=Paper=
{{Paper
|id=None
|storemode=property
|title=Visualizing Syllogisms: Category Pattern Diagrams versus Venn Diagrams
|pdfUrl=https://ceur-ws.org/Vol-854/paper3.pdf
|volume=Vol-854
|dblpUrl=https://dblp.org/rec/conf/diagrams/Cheng12b
}}
==Visualizing Syllogisms: Category Pattern Diagrams versus Venn Diagrams==
https://ceur-ws.org/Vol-854/paper3.pdf
Visualizing Syllogisms:
Category Pattern Diagrams versus Venn Diagrams
Peter C-H. Cheng
Department of Informatics, University of Sussex, Brighton, UK
p.c.h.cheng@sussex.ac.uk
Abstract. A new diagrammatic notation for Syllogisms is presented: Category
Pattern Diagrams, CPDs. A CPD configures different styles of line segments
to simultaneously assign quantification values to categorical variables and rela-
tions among them. The design of CPDs attempts to coherently visualize the
structure of syllogisms at various conceptual levels. In comparison to Venn
Diagrams and conventional verbal expressions of syllogisms, the potential
benefits of CPD may include: a relatively straightforward inference method;
simple rules for evaluating validity; applicability to multiple (>2) premise syl-
logisms.
1 Introduction: the project and programme
This paper is part of a project that is attempting to develop a novel set of related dia-
grammatic notations for various systems logic. The project’s objective is to design a
family of diagrammatic notations that share a common representational scheme for
encoding logical states of affairs and a common inference method. The focus here is
syllogisms, with the introduction of Category Pattern Diagrams, CPDs. The design
of CPDs builds directly upon our previous work on Truth Diagrams for propositional
logic [1,2]. In turn, CPDs are being used as intermediate stage to develop a related
notational system for full predicate calculus. The overarching aim of the project is to
show how re-codifying systems of logic in closely related notational systems may
reveal the similarities and differences in the conceptual structures of those logics.
The project is, in turn, part of a larger Representational Epistemic research pro-
gramme that is studying how notational systems encode knowledge and the potential
cognitive benefits that novel codifications of knowledge may confer on higher forms
of thinking [3-6]. The core principles of the Representational Epistemic approach
address how to design representational systems for knowledge rich topics. They
claim that directly encode the fundamental conceptual structure of a topic in coherent
notational schemes will provide semantically transparency and thus enhance problem
solving and conceptual learning in multiple ways [3-5]. Previous knowledge domains
that have been re-codified as part of the programme include electricity, probability
3rd International Workshop on Euler Diagrams, July 2, 2012, Canterbury, UK.
Copyright © 2012 for the individual papers by the papers' authors. Copying permitted for private and
academic purposes. This volume is published and copyrighted by its editors.
33
Fig. 1. Truth Diagram demonstration of the validity of Modus Tollens
theory and (school) algebra [3-5]. The project on logic is extending the scope of the
programme by providing further stringent test cases for the Representational Epis-
temic claims.
The previous work in the logic notation design project developed Truth Diagrams,
TDs, to re-codify propositional logic (and Boolean Algebra) [1,2]. Fig. 1 shows an
example in which the validity of Modus Tollens is demonstrated. The details of TDs
are not essential to consider here, rather it is the overall form of the notation that is of
concern, because the design of CPDs aims to adopt a similar representational scheme
and inference method. In TDs letters are labels for variables and configurations of
line segments assign truth-values to propositional variables and relations among those
variables. Solid lines stand for True and dashed for False. Fig. 1.1, 1.2, 1.4 and 1.6
are unary or binary relations of variables involving P and Q. The inference method
creates a composite diagram, Fig. 1.4, by combining the premise diagrams, Fig. 1.1
and 1.2, using a diagrammatic operator, Fig. 1.3, which specifies the types of the lines
to draw in the composite diagram given the permutations of line types in the premise
diagrams. The validity of the inference is determined by comparing the structure of
the composite diagram with the diagram for the given conclusion, Fig. 1.6, using a
simple set of diagrammatic validity rules, Fig. 1.5, which specifies correct correspon-
dences between the types of lines in the two diagrams. TDs constitute an efficient
method to reveal how the propagation of patterns of truth-values determines the struc-
ture and validity of interferences. Taken together, the structure of the diagrams, the
composition operators and the validity rules provide a novel, complete and sound,
system that reveals conceptual structures (symmetries and regularities) on multiple
levels that are typically hidden by standard formula notation [1,2].
The specific aims of this paper are: (a) to introduce Category Pattern Diagrams,
CPDs, as a notation for syllogistic reasoning, that adopts a similar representational
scheme and inference method to TDs; (b) to compare CPDs with syllogistic infer-
ences using Venn diagrams and the traditional verbal approach; (c) to examine how
codifications of syllogisms in these alternative notational systems provides quite dif-
ferent perspectives on the underlying conceptual structure of syllogisms, with varying
degrees of coherence, and the impact this has on the ease of making inferences. Thus,
the paper has the following sections: 2 is a brief reminder about syllogisms; 3 de-
scribes the graphical structure of CPDs; 4 gives the procedures of composing premise
diagrams in to a result diagram; 5 provides the method to determine whether the result
diagram correctly implies the given conclusion diagram; 6 extends the approach to
multi-premise syllogisms, sorites; and, 7 discusses the overall efficacy of the CPD
encoding of syllogisms and considers implications for the design of notations to en-
code logic.
34
2 Syllogisms: a brief reminder
See [7] and [9], for example, for full introductions to syllogisms; but as a reminder,
consider syllogisms S1 and S2.
S1. No diagrams are sentential notations
All Venn Diagrams are diagrams
No Venn Diagrams are sentential notations
S2. All Category Pattern Diagrams (CPDs) are diagrams
All diagrams are effective representations
No effective representations are poor systems for learning
Some poor systems for learning are sentential notations
No CPDs are sentential notations
S1 is a classical two-premise syllogism, consisting of a major premise, a minor
premise and a conclusion. The middle term, M, occurs in both premises and the
predicate and subject, P and S, are the major and minor terms of the major or the
minor premises, respectively. (M is a subject in the major premise and a predicate in
the minor premise.) The quantity and quality of S1’s major premise happens to be
universal and negative (No M are P); such propositions are labelled ‘E’. S1’s minor
premise is universal and affirmative (All S are M); labelled ‘A’. The conclusion is
also an E proposition (No S are P). Particular affirmative propositions are labelled
‘I’ and particular negative propositions labelled ‘O’. The mood of a syllogism is its
particular permutation of proposition types for the two premises and the conclusion:
S1’s mood is EAE. S always precedes P in syllogism conclusions. The four possible
permutations of the order of the premise variables are called Figures; S1 is of Figure
type 1: M-P, S-M. The Mood and Figure type of S1 may be summarised as ‘EAE-1’
and like all valid syllogisms has been given a name, “Celarent” [1,2].
To determine the validity of syllogisms in verbal form, one may apply five rules
concerning the quality and quantity of the propositions. The quality rules state: (QL1)
no conclusion may follow when both premises are negative; (QL2) a conclusion is
negative when either premise is negative; (QL3) a negative conclusion cannot follow
from two affirmative premises. The quantity rules rely upon the notation of distribu-
tion, which is the extent to which all the members of a category are affected in a
proposition [7]; e.g., S is distributed in All S are P, but variable P is not. The quantity
rules state: (QN1) the middle term must be distributed in one or both premises; (QN2)
if a term in the premises is not distributed, then it must not be distributed in the con-
clusion. These rules are challenging to understand and apply, and explanations of
why they govern the validity of inferences are not straightforward to give.
Venn Diagrams, e.g. Fig. 2, provide a more comprehensible means to assess the
validity of syllogisms. First, a diagram is drawn with three fully intersecting circles
to represent all the possible combinations of sub-sets, Fig. 2.3. Then, beginning with
any universal premises, Fig. 2.1, corresponding regions in Fig. 2.3 are shaded for
empty sets. Subsequently, for any particular premises, Fig. 2.2, a cross is drawn in
any corresponding non-shaded region of Fig. 2.3. Care is needed to correctly locate
35
(1) (2) (3) (4)
E I M O
M P S M S P
S × P
× ×
Fig. 2. Venn Diagram for the Ferio syllogism
the shading and crosses to take into account the term not mentioned in each premise.
The inference is valid if the conclusion, Fig. 2.4, can be read directly from the pattern
of shading and crosses in the three-circle diagram.
S2 is a sorites, a multiple premise syllogism. Their general form is P1–P2, P2–P3,
…, Pn-1–Pn ⇒ P1–Pn. The particular form of S2 is: All C are D, All D are E, No E are
P, Some P are S ⇒ No C are S. Although Venn Diagrams can be systematically
drawn for four and more sets [8], the diagrammatic benefits of that approach appear to
be reduced for larger numbers of premises.
From this brief overview, it is clear that to re-codify syllogisms CPDs must do
many things: identify the categorical variables; denote whether things belong to each
category or not; specify relations among the variables, i.e., the mode and Figure; sig-
nify the quantification of the multiple subsets defined by those relations; have the
potential to represent multiple premises (>2); provide a method to infer the quantity
values of variables in the combined relations; establish a procedure to determine
whether inferences correctly imply the given conclusion.
3 Graphical structure of CPDs
Fig. 3 shows examples of CPDs for unary, binary and ternary relations of categorical
variables. Each diagram represents a state of affairs relating the categories identified
by the letters. The letter is a label for a category and the lines run between the specif-
ic positions relative to the letters. In a unary variable diagram, Fig. 3.1, horizontal
line segments are positioned above and below the letter. For the binary relation, Fig.
3.2, four lines run between the letters with their ends located at the four possible com-
binations of positions above or below each letter. We will call such line segments
connectors. In the ternary relation diagram, Fig. 3.3, the eight connectors are com-
posed of two binary connectors joined near the middle letter and with free ends asso-
ciated with the other two letters. The eight connectors are arranged as four pairs: (1)
an inverted triangular pattern; (2) an upright triangular pattern; (3) a descending paral-
lelogram pattern, which slopes downwards from left to right; (4) an ascending paral-
lelogram pattern. As will be seen below, this design of the ternary CPDs is intended
(1) (2) (3)
member — member
R R T R T U
not member = not member
Fig. 3. Unary, binary and ternary Category Pattern Diagrams
36
to support judgments about the validity syllogisms.
The position of the ends of a connector, top or bottom, refers to possible member-
ship or possible non-membership of the category, respectively. Fig. 3.1 and 3.2 in-
clude labels making this explicit. Each connector is a particular case, a combination
of inclusion or exclusion of things in the subsets of the variables of the relation. The
unary relation has two cases and the binary relation has four cases.
Consider examples of some cases in the ternary CPD of Fig. 3.3. In the top invert-
ed triangle the upper straight connector refers to the case of the membership of all
variables, where as the \/ shape connector is the case of subsets R and U membership
but T not. In general, the local altitude of the middle point or free ends of a connector
within a triangle or parallelogram indicates the membership status of a subset, with a
high position in the shape for membership and low position for absence. In the de-
—
scending parallelogram the \ connector refers to the membership of R and T and the
absence of U, whereas the \_ connector refers the membership just of R.
The line style of the connector assigns a quantity to a case. There are three styles
for three quantifiers: (1) a single solid connector is some – at least one instance of the
case; (2) a dashed connector is none – no instance of the case; (3) a solid double-line
connector means no information (no-info) – the quantification of the case is not
known; it may either be some or none. In Fig. 3.1 the top some connector specifies
that something is a member of R and the bottom no-info connector means it is not
known if things are excluded from R or not. In Fig. 3.2 the three double-line connect-
ors means the only specific information provided relates R and not T, and its solid
single-line connector says that at least one thing is a member of R is not a member of
T: in other words, the diagram reads Some R are not T. Consider three cases in Fig.
3.3. The double-line top connector of the upper triangle pair says that there is nothing
known about the assignment of members to the intersection of R, T and U. In the
descending parallelogram the solid line of the lower \_ shape connector indicates that
at least one thing is a member of R but it is absent from T and U. The dashed line of
—
the / connector says there is nothing that is T and U and not R.
Each connector in a CPD is equivalent to a region in a Venn diagram.
Fig. 4 shows CPDs diagrams for the four syllogistic propositions, A, E, I and O
(and gives their verbal expressions). Notice that all the CPDs have three unknown
connectors (double-lines) and either a single some or a none connector to constitute
the particular and universal propositions. The intersection of the two sets is the top
connector; the two exclusive subsets of the variables are the ascending and descend-
ing diagonals; the exclusion of both sets is the bottom connector. When the order of
the terms in a proposition is swapped, the order of the letters in the CPD is simply
reversed. Equivalently, the pattern of the lines may be reflected with the letter posi-
tions fixed. (If both the letters and lines are reversed, the proposition is unchanged.)
Notice that the patterns of lines in E and I are symmetrical, which has interesting
implications for the validity of certain syllogisms; as will be seen below.
37
A E I O
S P S P S P S P
All S are P No S are P Some S are P Some S are not P
Fig. 4. The four types of syllogistic propositions as binary CPDs
Turning to ternary CPDs, Fig. 5.2 shows a generic ternary relation CPD with num-
bered connectors that show the corresponding regions of the Venn diagram in Fig.
5.1. Fig. 5.3 and 4 are two examples of specific ternary relations. In Fig. 5.3 the
descending parallelogram says there is nothing that is S and not P, whatever the case
with M. The ascending parallelogram says that no-info holds for not S and P, for both
values of M. The top and bottom triangles both possess cases in which either there
are no instances present or that no-info occurs, for different values of M. Fig. 5.4
shows other patterns of connectors including the assignment of some to one case.
With a little experience, identifying individual cases in CPDs appears to be as easy as
finding sub-sets in a Venn diagram. Similarly, selecting pairs of cases for the same
values of S and P, as is required to judge the validity of a syllogism, also appears to
be comparable in both notations. However, we will see below that the CPDs and
Venn diagrams diverge when more than three propositions are considered.
That completes the overview of the syntax and semantics of relational CPDs. The
next section considers how to make inferences with CPDs.
4 Composition of Binary CPDs
Fig. 6 shows the CPD for the Celarent syllogism (EAE-1: No M are P, All S are M,
therefore No S are P; Figure type 1). In outline, the overall procedure for syllogistic
inferences with CPDs has two stages. First, given the two premises (Fig. 6.1 & 6.3),
the conjunction operator (6.2, see below for a full explanation) is applied to generate
the ternary result diagram (6.4). (The term result refers to the set of implications
derived from the premises as distinct from the given conclusion.) In the second stage,
the pairs of connectors of the result diagram are compared to the conclusion diagram
(6.5.E) to check that the result diagram fully and correctly implies the conclusion
diagram. (In Fig. 6 the desired conclusion (6.5.E) is highlighted but three others are
included for the discussion of invalid inferences below.) This stage compares the
types of connectors in the result diagram with the corresponding conclusion connect-
ors using a table of validity rules (6.6). This section describes the construction of the
ternary result diagram and the next section gives the procedure for testing validity.
(1) (2) (3) (4)
M 1
8 2
7 3
4
S 3 4 P S 5 M P S M P S M P
1
6
5 2 6 7
8
Fig. 5. Ternary CPDs
38
(1) (2) (3)
Premises
A C1 C2 C3 E
—
—
S M --- = M P
— ⋅?⋅ =
=
— --- =
(6)
(4) Result V1 V2 V3 V4
√ √ √ ×
S M P — = Any
— --- =
— --- =
⋅? ⋅ --- other
— --- = Comb’n
(5) Conclusion
A E I O
| | |
S /P S P S P S /P
(5.A) (5.E) (5.I) (5.O )
Fig. 6. EAE-1 Syllogism (and some alternative conclusions)
To construct a ternary result diagram we simply consider each of the eight con-
nectors in the CPD in turn. Fig. 7 shows two examples of the construction of two
connectors in the result CPD of Fig. 6.4. Three steps are required for each connector.
Step 1 – Fig. 7, arrows 1: determine the shape and position of the new connector from
the relevant connectors in the two premise CPDs. Step 2 – arrows 2: from the styles
of the pair of premise connectors find the relevant composition rule. Step 3 – arrows
3: the style of the new result connector is given by the output of the selected rule.
In preparation for step 1, the two premise diagrams are drawn so that the middle
term (M) will be in the centre of the new diagram and the subject term (S) on the left
and predicate term (P) on the right, see Fig. 6 and 8. M is in the middle because it is
common to both binary premise diagrams. The S and P arrangement will facilitate the
comparison of the result diagram with the conclusion diagram later (see below). Fig
8.1 illustrates this process for a ternary CPD of no particular mood (faint lines for
arbitrary connector types). If S is to the right and P to the left of M in the premise
diagrams, as in Figure type 1 syllogisms such as Fig. 6, they can simply be put to-
gether without further ado. If the premises are different syllogism Figures, then one
or both of the premise diagrams is reflected before they are combined; for example, in
a type 2 Figure syllogism, the M term occurs on the right of both the binary premise
diagrams, so just the P-M diagram needs to be reversed. Thus, all the possible moods
and Figures of syllogism handled.
39
(1) (2)
A A C1 C2 C3 E
C1 C2 C3 E
2
— —
— 2 —
--- = S M --- =
S M ⋅?⋅
2 M P ⋅?⋅ = M P
— = —
=
= 2
— --- = — --- =
1 1 1 1
3
Result Result 3
S M P S M P
Fig. 7. Composition of ternary CPDs – connector shape and style
Now, step 1 builds each ternary result connector from possible pairs of the premise
connectors. The top connector of result CPD, Fig. 7.1 (arrows 1), combines the top
connectors of the premises. The _/ shape result connector, Fig. 7.2 (arrows 1), is
assembled from the bottom and ascending diagonal connectors of the premises. In
general, for each connector in one binary diagram there are two possible associated
connectors in the other diagram. Fig. 8.2 shows how the four pairs of connectors in
Fig. 3.3 and 6.4 are obtained from the binary diagrams in Fig. 8.1. Each of the four
patterns in Fig. 8.2 corresponds to a particular case of S and P values, but the values
of M differ.
In Step 2, we find the quantification value for the new connector by looking up the
values of the premise binary connectors in the composition operator look up table in
Fig. 9. A copy of this table is reproduced between the two binary diagrams in Fig. 6
and 7 for convenience. Given the three possible types of each of the two premise
connectors, 32=9 permutations are possible. The table determines mappings from
pairs of premise connector types at the top of each column to the result connector type
at the bottom. The ‘ ? ’ symbol in Fig. 9 means any type of connector. (C1) The
result of the operator will obviously be a some connector when both premises are
some connectors. Whenever one premise connector is a some connector and the other
a no-info connector, the result is also a some connector, because just one premise
possessing a member will ensure that the new case contains a member. (C2) When
both of the connectors are no-info types, combining them provides no new infor-
mation; therefore, the result is also a no-info connector. (C3) Given a single none
connector, or pair of them, the result must be a none connector, because the presence
of any members of the new com-
bined category is forbidden. For (1) (2)
example, in Fig. 7.1 rule C2 ap- S M M P
plies to the left no-info and the S M P S M P
right none connectors, so the new
connector will be have none style. S M P S M P S M P
In step 3, we simply draw the
result connector in the style given Fig. 8. Composing ternary CPDs
40
by the output of the rule selected in step 2, in
Premise 1 Both
the position determined in step 1; in Fig. 7.1 connector none no-info
(arrow 3) this is a dashed top connector. In Fig. combinations
7.2 the bottom and the ascending diagonal
C1 C2 C3
premise connectors will give a _/ shaped result
connector (step 1), rule C3 applies because both Both some —
—
--- =
premise connectors are both no-info (step 2), so ⋅?⋅ =
Some & —
the result connector is drawn in position as a no-info =
no-info connector (step 3). Repeating the steps — --- =
for the other six connectors completes the result
CPD. As the two premise CPDs in Fig. 6 both Some none no-info
possess just no-info or none connectors the Result connector type
resulting ternary CPD contains only connectors
Fig. 9. Composition operator rules
of these types.
5 Determining Validity
The second stage of the CPD approach compares the result diagram with the given
conclusion diagram to establish whether, or not, each case of possible assignments of
values of S and P in the conclusion is validly implied by the two possible cases for the
same assignment of S and P in the result. As noted, the pairwise design of the con-
nectors in the ternary CPDs supports these comparisons. Fig. 10 shows the corre-
spondence between the pairs of result connectors and the conclusion connectors: the
upper result triangle maps to top conclusion connector; the descending parallelogram
to the descending connector; the ascending parallelogram to the ascending connector;
the bottom triangle to the bottom connector. The subsets of S and P are the same in
the result and conclusion for each matching case.
For each of these matches, we now determined whether the types of the two result
connectors correctly imply the type of the conclusion connector. Fig. 11 provides a
look up table for valid matches, where each column is a validity rule. (V1) If either or
both of the result connector types are some, then the conclusion connector is some,
because the presence of any member in the result implies the conclusion will have a
member. (V2) Two none result connectors imply a none conclusion, because the total
absence of any category members in the result implies an absence of members in the
conclusion. (V3) Two no-info connecters,
or one with a none connector, implies a no- Result Conclusion
info conclusion connector, because these
combinations provide no information about E
whether there are category members or not. S M P S P
(V4) No other permutations of result and
conclusion connectors are valid. Given that
each of the three connectors may be one of
three types, a total 33 different permutations Fig. 10. Matching result connectors to
exist, so the four rules of Fig. 11 constitute the conclusion connectors
41
a concise encoding of the 27 possible ways result connectors may, or may not, validly
imply the conclusion. Again, this conciseness may be attributed to the representation
of the possible quantification values as three styles of lines.
Now applying the validity
rules to Fig. 6, the none top con-
nector of the target conclusion
(Fig. 6.5.E) is correctly implied
by the result, because the upper
triangle has two none connectors
(Fig. 6.4) – Rule V2. The de-
scending parallelogram correctly
implies the respective no-info
descending conclusion connector,
because the result parallelogram
has one no-info connector and Fig. 11. Validity rules
one none connector – V3. This
rule applies to the ascending parallelogram in the same fashion. It also applies to the
bottom triangle but in respect to the two no-info connectors. Therefore, as all four of
the result connector pairs correctly imply their conclusion connectors, the overall
inference is valid. Had just any one of these matches been invalid, the overall impli-
cation would have been invalid.
The other types of proposition, A, I and O, are shown in Fig. 6 as alternative con-
clusions, which we now demonstrate are not implied by the conjunction of premises E
and A; i.e., EAA-1, EAI-1 and EAO-1 are not valid syllogisms. The bars on the con-
clusion connectors in Fig. 6.5.A/I/O identify those that are not satisfied in the result
diagram. In the case of the A conclusion, the top no-info connector is not implied by
the pair of none connectors in the upper triangle (V4 true, V3 violated), and the de-
scending none diagonal is not implied by a single none connector in the parallelogram
(V4 true, V2 violated). For the I proposition the top some connector is not implied,
because there is no some connector among the pair of in the upper triangle of the re-
sult (V1 violated), and similarly for the some descending connector in the O proposi-
tion (Fig. 6.5.O).
Fig. 12 derives the valid Ferio, Festino and Ferison and Fresison syllogisms (EIO-
1, 2, 3, 4), and has three points of interest. (1) The presence of the some connector
yields a some connector or a none connector in the result CPD when it is combined
with a no-info or a none connector, respectively, from the other premise. (2) The
match of the some connector in the result diagram and the O conclusion satisfies V3,
but the A, E and I conclusions neatly show how different forms of mismatch are easi-
ly spotted. (3) Both premise diagrams are symmetric, because their only non no-info
connectors are the top lines, which means that the overall configuration of the result
ternary CPD is invariant: the orders S, P and M does not matter, which is why the EIO
mood is the only one that is valid for all four Figures. By the same reasoning, this
explains why valid syllogisms often occur in pairs; they have an E or I as a premise.
42
Premises E C1 C2 C3 I
—
—
S M --- = M P
— ⋅?⋅ =
=
— --- =
V1 V2 V3 V4
Result √ √ √ ×
— = Any
— --- =
S M P
— --- = other
⋅ ?⋅ ---
— --- = comb’n
Conclusion
A E I O
| |
S
S /P S /P
/P S P
Fig. 12. EIO-1/2/3/4 syllogism (and some alternative conclusions)
Is the CPD approach for classical syllogisms complete and sound? All the 256
possible combinations of mood and figures have been examined. CPDs are complete
because all 15 valid syllogisms [7] are found to be valid in the approach. It is sound
because none of the 241 invalid syllogisms [7] are found to be valid. (As the compo-
sition and validity rules are few in number and simple, a spreadsheet was setup to test
all 256 syllogisms en masse.)
6 Sorites
The CPD approach extends beyond classical syllogisms. Fig. 13 shows two examples
of sorites, or polysyllogims. In each, the sequence of premises is on the left and the
conclusion on the right. A ternary CPD has eight connectors, and as each additional
proposition doubles the number of cases, quaternary and quinary CPDs will have
sixteen and thirty-two connectors, respectively, so would consequently be cumber-
some to draw. However, given the relative simplicity of the composition rules and
validation rules it is not essential to expand the row of premise CPDs, but rather we
may consider possible paths along connectors from the first variable through to the
last. The composition rules in Fig. 9 may be applied iteratively to a sequence of con-
nectors. The top row of Fig. 13.1 are all no-info connectors, so Rule C3 (Fig. 9)
yields an overall no-info path. The \ _ _ shaped path has one none connector and two
no-info connectors, so its overall path is none. Is this sorite valid? In an equivalent
fashion to Fig. 10, all the paths through the premises from a specific start point to a
specific end point are compared to the conclusion connector that has corresponding
points; for example, paths from the top-left to bottom-right through of the sequence of
43
(1) (2)
A A A A A A E I E
P Q R S P S C D E P S C S
Fig. 13. Two sorites as CPDs
premises (top P to bottom S) corresponds to the descending diagonal connector of the
conclusion (top P to bottom S). The rules in Fig. 11 are used to judge whether all the
types of these paths correctly imply the conclusion connector type. The top connector
of the conclusion of Fig. 13.1 is a no-info connector and by inspect we can see that all
paths (———, \/—, \_/, —\/) from the top left to the top-right of the premises are
either no-info or none paths by rules C2 and C3. Thus, all four paths satisfy V3.
Similarly, the ascending connector in the conclusion is correctly implied by the four
paths from the bottom-left to top-right, because there is at least one no-info path and
the rest are none paths, applying C3, C2 and V3. The same is true of bottom conclu-
sion connector and the bottom-left to bottom-right premise paths. The descending
line in the conclusion is a none connect. Again by inspection, we see that all four
paths from the top-left to the bottom-right contain one or two none connectors, so by
rule C2 all the paths are none paths, which means that the conclusion is correctly
implied (V2 satisfied). As all the conclusion connectors are correctly implied the
overall inference is valid.
Our second syllogism above, S2, is a four-premise inference with a mixture propo-
sition types (A, E and I). The letters of the variables in Fig. 13.2 have been chosen to
match the terms in the S2. Although this example is more complex than Fig. 13.1,
testing its validity is relatively straightforward. Consider the top none connector of
the conclusion. Rule V2 say that all premise paths must be the none type for this to
be correct, however we immediately see that there is a path consisting only of no-info
——
connectors, \/, so this case is not valid, and in turn the overall inference is invalid:
QED. (Testing the other cases is not arduous. All the other premises paths corre-
spond to three no-info connectors in the conclusion. By inspection all the cases in-
clude at least one no-info path (C3) and none paths as the only other type (C2), so all
have a mixture of no-info and none paths, therefore all three conclusion connectors
are correctly implied, because the conditions for V3 are met. Nevertheless, the sorite
is invalid, because the validation of top conclusion connector failed.)
This inspection method may, of course, be applied to two-premise syllogisms, and
is simpler than constructing of the ternary result diagram (Fig. 6 and 12). However,
the ternary result diagrams are nevertheless worth considering, because they provide
an explicit introduction to the analysis of the structure for binary CPDs sequences that
is needed to familiarize learners on the composition of connectors and about the
matching of multiple connectors to test validity. Quaternary CPDs can be drawn with
four groups of distinct patterns of four connectors that serve the same role as the four
pairs of connectors in ternary CPDs. However, they are cumbersome, because they
include 16 distinct lines. Clearly, higher order CPDs will be impractical to draw.
Fortunately, this limitation of CPDs is mitigated by the potential to use the inspection
44
method on linear sequences of binary CPDs for many purposes when dealing with
multi-premise syllogisms.
7 Discussion
Two of the aims of developing Category Pattern Diagrams were (1) to investigate
whether a new notation for syllogisms could be designed using a similar representa-
tional scheme and inference method to that devised for propositional logic Truth Dia-
grams (c.f., Fig. 1), and (2) to examine whether the possible benefits of the CPD nota-
tion were similar to those of the TD notation.
CPDs have been successfully developed using a scheme in which assignments of
values to variables, and values to relations among variables, is based on the position,
shape and style of line segments running among letters for categories. CPDs used
three styles of lines for some, none and no-info connectors, whereas TDs have two
styles for truth-values. Unlike the many diagrammatic composition operators of TDs,
there is just a single composition operator for CPDs, as syllogisms merely concern
conjunctions of propositions. Although there are many possible permutations of val-
ues for a pair of connectors, the CPD composition operator includes just three simple
rules. Similarly, the method for testing the validity of an inference consists of just
four simple rules to compare connectors in the conclusion and combined diagram of
the premises. When one is new to CPDs, an explicit result diagram may be drawn in
order to work methodically through all the permutations of connectors (e.g., Fig. 6
and 12). However, when one is familiar with the system, the validity of an inference
may be determined by inspecting paths running through the sequence of premises
(e.g., Fig. 13). This approach is feasible because (i) the small number of simple com-
position rules enables one to mentally compute the overall type of a path traced along
successive connectors and (ii) the small number of simple validity rules means that
the implications of a group paths can be readily judged in relation to a conclusion
connector.
The simple rules of the CPD approach stands in marked contrast to the convention-
al verbal approach to the evaluation of syllogisms that relies on the three quantitative
and two qualitative rules given in section 2. Because the quality rules, QL1-3, are
stated in terms of negatives or even double negatives, this inevitably makes them
somewhat tricky to apply. (They may be restated in positive terms, but at the cost of
introducing awkward disjunctions to work through.) The same comment holds for
quantity rule QN2. Further, both quantity rules are also challenging to apply, because
they not only concern the distribution of terms among the premises and conclusions,
but very notion of distribution is conceptually demanding to apply to all the terms in
all four types of syllogistic proposition. Inferences with CPDs works at a more ele-
mental level, with judgments about the overall validity of an inference depending
upon simple comparisons of whether the assignment of values to conjunctions of
variables are compatible, which is done by visually matching the styles of simple
patterns of line segments.
45
A similar claim holds for Venn Diagrams, as the assessment of the validity of an
inference revolves around whether the presence of a cross or the shading of particular
region in the three circle diagram are consistent with the conclusion. Whether CPDs
or Venn diagrams, in themselves, are better visualization for classical two premise
syllogisms will depend on particular representation design issues. One such is the
explicit representation of the absence of information in CPDs (i.e., the no-info con-
nector) versus the implicit encoding in Venn Diagrams (i.e., no × and no shading).
Another issue is the efficacy of representing sub-sets using spatially contained regions
versus distinct line segments. Such design issues will require empirical tests with
users. However, an advantage of CPDs over Venn diagrams is in relation to multi-
premise syllogisms. Venn himself show how to draw his diagrams for four and more
sets, but even with more simpler modern designs (e.g., [8]), the difficulty of dealing
with large numbers of premises increases more rapidly for Venn Diagrams than with
CPDs. The complexity of constructing the diagram and interpreting its relations ap-
pears to grow with the power of the number of sets. In CPDs the difficult arises with
the growing number of paths, but this is mitigated by the multiple constraints that the
construction rules and validity rules usefully offer. For example, composition rule C2
means all combinations of paths up or down stream of a none connector in a sequence
of binary premise diagrams will be none paths. Finding just a single no-info or some
path corresponding to a none connector in the conclusion invalidates the whole infer-
ence.
The comparisons of CPDs to the verbal and Venn diagrams approaches allow some
observations to be made about the general nature of how notations systems might
effectively codify logic. First, although both CPDs and Venn Diagrams are graphical
representations they use quite different schemes to encode the same concepts, which
again supports the theoretical claim that it is the nature of the relation between the
conceptual structure of the ideas being encoded and the characteristics of a notational
that largely determines the efficacy of a representational system [3-6]. It is not mere-
ly that a graphical representation is spatial or geometric in nature that provides poten-
tial benefits to reasoning, but how particular diagrammatic properties encode and
interrelate the concepts. Although the spatial containment on the plane provides an
initially compelling device to encode a small number of set memberships, the scheme
becomes rather less efficacious with larger numbers of sets.
The second observation is that the composition and validity rules of CPDs operate
at the “elemental” level of the assignment of fundamental quantity values (some, no-
info and none) to the “atomic states” of member and non-membership of the subsets
of variables and relations. As a consequence the basic rules of the system are simple
and relatively few in number. It is therefore possible to hypothesize that the concep-
tual difficulties we face in order to understand syllogisms does not arise from the
intrinsic nature of the topic, but rather is due to the complexity generated by
combinatorics of these fundamental elements in situations with multiple terms. The
design of CPDs appears to demonstrate that directly encoding the fundamental con-
cepts of the syllogism domain in the primary representational schemes of a notational
system creates an effective codification of the topic (potentially). As such, this would
46
be a further example of the core Representational Epistemic principle, which was
previously demonstrated in a range of other knowledge rich topics [3-6].
The third observation concerns how the direct encoding of the fundamental con-
cepts supports reasoning with the new notation. It has previously been theorized such
codifications of knowledge produce a semantically transparent system, in which many
of the concepts at different levels of granularity, levels of abstraction and in alterna-
tive perspectives are readily accessible in the same of expressions of the notation [3-
6]. It appears that this claim is also true for CPDs, as they provide explicit access to
multiple types of information that are variously used to make inferences with and to
explain syllogisms. These include: the identification of categories (labels); distin-
guishing the subsets of variables (high and low position); specification of relations
among variables (connector shapes); assignment of values to the variables (position-
ing of connectors relative to letters); the type and order of propositions, or moods
(arrangement of binary CPDs); the ordering of the variables within a proposition, or
Figures (arrangement of unary CPDs in each binary CPD). As the composition opera-
tor and validity rules apply directly to patterns of connectors their effect on the cate-
gorical state of affairs tends to be plain. Further, by examining overall patterns of
connectors for different combinations of mood and Figures one can gain a sense of
regularities that follow from the underlying categorical constraints (e.g., the impact of
the symmetry of the E and I) and also the implications of concepts such as distribution
(e.g., by adding symbols to explicitly show the distributional status of terms).
The next challenge for the project is to extend CPDs beyond syllogisms to cover
predicate logic in full.
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