I. Some S
(is) P + S + P
O. Some S
(is) unP + 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