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Can the Laws of Form Represent
Syllogisms?
John Mingers
Kent Business School
Working Paper No. 293
January 2014
1
Can the Laws of Form Represent
Syllogisms?
John Mingers
Kent Business School, University of Kent
[email protected]
Abstract
The Laws of Form was created to represent propositional logic and Boolean algebra.
But Spencer Brown (and later commentators) also claimed that it could represent
Aristotelian syllogistic logic although, as he showed in his book, at least one invalid
syllogism appeared to be valid. This paper explores the extent to which the Laws of
Form can correctly deal with all syllogisms. There are in fact 256 possible syllogisms
and only fifteen of them are uncontroversially valid (a further nine are valid if certain
existence assumptions are made). Using truth tables implemented in a spread sheet, all
256 syllogisms were evaluated and it was discovered that, in fact, 83 invalid
syllogisms appear to be valid when simply represented in Laws of Form notation.
(The issue about the application of Spencer Brown’s Interpretative Theorem 2 will be
explored in the paper). This is clearly a significant number. Further investigation show
that the problem might be caused by the way that “some/some not” propositions are
conventionally represented and a variety of alternative are explored, some related to
free logic. One particular interpretation reduces the number of wrongly categorised
syllogisms to only seventeen and, surprisingly, fifteen of the 17 are mirror images of
the fifteen valid ones.
Key Words: Laws of Form, syllogisms, free logic
1
1. Introduction
The Laws of Form (LoF) were developed by George Spencer-Brown (1972) as a way
of representing graphically the laws of logic, and in particular the propositional
calculus or Boolean algebra. They provide an extremely elegant and parsimonious
form of representation for both the algebra and the arithmetic underlying it. In fact,
LoF uses only a single explicit symbol, the cross or marked state, together with a
blank space as an unmarked state. It has been called the calculus of indications or
distinctions (Varela, 1979;Kauffman, 1978). It has a precursor in Charles Sanders
Peirce’s existential or entitative graphs (Engstrom, 2001;Kauffman, 2001).
The basic form of logic is called propositional logic because it deals with propositions,
expressed in sentences, which may be either true or false. A logical calculus, or
Boolean algebra, concerns the way that propositions can be combined together using
connectives such as “and”, “or” or “implies”. This paper will explore the extent to
which LoF can, or cannot, represent a more developed form of logic – the syllogism.
This is a step on the way to more modern forms of logic such as first-order predicate
logic. Put briefly, a syllogism consists of three propositions: the first two are the
premises and the third is the conclusion which may or may not follow logically from
the premises. Each proposition must be of the form “all s are/ are not m” (universal)
or “some s are/ are not m” (particular). One of the most basic examples is: “All s are
m; all m are p; therefore all s are p”. This syllogism is valid as the conclusion does
follow from the premises. Spencer-Brown himself put forward the idea that LoF could
represent syllogisms and showed that it dealt with the one above correctly. However,
it represented the following one incorrectly: “Some s are m; some m are p; therefore
some s are p”, giving a true result when it is in fact false. Spencer-Brown discusses
this example extensively on pages 124-134. We will try and discover the reasons why
LoF does not work for all syllogisms (ignoring for the moment interpretive theorem 2)
and if the problems can be rectified.
We should note at an early stage that one of the primary difficulties is with what is
known as “existential import”. In the main, modern logicians and mathematicians take
particular propositions (some s are p) to mean that that there actually exists at least
one s that is p; but universal propositions (all s are p) do not make this assumption, i.e.,
there may be no s’s and the proposition can still be true.
For reasons of space, we will assume that the reader has a basic knowledge of LoF. If
this is not the case, then there is no real substitute for reading Spencer-Brown’s book
which is in many ways like no other. There is a secondary literature, for example,
(Baecker, 1999;Banaschewski, 1977;Engstrom, 2001;Orchard, 1975;Mingers, 2006,
chapter 4).
LoF begins with two initials or axioms, and from these deduces nine other main
2
consequences. These are summarised in Appendix 1i. In the first section we will
demonstrate the relationship between LoF and truth tables as these will form an
important part of the research method. In section two we will overview syllogisms
and demonstrate the problems of representing them in LoF. In section three, the
substantive part of the paper, we will consider a variety of ways of trying to overcome
these difficulties.
2. Syllogisms
Although the theory of syllogisms are somewhat dated, having been superseded by
predicate logic, they are nevertheless important, and being able to represent them
correctly in LoF would be extremely significant.
A syllogism consists of three propositions – two premises followed by a conclusion.
Each proposition must be in one of four forms (Copi and Cohen, 2009):
“All s are m”
(A);
“All s are not m”
(E) (the same as “no s are m”);
or particular:
“Some s are m”
(I);
“Some s are not m”
(O)
Where s and m are terms denoting particularly categories of things.
Either universal:
A valid syllogism is a type of argument that is valid because of its form alone – i.e.,
because of the particular relationships between the terms and the propositions in it,
regardless of the content of the propositions or the truth of the premises. Thus the
form of a syllogism is that the conjunction of the premises materially implies the
conclusion. Note that the premises being false does not imply that the conclusion must
be false – it could be true for other reasons.
The letters after the examples allow us to classify the syllogisms. The valid syllogism
mentioned in the introduction in which each proposition is universal is the most basic
valid syllogism and can be classified as AAA. All the valid syllogisms have been
given names which reflect the types of propositions and this one is called Barbara.
The conclusion consists of a subject (s) and a predicate (p). m is called the middle
term and must occur in both premises. Formally, syllogisms are written with the major
premise first which links m to p, and the minor premise, which links s to m, second.
Thus Barbara should be written as: “All m are p; all s are m; thus all s are p”.
Given that each proposition can be of four kinds, and there are three of them in any
syllogism, that means there are 64 different possible basic syllogisms. Each of these is
called the “mood” of a syllogism (Smith, 2012). However, there are also four different
ways in which the terms may appear within the premises. These are called “figures”.
Thus:
3
Figure 1
m - p;
s-m
s-p
Figure 2
p-m
s-m
s–p
Figure 3
m-p
m-s
s–p
Figure 4
p-m
m-s
s-p
In each case the middle term appears in different places in respect of subject and
predicate.
Thus in total there are 256 possible syllogisms taking into account both mood and
figure. Barbara could properly be written as AAA-1.
Of these 256, there are 15 which are uncontroversially valid. These are shown in
Table 1 with their accepted names. In this Table, I have distinguished the pattern of
the premises in the rows – both universal, mixed universal and particular, and both
particular; and the conclusion in the columns, either universal or particular. This gives
six categories – NW, NE, WW, EE, SW, SE. I have then allocated all the valid
syllogisms into the appropriate categories. We can see that three categories have no
valid syllogisms – those with both premises particular, whatever their conclusion (SW
and SE), and the ones with mixed premises but universal conclusions (WW).
Conclusion
Premises
Universal (A, E)
Particular (I, O)
Both Universal
NW: Total: 32
NE: Total: 32
(A, E)
Barbara
Barbari
Darapti
Bamalip
Celarent
Celaront
Fesapo
Cesaro
Camestres
Camestros
Calemos
Calemes
(These are the nine contentious ones depending on
Cesare
existence assumptions)
(none incorrect)
(none incorrect)
Mixed
WW: Total: 64
EE: Total: 64
universal (A,
(39 incorrect)
Darii
Datisi
E) and
Disamis
Dimatis
particular (I,
Ferio
Festino
O)
Baroco
Ferison
Felapton
Fresison
Bocardo
(none incorrect)
Both
SW: Total: 32
SE: Total: 32
particular (I,
(25 incorrect)
(19 incorrect)
O)
Table 1 A Classification of the 256 possible syllogisms showing how many there are in
each combination and naming the valid ones. This also shows, in brackets, the number
found to be incorrectly valid in LoF (see Section 4.3)
4
In the NW corner we find those valid syllogisms with universal premises and
conclusions. There are five. In the EE box we find the remaining ten valid syllogisms,
with mixed premises and a particular conclusion. In the NE we find a further nine
syllogisms whose validity is contingent, depending on whether or not we assume that
examples of the classes do actually exist. This issue of existence is extremely
important and we need to discuss it in more depth now.
2.1 Universals, particulars and existence.
The basic question is, does a proposition such as “all s are m” imply that there are
actually any s’s? One might think yes, but consider the propositions: “all unicorns are
white” and “all shoplifters will be prosecuted”. As far as we know, there are no
unicorns, and there may be no shoplifters, and yet these propositions do seem to be
valid. This has actually been a long-standing debate within philosophy (see, for
example, (Copi and Cohen, 2009, p. 229) for a good discussion). In traditional
Aristotelian logic it was assumed that both particular and universal propositions did in
fact assume existence, i.e., that the class was non-empty. However, this approach
brings with it problems of discussing empty, or potentially empty, classes and so,
since the time of Boole (1854), it has been assumed that particular propositions do
have existential import but universal ones do not. In other words, the proposition
“some s are m” implies that s do indeed exist, while “all s are m” does not since it is
still true even if there are no s.1
In relation to the nine contentious syllogisms, this creates a problem. The nine all
have two universal premises but a particular conclusion. Thus the premises, even if
true, do not guarantee existence but the particular conclusion assumes it. In short,
these syllogisms are valid only if we can assume that the objects do actually exist.
Thus for most modern logicians these syllogisms are considered to be false leaving
only fifteen properly valid ones.
3. Interpreting the Laws of Form for syllogisms
In order to express a syllogism in LoF we need to interpret each of the four types of
proposition within the LoF notation. A list of the two initials and nine consequences
of LoF is shown in Appendix 1. By way of recap, there is only one explicit symbol in
LoF, which is the cross ┐, which represents the marked state. In this paper we
represent the cross by a pair of brackets (). The other, implicit, symbol is the blank
space which represents the unmarked state. In order to interpret the notation for logic,
we need to decide whether the cross is to represent true or false. It can be done either
way (see (Engstrom, 2001)) but the most commonly chosen (Spencer-Brown,
1
There are some modern forms of logic that do not make this assumption, called “free logics”
(Lambert, 1963;Lehman, 1994)
5
1972;Meguire, 2011) is to make the empty cross represent truth, which we shall do
here.
Given this, we can see how all the basic connectives of propositional logic (and
Boolean algebra) are represented in the primary algebra (pa) of LoF in Table 2. To
highlight the most common (where a and b here “stand for the possible truth values of
the various simple sentences in a complex sentence” (Spencer-Brown, 1972, p. 114)):




Negation
Disjunction
Conjunction
Implication
conditional)
NOT a
a OR b
a AND b
a implies b
(a)
ab
((a)(b))
(a)b (This is material implication or the
I would like to be clear at this point, why Spencer-Brown argues that this is a valuable
thing to do. In his own words:
“The power granted to us through this simplicity, although great, is itself small
compared with the power available through the connexion of the primary algebra
with its arithmetic. For this facility enables us to dispense with the whole set of
lengthy and tedious calculations, and also with their no less troublesome alternatives,
such as the exhaustive ... procedures of truth tabulation, and the graphical … methods
of Venn diagrams. … With such a huge gain in the formal clarity of expressions, the
invalidity of a false argument is similarly open to immediate confirmation” (p.
115-117)
In other words, the benefit is in being able to express logical arguments and
syllogisms in the primary algebra, and then calculate the results using the arithmetic,
quite easily.
3.1 The syllogism
We saw in the previous section that the syllogism as a whole is in the form of “a and b
implies c” where a, b and c are universal or particular propositions. We take, therefore,
a variable such as “a” as representing the proposition “x is a member of class a” or,
more specifically, the truth valus of that proposition. A syllogism can then be
expressed in the primary algebra as a conjunction of two propositions combined with
an implication to the conclusion:
(((a) (b))) c
By consequence 1 (C1 Appendix 1), the double crosses can be removed so we get:
(a) (b) c
6
In other words, whatever the particular form of the syllogism, it can always be written
simply as the first premise crossed, the second premise crossed, followed by the
conclusion uncrossed. We now need to consider how to represent the four types of
propositions within the pa. There is a standard way which we will describe in the next
section (Meguire, 2011;Spencer-Brown, 1972;Engstrom, 2001).2
3.2 Universal propositions
Let us consider the individual propositions, firstly the universals. The first problem is
that propositional logic applies to propositions as a whole, in terms of their truth or
falsity, and does not distinguish between internal parts of them. However, with a
syllogistic premise, it is actually the parts of the proposition (the s and m) that are of
concern. To deal with this we need to split the premise in two.
As we saw in the previous section, “all s are m” should be interpreted as conditional
or hypothetical: “if there are s’s, then they are all m’s”. Another way of expressing it,
which relates to both set theory and predicate logic, is: “for all objects x, if x is an s
then x is an m”. From this we can conclude that if we allow a letter “a” to stand for
the proposition “a particular entity x is a member of the set a”, (Engstrom, 2001, p. 35)
or equivalently “has the property a”, or simply “is a”, then “all a are b” can be
represented by the relation of material implication:
“all a are b” ~ if “x is an a” then “x is a b” ~ a implies b
(a)b
Similarly,
“all a are not b” (“no a are b”) ~ if “x is an a” then “x is not a b” ~ a implies
not b (a)(b)
Let us now combine these two to represent the Barbara syllogism in full and then
simplify it using the rules in Appendix 1:
=
=
=
=
=
((((b)c) ((a)b))) (a)c
((b)c) ((a)b) (a)c
((b)) ((a)b) (a)c
((b)) (b) (a)c
b (b) (a)c
() (a)c
= ()
(C1)
(C2)
(C2
(C1)
(C3)
We can see that we have represented Barbara in the pa, and then simplified it, using
the consequences of the pa, to show that it comes out as the marked state or, on our
interpretation, true.
2
Note that traditional syllogistic logic always puts the major premise first as was stated above. Spencer
Brown follows Russell in putting the minor premise first.
7
Let us consider one other example: “all c are b; no a are b; therefore no a are c”
(Camestres)
((c)b) ((a)(b)) (a)(c)
= (b) ((b)) (a)(c)
= () (a)(c)
= ()
Again, this is shown to be true.
(C2 twice)
(C1, C3)
3.3 Particular propositions
How should we represent particular propositions? This, I argue later, is more
contentious but to begin with I will introduce the standard interpretation. In logic,
saying “some a are b” is interpreted as saying “not (no a are b)” – i.e., it is the
negation of the universal proposition that no a are b. This is taken to mean that there is
at least one a that is a b, and is taken to include the universal case that all a are b. It
also has existential import – i.e., it presupposes that there actually is at least one a.
Given this interpretation, then it can be represented in the pa as
“Some a are b” ~ “not (no a are b)”~
is b”
~ a and b
((a)(b))
There exists an “x that is a” and an “x that
This can also be seen to be simply the negation of the universal proposition described
above that no a are b - (a)(b).
We can see from Table 2 that this is in fact equivalent to the conjunction a and b. In
other words, this proposition is only true in the case where some x is a and some x is b.
It is not true where x is not a, i.e., where there are no a’s.
Similarly, “some a are not b” is the negation of the universal “all a are b” and means
that at least one, and possibly all, a’s are not b.
“Some a are not b” ~ “not (all a are b)”~
that is not b” ~ a and not b
((a)((b)))
There exists an “x that is a” and an “x
~ ((a)b)
This is true only in the case where some x is a but is not b.
Here is a valid syllogism including particular propositions: “all b are c; some a are b;
therefore some a are c” (Darii)
=
=
=
((b)c) (((a)(b))) ((a)(c))
((b)c) (a)(b) ((a)(c))
(c) (a)(b) ((c))
() (a)(b)
= ()
(C1)
(C2, twice)
(C3)
Finally, let us look at an invalid syllogism, one of the nine contentious ones.
8
“all b are c; all b are a; therefore some a are c” (Darapti). This appears to be valid –
there are b’s that are both c’s and a’s and so the conclusion should follow, but because
of the lack of existential import from the premises it does not follow unless we add in
another premise to the effect that there are, in fact, some b’s.
In the pa we get:
((b)c) ((b)a) ((a)(c))
which cannot be reduced and thus does not show as
valid.
If we add in the existence of b (crossed as part of the conjunction) we get3:
((b)c) ((b)a) (b) ((a)(c))
= (c) (a) (b) ((a)(c))
(C2 twice)
= () (b) = ()
(C3)
Which is therefore valid.
Summary
In summary, it is possible to represent propositions and syllogisms in the pa, and, in
fact, all the fifteen main valid syllogisms come out as true. The nine contentious ones
come out as false, primarily because the universal propositions do not imply existence.
But, if an extra proposition asserting existence is added, then they become valid also.
3
It is not entirely clear that asserting the variable “b” does, of itself, imply the existence of a member
of the class b although our interpretation of b is “x is a member of set b” (Engstrom, 2001). A similar
approach is used in the Wikipedia entry on syllogisms (Wikipedia: The Free Encyclopedia, 2013).
9
Syllogism
Boolean
Primary
Primary
Boolean
operator
Algebra
Algebra
operator
True
()
T
F
T
F
T
F
T
F
T
F
Disjunction
T
F
Alternation
T
F
F
T
T
F
a OR b
ab
a(b)
False
(ab)
a NOR b
(a(b))
Relative
SOME b
is NOT a
ALL b is
b IMPLIES a
a
Conditional
T
F
complement
Material
F
T
b only
implication
T
F
T
F
ALL a is
a IMPLIES b
b
Conditional
F
Material
implication
NO a is b
a NAND b
(a)b
(a)(b)
NO b is a
((a)b)
Relative
SOME a
T
complement
is NOT b
T
F
a only
T
F
F
T
T
F
T
F
SOME b
T
F
is a
((a)(b))
a AND b
SOME a
Conjunction
is b
a IMPLIES b
(((a)b)(a(b)))
T
F
((a)b)(a(b))
Symmetric
and
((a)(b))(ab)
F
T
(((a)(b))(ab))
difference
b IMPLIES a
F
T
Exclusive
Material
T
F
OR
equivalence
a different to
Biconditional
b
a
b
a
b
Syllogism
T
F
T
F
F
T
F
T
T
F
F
T
T
F
F
T
(a)
NOT a
(b)
NOT b
Table 2 The sixteen logical connectives of Boolean algebra and their LoF interpretations
where a and b are propositions or simple sentences
10
4. Problems with the LoF representation
4.1 The problem of existential syllogisms and Spencer Brown’s solution
We have seen that the valid syllogisms can be represented in the pa of LoF, but there
is a problem that is highlighted by Spencer-Brown. If we represent the following
syllogism, which is invalid, it too comes out as valid: “some b are c; some a are b;
therefore some a are c”.
=
=
(((b)(c))) (((a)(b))) ((a)(c))
(b)(c) (a)(b) ((a)(c))
(b)(c) (a)(b) ()
= ()
(C1 twice)
(C2 twice, C3)
The first question is why has this come out as valid? Spencer-Brown discusses this
problem in terms of the existence assumptions in some detail (pp. 126-135). He
begins by pointing out that we could, if we wished, interpret universal propositions
existentially by saying that if a’s existed they would be b’s and it was just an empirical
matter as to whether or not one actually did exist. Equally, we could interpret
particular propositions non-existentially, so that saying “some a are not b” allows the
possibility without actually requiring the existence of one to prove it. On this
interpretation, the invalid syllogism could be seen to be valid in that it allows that a’s
can be b’s and b’s can be c’s and that therefore possibly an a could be a c without
actually requiring that there is one in existence.
From this he suggests that “We have observed that as long as inferences or equations
in class logic are universally interpreted, the primary algebra can be freely used to
determine them. … it is the denials of such statements, when we wish to interpret them
existentially, that present the difficulty” (p. 130). This still leaves the problem of what
to do with the problem syllogism since he does accept that it, and others like it, are in
fact invalid. His answer is to go back to the basic Barbara syllogism which is all
universals. Now it is possible to transform this syllogism in several ways and still
maintain its validity (Kauffman, 2013b). In particular, one can carry out three
operations that maintain the validity of the LoF expression:



Swap or transpose the conclusion with one of the premises, negating both.
This effectively changes both conclusion and premise from a universal to a
particular, or vice versa.
Permute the variables within a premise which changes the figure of the
syllogism.
Substitute all the occurrences of a variable with its negation which changes the
mood but not the figure.
Spencer Brown shows that if you transpose the conclusion (of Barbara) with each of
its premises in turn you derive Bocardo and Baroco which are therefore alternative
11
forms of the basic universal argument even though both have particular components.
Spencer-Brown then states, with no further proof or demonstration:
“Interpretative theorem 2: An existential inference is valid only in as far as its
algebraic structure can be seen as a universal inference” (p. 132)
What this means is that syllogisms that contain existential (i.e., particular as opposed
to universal) propositions are only valid if they can be derived from Barbara.
He continues: “This single rule takes care of all the separate rules for syllogisms,
their parts and their extensions. It even includes the provision that there shall be no
more than one particular premiss {sic}, for with more than one, no representation as a
universal argument is possible.” (p. 133)
This latter provision would rule out the problem existential syllogism above as that
has two particular premises and so, when transposed, is always left with at least one
particular premise rather than two universal ones.
And he concludes: “In this prototype {Barbara – JM}, not only can we transpose each
complex, but we can also independently cross each literal variable, finding by a
combination of these means, a set of 24 distinguishable valid arguments. Formally
there is no difference between them. If we distinguish any we should distinguish all. In
fact not all twenty four are distinguished in logic, which arrives somewhat arbitrarily
at the number fifteen” (p. 133)
Essentially, this is his way out of the invalidity problem. He recognises that some
invalid syllogisms, when transcribed into the pa, appear to be valid, i.e, come out as
the marked state and therefore “true” when simplified, but has created a rule such that
they are only in fact to be considered as valid if they can also be derived from the
basic Barbara syllogism as described above.
4.2 Evaluation of Spencer Brown’s solution
To what extent is Spencer Brown’s solution, in particular Interpretive Theorem 2
(IT2), a satisfactory resolution of the problem that certain invalid syllogisms, when
translated into the pa, turn out to be valid?
In brief, it is not a satisfactory solution for several reasons:
First, it does not resolve the fundamental problem, it side-lines it. This is similar to
Bertrand Russell’s reaction when faced with the fact that self-referential propositions
generated paradoxes within his logical system. At first he thought that it would be
easy to solve but when he failed, he created the theory of logical types to ensure that
such constructions could not occur within his system. This does not deal with the
problem since they still occur in many other types of systems (Mingers, 1997). In a
12
similar way, Spencer-Brown is not solving or resolving the problem with the pa
interpreted for syllogisms, he is adding in a separate restriction to correctly classify
the results. This surely goes against his own motivation for simplicity and clarity, as
spelled out at the beginning of Appendix 2 and quoted above.
Second, this means that the LoF cannot actually fulfil its function of allowing us to
“calculate” logical consequences correctly. We would have to apply the standard
procedure and then, if the result were “true”, have to apply another, quite complex,
procedure to check if the expression is derivable from Barbara or not. This seems to
go against much of the discussion on Appendix 2 where Spencer-Brown argues that
the LoF is very helpful in allowing us to draw conclusions from complex sorites. “All
forms of primitive implication become redundant since both they and their derivations
are easily constructed from, or tested by reduction to, a single cross. … With such a
huge gain in the formal clarity of expressions the invalidity of a false argument is
similarly open to immediate confirmation.” (p. 117)
Third, his discussion of the invalid syllogism centres around the problem of existence
but in fact existence is not at issue since all the propositions in this syllogism are
particular ones and therefore assume existence. Rather, the problem is with the
quantifier “some”; there are some b’s that are a’s, and some b’s that are c’s, but they
may not be the same b’s. In general, from two particular propositions we cannot infer
anything because the propositions do not specify which specific members of the class
are involved. This is different from a universal proposition where all the members of
the class are included. In logic, this problem is called the “fallacy of the undistributed
middle” (Copi and Cohen, 2009, p. 246). It is necessary that the middle term be
distributed in (apply to all members of) at least one of the classes in the premises.
With particular propositions this is not the case.
Finally, we need to consider if IT2 actually does what it claims – separate valid from
invalid syllogisms - since Spencer Brown does not demonstrate or prove this (at least
not in the published book). It is indeed possible to generate 24 syllogisms from
Barbara using transposition (which generates Bocardo and Baroco) and then negation.
Each term in the syllogism can be negated or not in combination. This gives 8
possibilities for each of the three transpositions – Barbara, Baroco and Bocardo –
giving 24 combinations in all. However, are these in fact the same 24 valid syllogisms
recognised in logic? I have explored all 244 and the results are shown in Table 3.
4
With the help of a document provided by Louis Kauffman (2013a, 2014)
13
Kauffamn
a
b
c
I Barbara
II Bocardo
III Baroco
.1
a
b
c
Barbara
Bocardo
Baroco
.2
not a
b
c
Barbara
Bocardo
Baroco
.3
a
not b
c
Barbara
Bocardo
Baroco
.4
not a
not b
c
Barbara
Bocardo
Baroco
.5
a
b
not c
Celarent
Disamis
Festino
Cesare
Dimatis
Reference
.6
not a
b
not c
Calemes
Bocardo
Fresison
.7
a
not b
not c
Camestres
Ferison
Darii
Calemes
Fresison
Datisi
Festino
Ferio
.8
not a
not b
not c
Barbara
Baroco
Bocardo
Table 3 The 24 syllogisms obtainable from transforms of Barbara
The three main columns of Table 3 show Barbara, Bocardo and Baroco, where the
latter two have been obtained by swapping and negating a conclusion with a premise.
The rows are then obtained by negating different combinations of the terms a, b, c as
shown in the columns headed a, b, c. The Table then shows the result of each possible
combination. In each column, the first four rows simply repeat the basic syllogism
form with different variants of a and b. It is only when the conclusion (c) is negated
that we generate different syllogisms. Where more than one is generated in a box they
are obtained by permuting the terms within one of the propositions.
The main conclusions I come to from these results are:
i.
ii.
iii.
The transformations produced 24 syllogisms but they were not all distinct - as
can be seen, many repeat in several places.
They did in fact generate the 15 valid syllogisms although in some cases a
further permuting of the terms in a premise was needed to get them (the boxes
with more than one entry).
They did not generate any of the nine contentious syllogisms, nor would this
be possible since these all come out as invalid when put into LoF notation and
so could not be derived from Barbara.
This does cast doubt on IT2. Certainly Spencer Brown is wrong to claim that 24 distinct valid
syllogisms can be generated as only fifteen can be, and it is not correct for him to say that the
fifteen are arrived at “arbitrarily” as he does in the quote above. The other side of the claim of
IT2 is that it correctly classifies all the invalid syllogisms as invalid. No proof or demonstration
of this is offered and I have not pursued this myself.
14
Given these problems with IT2, the question now is, how many other invalid
syllogisms appear to be valid when represented in the LoF pa, and can we do anything
to avoid that?
In order to answer the first question, we would need to formulate all of the 256
syllogisms into the pa and then manually test whether they come out as true or false.
Apart from being very time-consuming it would be error-prone in terms of
simplifying the expressions. For these reasons, it was decided to do this testing
automatically in an Excel spread sheet using truth tables, which are isomorphic to the
pa as can be seen from Table 2.
4.3 Truth tables, syllogisms and spread sheets
A truth table for two propositions, a and b, is shown in Table 4
a
b
a OR b
a AND b
a implies b
(some a is b)
(all a is b)
T
T
T
T
T
T
F
T
F
F
F
T
T
F
T
F
F
F
F
T
Table 4 Truth Table with some propositions
This shows the result of a logical operation such as “or”, “and”, “implies” for all the
possible truth values of a and b. We can represent the results of a syllogism as a whole
with a large enough truth table. An example for Barbara is shown in Table 5.
Row
a
b
c
All b are c
All a are b
(P1)
(P2)
P1 and P2
All a are c
Result
(implication)
1
T
T
T
T
T
T
T
T
2
T
T
F
F
T
F
F
T
3
T
F
T
T
F
F
T
T
4
T
F
F
T
F
F
F
T
5
F
T
T
T
T
T
T
T
6
F
T
F
F
T
F
T
T
7
F
F
T
T
T
T
T
T
8
F
F
F
T
T
T
T
T
Table 5 Truth table for a valid syllogism (Barbara)
After the a, b, and c columns, the next one shows the results for “all b are c” (b
implies c); the second for “all a are b”; the third shows the conjunction of the
premises; and the fourth shows the conclusion, “all a are c”. Now, by material
15
implication the syllogism is valid if, each time the conjunction of the premises is true
the conclusion is true also. This occurs in rows 1, 5, 7 and 8 and in each case the
conclusion is indeed true (highlighted in bold). Given how material implication works,
it does not matter if the conclusion is true or false when the premises are false. This
shows that a truth table can show the validity of a syllogism, given the particular ways
that we have represented the four kinds of propositions.
We will now show an example of a syllogism that is not valid (Table 6):
“All b are c; no a are b; therefore no a are c”. From the premises no conclusions about
a and c can be drawn.
a
b
c
All b are c
no a are b
(P1)
(P2)
P1 and P2
No a are c
Result
(implication)
T
T
T
T
F
F
F
T
T
T
F
F
F
F
T
T
T
F
T
T
T
T
F
F
T
F
F
T
T
T
T
T
F
T
T
T
T
T
T
T
F
T
F
F
T
F
T
T
F
F
T
T
T
T
T
T
F
F
F
T
T
T
T
T
Table 6 An invalid syllogism
The premises are true in rows 3, 4, 5, 7 and 8. The conclusion is true in each of these
but not in row 3; thus the syllogism is invalid.
A truth table can be implemented in Excel which contains the basic logical operations
as functions. Although it has “not”, “or” and “and”, it does not have “implies” which
is the basis of the four propositions. However, it is possible to develop this in special
functions (e.g.,a implies b can be represented as not “a or b”) and this enabled us to
create functions for “all”, “some” etc. and then use these to construct the tables.
The first table had two universal premises (AA) and then had each of the four possible
conclusions. It was programmed so that the table itself registered if the result was true,
i.e., the syllogism was apparently valid, or if it was false. This table was then copied
64 times and the premises changed to produce all the possible variations within a
particular figure. This set of tables was then replicated, changing the order of the
terms to produce all 64 tables for the other three figures so that in the end the spread
sheet contained all 256 possible syllogisms, each registering whether it was valid or
not. Although this took a considerable amount of time, at least it meant that the results
were more likely to be correct. Samples of the tables were checked manually. The
other big advantage of this method was that when we tried experimenting with
different logical interpretations of “some” or “all” we only needed to re-program the
functions and all the results were calculated instantly.
16
4.4 Initial results for the standard interpretation
The results generated by the spread sheet for the standard interpretation of syllogisms
were quite surprising. They showed that 83 of the 256 possible syllogisms were
calculated incorrectly, being judged to be valid when they were actually invalid. This
is almost a third of all the possibilities. Clearly there is something significantly wrong
with this interpretation in LoF. We have shown the distribution of these false results in
Table 1.
Considerable analysis was done to see if any patterns emerged in the types of
incorrect results. In terms of the data in Table 1, we can see that no false ones
occurred when both premises were universal. The largest number, 39 occurred when
the premises were mixed and the conclusion universal. These syllogisms are all
invalid because one cannot deduce a universal conclusion from a particular premise.
There were no false ones in the box where the premises are mixed and the conclusion
is particular, which is where many of the valid syllogisms lie. The other false ones are
in the boxes where both premises are particular. There are no valid syllogisms in these
boxes. So, the first conclusion is that the problems only occur when particular
premises and/or conclusions are involved.
We also looked at the distribution across figures but this was very even – 20 each in
figures 1, 2 and 3, and 23 in figure 4.
The next step was to look in more detail at precisely why, technically, the result was
wrong.
Consider the following: “All b are c; some a are b; therefore all a are c”. If the
conclusion were “some a are c” it would in fact be valid (Darii) but it is not valid with
the universal conclusion, yet it appears to be both in the pa and in truth tables.
In the pa this is:
=
=
=
((b)c) (((a)(b))) (a)c
((b)c) (a)(b) (a)c
(c) (a)(b) (a)c
() (a)(b) (a) = ()
(C1)
(C2)
(J1, C3)
Here we can see that the “some” premise begins crossed as it is a negation of a
universal premise. However, it then get crossed again in the conjunction but the
double crossing cancels out (C1) to leave the individual terms (a) and (b) open, i.e.,
not under a cross. They can then easily combine with other terms as they do in the
next row, finally resulting in (). In general, the particular premises become open and
thus combine more easily leading to more “true” expressions.
As a truth table (Table 7):
17
a
b
c
All b are
c (P1)
T
T
T
T
F
F
F
F
T
T
F
F
T
T
F
F
T
F
T
F
T
F
T
F
T
F
T
T
T
F
T
T
Some a
are b
(P2)
T
T
F
F
F
F
F
F
P1 and P2
All a are c Result
T
F
F
F
F
F
F
F
T
F
T
F
T
T
T
T
T
T
T
T
T
T
T
T
Table 7 Invalid syllogism that appears valid
We can see from the truth table that while “all be are c” has several true rows, “some a
are b” has only two because of the existence requirement, so that the conjunction of
the two premises has in fact only one which is true – row 1. This means that any
conclusion that it true in row 1 (where all of a, b and c are true) will appear valid. It
does not matter what its truth values are in any other row.
So, whichever way we look at it, the occurrence of particular premises, in the form we
are interpreting them, makes it easier for expressions in LoF to become true and thus
for invalid syllogisms to be seen as valid. Apart from the technical problems with
particular propositions, I also want to argue that there are semantic problems in
interpreting particular propositions the way we have.
The first is the question of existence. As we saw earlier, it is generally accepted that
particular propositions imply existence. It used to be accepted that universal ones did
as well, but this changed after Boole and Russell. However, this seems to me to
simply be inconsistent. There are many terms, in general, that quantify a set – “all”,
“some”, “many”, “a few”, “50%”, “none”. Why is it that some should be taken to
imply existence and others not? It is argued that we could not talk about “all unicorns”
or “all shop-lifters” if all implied existence, but currently we also cannot talk about
“some unicorns” or “some shop-lifters”. As well as this, as is recognised (Copi and
Cohen, 2009), the current scheme invalidates the traditional “square of oppositions”.
So we might expect that “all a are b” logically implies that “some a are b” (called
subalternation) but in fact it does not because the universal does not imply existence. I
argue that we should be consistent – either all and some both imply existence or they
both do not but it is inconsistent to mix them.
Given the choice, I come down on the side of assuming that neither type of
proposition implies existence. There is in fact a good deal of debate about existence in
the literature which is beyond the scope of this paper. Logics have been developed
which do not imply existence and are known as “free logics” (Lambert, 1963;Lehman,
1994). Ornstein (1999) has written an interesting paper developing a form of
18
Aristotelian syllogism which fits within first order predicate logic and can itself be
seen as a free logic.
The second semantic issue concerns the usage of “some”. As so far defined, “some” is
represented as “not none”; that is as “at least one”. This includes the possibility of
“all”. So “some” is taken to mean at least one and possibly all. However, in common
usage I suggest that this is not the case. If someone says, “some of the committee
members were at the meeting” I argue that they are actually meaning “only some”,
excluding all. Given that they must know how many members were actually present
then, if in fact all were there, they would say all. The fact that they say some seems to
me to exclude all. So, “some”, in this instance anyway, really means at least one but
not all. In fact, one could argue that it actually means more than one since if only one
were present the statement would be “only one of the committee were there”.
A corollary of this is that the proposition “some members were present” is compatible
with, and could be said to imply, that some members were not present (generally
called subcontraries). Or, put another way, “some a are b” and “some a are not b” can
both be true but all and none cannot both be true. In practice the situation is complex
as terms in real conversations may have several possible implications. For example,
the phrase “you must eat some of your vegetables” could be taken to include all, in
contrast to the previous example. This can be seen as an example of what Grice (1975)
calls “implicature”, i.e., that in real conversations terms may have implications that
are logical or rational but go beyond that which may be formally assigned to them.
In summary, I have pointed out that defining the particular propositions as negations
of universal propositions causes technical problems; and that there are issues around
the semantic meaning assumed for “some”.
5. Alternative interpretations of particular propositions
Let us consider if there could be an alternative interpretation of “some” and
“some/not”.
Table 8 shows a basic truth table for two propositions. Let us assume that they
correspond to the truth or falsity of statements “x is an a (e.g., member of the
committee)” and “x is a b (e.g., “present at the meeting)”. Which of the four possible
combinations are compatible with, i.e., not contradicted by, the revised definition of
some discussed above?
19
a
b
Some a are
Some a are not
b (revised)
b (revised)
T
T
T
T
T
F
T
T
F
T
T
F
F
F
F
T
Table 8 Revised interpretation of some/some not
a and b both being true is clearly compatible. Next we have a true and b false – this
corresponds to x being a member of the committee but not at the meeting. As we
argued above, the term “some” implies that some are not, and hence this can be true.
The third row corresponds to x not being a member of the committee but being at the
meeting. If we do not assume that the particular implies existence then this too is
allowable. Finally we have x being neither on the committee nor at the meeting. If we
argue that this too is allowable, then all combinations would be true and it would just
be a tautology, so instead we say that while “some” is compatible with committee
members not at the meeting, and people at the meeting not being committee members,
it is not compatible with neither of these propositions being true.
This results in a particular interpretation for “some” which as can be seen from Table
8 is actually equivalent to “a OR b”, the inclusive OR. What should we make of
“some a is not b”. If “some a is b” is interpreted as “a OR b”, then it perhaps makes
sense to interpret “some a is not b” as “a OR not b” which gives the second column in
Table 8. This interpretation gains support from consideration of Zellweger’s (1997)
“logical garnet”, a sophisticated geometrical representation of the relations between
the 16 basic logical connectives (Kauffman, 2005, 2001). The question then is,
whatever the rights or wrongs of this particular interpretation, what effects does it
have in terms of producing valid or invalid syllogisms?
5.1 Revised results
The revised interpretations were run though all 256 syllogisms with the following
results:
1. All the fifteen valid syllogisms are still valid.
2. The nine contentious syllogisms are all invalid
3. The problem case that Spencer-Brown discusses now comes out as invalid as
it should.
4. The vast majority of the rest of the syllogisms are analysed correctly.
5. But unfortunately there are still seventeen that come out wrong.
So, at one level this is a huge improvement – from 83 to 17 incorrect ones – but it is
still not wholly correct.
20
In this section we will focus on the particular syllogisms that are still wrong in the
new interpretation. The seventeen are split roughly evenly across the figures – four in
figures 1 -3 and five in figure 4. If we look at the syllogisms in a particular figure then
we can see that there is a definite pattern
Table 9 shows the results for figures 1 and 2.
Figure 1
Figure 2
Premise 1
Premise 2
Conclusion
Premise 1
Premise 2
Conclusion
I
E
A
O
A
A
O
E
E
I
E
A
I
O
I
O
I
I
O
O
O
I
O
I
Table 9 Invalid syllogisms in figures 1 and 2 (revised interpretation)
In figure 1, each of the four possible conclusions appears once and there is a clear
pattern in the premises. In figure 2, the second premise includes each of the four
possibilities with the others appearing in pairs. In figure 3 (not shown) it is premise
one that contains each of the four possibilities while in figure 4 each of the four
appears in each of the premises and the conclusion. Whatever is happening here, there
is clearly an underlying structure – it is not simply random.
We next classified these syllogisms in a different way, one that has been used to
classify the valid syllogisms (see for example the Wikipedia entry on syllogisms
(Wikipedia, 2012)). In Table 10, the rows represent the four figures, and the columns
represent (in no particular order) syllogisms with the same mood, i.e., combination of
propositions.
Figure
1
2
OOO
IOI
IOI
OEE
OII
3
4
IEA
IEA
OEE
OII
EOE
IEA
EOE
IEA
OAA
AOA
Table 10 The invalid syllogisms (two are missing from this table – AAO in figure
4 and OOA in figure 4)
At first sight, there does not seem to be a particular structure. However, notice that
there is only one pattern that occurs in all four figures – IEA. Now, if you look at the
similar table of valid syllogisms in Wikipedia (Wikipedia, 2012) you find there is also
only one type represented in all four figures and that is EIO. If we swap E for I and O
for A then our invalid IEA becomes transformed into the valid EOI. Moreover, and
quite startlingly, if we make the same swaps in all the fifteen invalid syllogisms in
Table 10, then they all become transformed into the fifteen valid syllogisms shown in
21
Table 11! In other words, replacing each term in the invalid syllogisms by their
contradictory generates the fifteen valid syllogisms.
Figure
AAA
EAE
1
Barbara
Celarent
2
Cesare
AEE
IAI
Darii
EIO
Festino
Datisi
Calemes
AOO
OAO
Ferio
Camestres
3
4
AII
Disamis
Ferison
Dimatis
Fresison
Baroco
Bocardo
Table 11 The fifteen valid syllogisms.
This is very unexpected. There seems to be no intrinsic reason why making the
change that has been made to the interpretation of “some” and “some..not” should
generate as invalid syllogisms mirror images of the fifteen valid syllogisms. We
should point out that there are two further invalid ones – AAO and OOA. These do
not have counterparts in the valid ones, and indeed when they are transformed in a
similar manner they are then mirror images of themselves.
It is not clear what we should make of this. It was completely unexpected when these
relations were discovered, and although it must be generated by some underlying
structural relation it is not clear what this is. Nor is it clear how this might help
overcome the problem – we are still left with seventeen syllogisms incorrectly
classified in LoF.
5.2 Other interpretations
We did experiment with other possible interpretation of the particular and universal
propositions and with altering conjunction/implication as the basic form of the
syllogism (as suggested by Spencer-Brown (1972, p. 119)) and the results for the
propositions can be seen in Table 12.
22
All
None
Some
Standard Version
1st Revision
2nd Revision
3rd Revision
a implies b
a implies b
a and b
A equivalent to b
(a)b
(a)b
((a)(b))
(((a)b)((b)a))
a implies not b
a implies not b
Not (a implies b)
Neither a nor b
(a)(b)
(a)(b)
((a)b)
(a)(b)
Not (none)
a or b
a or b
a and b
a and b
ab
ab
((a)(b))
Not (all)
a or not b
a or not b
a or b but not both
Not (a implies b)
a(b)
a(b)
a eor b
((a)(b))
Some not
((a)b)
Number of invalid
83
((a)b)((b)a)
17
97
73
results
Table 12 Results from various interpretations of the types of propositions.
It is clear that the 2nd and 3rd revisions are no better than the standard version and that
it is only the 1st revision that produces a significant improvement.
6. Conclusions
This paper has investigated the extent to which the Laws of Form can properly
represent syllogistic logic. Although this was suggested as possible by Spencer-Brown
and several other researchers, it was evident even in Spencer-Brown’s original work
that, while the valid syllogisms were properly represented, certain invalid syllogisms
also were classified as valid. Moreover, Spencer Brown’s suggested way round this –
Interpretative Theorem 2, was shown to be unsatisfactory. The first stage of this
research discovered the extent to which LoF misclassified syllogisms – in fact 83 out
of a total of 256 – a very significant proportion. The second stage considered various
possible reasons for this and highlighted particularly the standard interpretation of
“some” and “some not” in LoF.
Based on this, an alternative interpretation was proposed and investigated. It was
found to be significantly better than the traditional interpretation but still seventeen
syllogisms were misclassified. A major discovery was then made, that fifteen of the
seventeen incorrect syllogisms were actually transforms of the fifteen valid syllogisms.
One could be transformed into the other by swapping “all” for “some not” and “none”
for “some” and vice versa. The exact implications of this for the question at hand is a
topic for further research.
Two other alternative interpretations were tested but were found to be no better than
the traditional interpretation.
23
In conclusion, progress has been made on the problem of applying the Laws of Form
to syllogisms although a completely satisfactory result has not yet been reached.
24
Appendix 1 Summary of Laws of Form
Arithmetic Initials
()()
(())
= ()
=
I1
I2
=
=
J1
J2
Algebraic Initials
((p)p)
((pr)(qr))
((p)(q))r
Consequences
((a))
= a
(ab)b
= (a)b
()a
= ()
((a)b)a
= a
aa
= a
((a)(b))((a)b) = a
(((a)b)c)
= (ac)((b)c)
((a)(br)(cr)) = ((a)(b)(c))((a)(r))
(((b)(r))((a)(r))((x)r)((y)r))
= ((r)ab)(rxy)
25
C1
C2
C3
C4
C5
C6
C7
C8
C9
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i
In the current version of the paper, for typographical ease, the LoF is represented using a common
bracketed notation which is not the same as used in the original. In the final version I hope that the
original notation will be used.
27
http://www.kent.ac.uk/kbs/research-information/index.htm