Reclaiming the Pedagogical Use of Diagrams
for Inculcating Norms of Logic
Introduction
In an introduction to logic, the Venn diagram plays both a logical and a
pedagogical role. Its logical role is to test the validity of categorical syllogisms; its
pedagogical role is to help the student understand and accept the evaluations.
Blurring the distinction between the two roles is standard practice. Most logic
teachers expect the student to rely on self-ascriptions of perceptual intuitions—such
as, seeing that a diagrammatic region is empty—for a proof and an understanding of
an argument’s evaluation. According to Neil Tennant (1986, p. 304) it is as if the
Venn diagram were a general model in which the premises of a syllogism are true and
one can check whether the conclusion must be true as well.
Jon Barwise and John Etchemendy’s (1990) development of Hyperproof (a
computer program that teaches analytical reasoning by using an integrated
combination of sentences and diagrams) has spurred a number of logicians to
formalize systems of diagrammatic proofs (Allwein & Barwise, 1996). Sun-Joo Shin
(1994 & 1996) has developed a formal system of Venn diagrams (VENN) to assure
us that ascriptions of perceptual intuitions of Venn diagrams can assist in proofs of
validity. She uses ascriptions of perceptual intuitions of the syntax of the system to
replace the common ascriptions of perceptual intuitions that logic teachers rely on to
blur the logical and pedagogical use of the diagrams (Shin, 1994, p. 176).
Shin’s conception of representational competence leads her to restrict the
information content of ascriptions of perceptual intuitions to facts of syntax.
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Representational competence is, according to Shin (1994, p 8), an ability to follow the
rules that govern the syntax and the semantics of a system of representation. This is a
common conception among contemporary logicians who aim to make diagrams
acceptable elements of proofs. It also leads her to conjecture that Venn’s (1971
reprint, pp. 119-124) unique insight—that his diagrams contain and represent
compartments, which may, or may not, be occupied by classes of things—is a
confusion of syntax with semantics (Shin, 1994, p. 20). This is important because
Venn’s insight is the basis of our pedagogical use of Venn diagrams.
There is an introductory logic textbook (Klenk, 2002) that employs the syntax and
semantics VENN formalizes, but it does not adhere rigorously to the constraints of
VENN. Doing so would eliminate some of the current benefits of using Venn
diagrams. First, a formal diagrammatic proof of validity would consist of a sequence
of diagrams somewhat longer than teachers currently use. It would begin with the
diagrams of the premises of a categorical syllogism and terminate in the diagram of
the conclusion of the syllogism, but each diagram after the premise diagrams would
have to be a transformation of previous diagrams according to one of the six
transformation rules of VENN. Second, ascriptions of perceptual intuitions would
serve only to shorten some of the proofs to the customary length: both premises must
be universal categorical propositions. Third, there would be no proofs of invalidity.
Finally, a pedagogical use of the diagrams would seem inappropriate. The common
perceptual intuitions students are expected to ascribe to themselves are not of the
syntax or the semantics of VENN. Given only the syntax and semantics of VENN to
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work with, the teacher would seem to be at a loss for words to describe their factual
content.
If Shin’s conjecture of a confusion of syntax and semantics at the basis of the
pedagogical use of Venn diagrams is tenable, then the ascriptions of perceptual
intuitions logic teachers currently rely on to blur the logical and pedagogical use of
the diagrams also involve a confusion of syntax and semantics, and they would
accordingly be meaningless. Moreover, logic teachers will be compelled to give up
the belief that Venn diagrams have a pedagogical role in the curriculum (other than
being a heurist to giving proofs). There would be support for Tennant’s (p. 305)
beliefs that the diagrams are actually illusory models and have no peculiar and
irreducible role to play in understanding the language of the logic.
It is therefore imperative that we reconsider the use of Venn diagrams in teaching
logic. We stand to lose a convenient tool for teaching some of the norms of logic.
The student’s self-ascriptions of perceptual intuitions not only help the student
understand the logic’s evaluations of arguments, they also serve to convince her that
the evaluations are correct. Accordingly, I do two things in this paper. I provide a
critical review of VENN. Logic teachers should be aware that VENN has set a
narrow standard for the use of ascriptions of perceptual intuitions in proving the
validity of categorical syllogisms. They should also consider using VENN as an aid
for reforming the practice of blurring the logical and the pedagogical roles of the
diagrams.
My primary undertaking is a theoretical justification of the pedagogical use of
Venn diagrams. There are two reasons why I believe that this is important. First,
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Shin’s work leaves us with a considerable reduction in the visual power teachers have
come to expect from Venn diagram proofs. If, in addition, Venn diagrams cannot aid
the understanding of the logic, it would be prudent to stop teaching them and to
devote the time to the predicate logic. Second, I empathize with the logic teacher. I
find it hard to believe that the ascriptions of perceptual intuitions logic teachers
currently rely on in the pedagogical use of Venn diagrams are meaningless even
though I have no pre-theoretical intuition about their factual content.
To respond to Shin’s conjecture that a confusion of syntax and semantics is at the
basis of the pedagogical use Venn diagrams, I adopt an alternative view of
representational competence. It recognizes that some diagrams are open to multiple
interpretations and that comprehending them depends on perceptual and conceptual
skills that bring forward diagrammatic facts that are neither facts of syntax nor facts
of semantics (Goodman & Elgin, 1988, pp. 101-120). I then explain how the selfascriptions of perceptual intuitions students use to form an understanding the logical
evaluations of categorical syllogisms involve the perceptual skill of seeing that
something is the case and a conceptual skill that is associated with comprehending
diagrammatic illustrations. The conceptual skill is the ability to caption illustrations.
Semantically speaking it involves an extensional mode of reference known as
mention-selection (Scheffler, 1979, pp. 31-36).
My approach is to treat the terms logic teachers apparently misapply to Venn
diagrams (according to Shin’s point of view) as diagrammatic captions. By doing so,
we find that the ascriptions of perceptual intuitions teachers rely on for a pedagogical
use of the diagrams have factual content. A mechanism known as pedagogical
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transfer (Scheffler, 1979, p. 56) can explain how the factual content is involved in
forming an understanding of the interpretations of the diagrams in VENN.
On balance, Shin’s work makes clear a need to reform the logical use of Venn
diagrams in an introduction to logic. On the other hand, her conjecture that a
confusion of syntax and semantics is at the basis of the pedagogical use of Venn
diagrams lacks full-fledged tenability. It rests solely on the computational conception
of representational competence she uses to justify her development of VENN. It does
not adequately support the claim that Venn diagrams have no peculiar and irreducible
role to play in understanding the language of the logic of categorical syllogisms.
A Naive Pedagogical Theory
When using a Venn diagram as pedagogical tool, the logic teacher relies on it to
occasion self-ascriptions of perceptual intuitions that can help the student understand the
Boolean formulations of individual categorical propositions and understand that when the
premises of valid categorical syllogisms (so formulated) are true their conclusions must
likewise be true. In each of these pedagogical settings, the intensional content of the
clauses used to describe the student’s perceptual intuition serve to aid the student’s
understanding. The keynote of the intensional intrusion is the content-clause syntax, the
‘that p’ idiom (Quine, 1990, p. 71). It signals that the content clause reflects the student’s
usage of the terms.
The teacher must accordingly have command of the intensional discourse. To
achieve it, teachers use such idioms as ‘x indicates that p’ and ‘x shows that p’ and rely
on the causal power of the diagrams to present the pertinent information content to the
student’s perception. Nevertheless, the intensional intrusion cannot be ignored as an
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appropriate perceptual intuition is expected. Command of the intensional intrusion will
ultimately rest on an ability of the teacher to interpret the sentences used to describe the
student’s self-ascribed perceptual intuitions. The current practice—for instance, using
‘the X∩Z region of the diagram is empty’—seems, however, to be based on nothing more
than Venn’s unique insight that his diagrams both contain and represent compartments
which may not be occupied by actual classes (Venn, p. 124).
A naïve pedagogical theory is easy to formulate. Treat the diagrammatic
representations of classes as their proxies, and teach the student to refer to the
diagrammatic representations “correctly” with the Boolean function terms and the labels
‘empty’ and ‘not empty’. With the right sequence of prompts and responses, the teacher
can conjecture that the student has had the appropriate perceptual intuitions and has
formed an understanding of the logic. The assumptions supporting the conjecture are the
following: the content clauses describing the perceptual intuitions have meaning, and
there is a transfer mechanism for using their meanings in understanding the Boolean
formulations of the categorical propositions.
In light of the development of VENN, maintaining that Venn diagrams do aid the
understanding of the logic of categorical syllogisms by setting before the eye relevant
information content that is not in the syntax or semantics of VENN, will require a more
sophisticated theory. We will ultimately need a justification of the assumptions.
VENN and the Logical Use of Venn Diagrams
VENN clears the way for the ascriptions of perceptual intuitions in proofs of
validity. It will be helpful to review some of the details of VENN and to make clear its
limitation on the logical use of ascriptions of perception intuitions.
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VENN has all the elements of standard logic. The atomic syntax includes a
variation on the syntax Peirce introduced to represent particular propositions:
rectangle
closed curve
shading
line
x
Fig. 1. Atomic syntax of VENN
The set of well-formed diagrams includes the standard set found in textbooks that
use the same syntax (Shin, 1994, pp. 57-64). New are a set of rules for transforming
diagrams into other diagrams (which define the set of diagrammatic proofs) and a modeltheoretic semantics (Shin, 1994, pp 62-93). New too are the proofs that VENN is a sound
and complete logic of categorical syllogisms (Shin, 1994, pp. 93-110).
VENN formalizes the intuitions that a shaded region represents the empty set and
that a region with an x-sequence represents a non-empty set (Shin, 1994, pp.68-69). A
set of representing facts in VENN is defined (Shin, 1994, pp. 68-69). There are two
kinds: that a region is shaded and that an x-sequence is in a region. A corresponding set
of represented facts is defined (Shin, 1994, pp. 69-70). They are that a set assigned to a
region is empty and that a set assigned to a region is not empty. Two homomorphisms
assign the representing facts to the represented facts. One homomorphism maps the
regions of a diagram to the subsets of some non-empty domain. The other
homomorphism assigns the representing facts to the empty set and non-empty sets across
the set assignments of the initial homomorphism.
The proper use of ascriptions of perceptual intuitions in VENN is to shorten the
sequence of diagrams formally required to prove validity for some categorical syllogisms.
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Logically speaking, its role is to make perceptual inferences possible (Shin, 1994,
pp.167-176). Perceptual inference is an intensional concept. A perceptual inference is
said to take place when a student extracts information content that is implicitly involved
in an explicit representation of information content (Shin, 1994, pp. 153-184, pp. 188189; Barwise and Etchemendy, 1990, pp. 21-22 , p. 25, & 1996, p. 80).
Two factors make perceptual inferences possible in VENN. One is the extended
homomorphism between the representing and the represented facts, and the other is an
ability to see that regions are shaded. In some situations, seeing that a region is shaded
eliminates the need for the extended sequence of diagrams formally required for a proof.
Yet both the syntax of VENN and the homomorphic representation relations are needed
to clear that way for perceptual inferences to shorten proofs: together they satisfy the
requirement that the irreducible intensional intrusion leaves the extensional side of the
logic unimpaired (Quine, 1990, p. 72).
Because perceptual inference is possible in VENN, Shin (1994, p. 154) can claim
that VENN is a visual system of logic. She acknowledges that if perceptual inferences
were not possible in VENN, it would be a trivial linguistic system of logic of no interest
to logicians (Shin, 1994, p. 153).
VENN does however employ a conventional linguistic element (Shin, 1994, pp.
178-184). This serves to conjoin the representation of a universal categorical proposition
with a representation of particular categorical proposition. As it happens, their
conjunctive information involves exclusive disjunctive information, and VENN cannot
represent exclusive disjunctive information diagrammatically. This is because in general
there is no perceptually obvious way to represent exclusive disjunctive information (Shin,
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1994, p. 180). So in VENN, a sequence of x’s linked by lines through regions is a
linguistic convention for representing exclusive disjunctive information. There is an
example of this below.
Despite the inability to represent exclusive disjunctive information
diagrammatically, VENN assures us that ascriptions of perceptual intuitions can
occasionally justify relatively brief proofs of validity. On those occasions, we can
conjecture that the student sees that a categorical syllogism is valid by looking at the
diagram of its premises. Of course, this will mean only that the student has perceptually
inferred the information content of the conclusion from the information content explicitly
represented in the diagram of the premises. It does not entail either that the student sees
the empty set or even that the student sees that a set is empty.
Unfortunately, in cases where one of the premises of a valid categorical syllogism
is particular, a student cannot perceptually infer the information content of the conclusion
from the information content explicitly represented in the joint diagram of the premises.
To avoid any confusion about why this is the case, let us now turn to some examples in
VENN.
To begin, let us consider the following categorical syllogism:
All dogs are mammals.
All hounds are dogs.
So, all hounds are mammal.
Its Boolean formulation is:
D∩−M = Ø
H∩−D = Ø
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H ∩−M = Ø
The diagrams of the individual premises are D1:
Fig. 2. D1
and D2:
Fig. 3. D2
Each diagram can represent information contained in the corresponding premise, and in
VENN that information is involved in the diagram’s information content. To grasp the
information content involved in D1 and D2 respectively, the student must see that the
regions are shaded and know the extended homomorphism setting up the representation
relation. The content of D1 is that the set assigned to the shaded region is the empty set.
The transformation rule Shin codifies for combining D1 and D2 justifies the first
step in the formal diagrammatic proof of the syllogism’s validity and it insures that the
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combined diagram represents their conjunctive information content (Shin, 1994, p. 88 &
pp. 96-97). The combined diagram is D3:
Fig. 4. D3: The unification of D1 and D2
As you can see, the syntax of D3 does not include any type of syntax that is not
contained in diagrams D1 and D2. Nevertheless, a relatively new representing fact has
emerged, namely that the H∩−M region is shaded. It is relatively new in the sense that it
is not in D1 and not in D2. In addition, a relatively new represented fact has emerged in
the information content of D3, namely that the set represented by the H∩−M region is the
empty set. This information is not involved in the information content of D1 and is not
involved in the information content of D2.
If a student claims to see that the syllogism is valid by looking at D3, the teacher
has some evidence that the student has perceptually inferred the information that the set
represented by the H∩−M is the empty set. To conclude that the perceptual inference has
taken place, the teacher must reasonably conjecture that the student has taken the
requisite psychological steps. First, the student must see that the H∩−M region is shaded
(Shin, 1994, p.159), and second, she must interpret it without much intellectual effort
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(Shin, 1994, pp.175-176). It is the ascription of the perceptual intuition of seeing that the
H∩−M region is shaded that justifies terminating the proof with D3. We do not need the
extended sequence of diagrams that terminates in D4.
Fig. 5. D4
D4 follows from D3 by the rule of erasure of a diagrammatic object.
Let us now consider a case where there is no ascription of perceptual intuition to
justify truncating a proof. Let us consider the following syllogism:
Some dogs are hounds.
All dogs are mammals.
So, some mammals are hounds.
The Boolean formulation of the syllogism is:
D∩H ≠ Ø
D∩−M = Ø
M∩H ≠ Ø
The premises diagrammed separately are D5:
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Fig. 6. D5
and D1 (see Figure 2, page 10).
The diagrammatic proof would proceed with the unification of D5 and D1. To
reflect our intuitions about the unification of the two pieces of information involved in
D5 and D1, the syntax required for representing exclusive disjunctive information is
required. The unified diagram is D6:
Fig. 7. D6: The unification of D5 and D1
To have a proof of validity for the syllogism the student must produce the
remaining sequence of transformed diagrams that terminates in D7:
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Fig. 8. D7
D7 represents explicitly that the set assigned to M∩H is not the empty set and the
extended sequence of diagrams is the proof that that information is involved implicitly in
D6. Only one diagram must be inserted between D6 and D7 to form the proof. It is
D6.5:
Fig. 9. D6.5
D6.5 follows from D6 by the rule of erasure of part of an x-sequence. D7 follows
from D6.5 by the rule of the erasure of a diagrammatic object.
A student cannot perceptually infer the information explicitly represented with D7
from D6 even if she should claim to see that the syllogism is valid by looking at D6. The
reason is the student cannot have the appropriate perceptual intuition by looking at D6:
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the student cannot see that there is an x-sequence in region M∩H in D6. This is because
there is no x-sequence in region M∩H in D6. Let me take a moment to explain why
precisely.
As I said earlier, there are only two types of representing facts in VENN: that a
region is shaded and that an x-sequence is in a region. An x-sequence in a region is an
unattached x in a minimal region of a diagram, as is in D5 and in D7, or a sequence of
two or more x’s attached with lines through regions, as is in D6. To avoid ambiguity
about what it means to say that an x-sequence is in a region, Shin interprets it precisely to
mean the smallest region containing an entire x-sequence. So while it is true that there is
an x in each of two minimal regions in D6, neither x in a minimal region in D6 is a
representing fact in VENN.
It should now be clear that in VENN ascriptions of perceptual intuitions provide
relatively brief proofs of validity (of the length normally found in an introduction to
logic) for only five of the fifteen valid forms of categorical syllogisms. A proof for any
of the remaining ten valid forms will require an extended sequence of diagrams
constructed according to the transformation rules of VENN.
Shin’s Conjecture
VENN packs a couple of surprises. First, as the second example above
demonstrates, it is no longer clear how we can expect students to see that valid syllogisms
containing particular premises are valid by looking at the joint diagram of the premises.
Second, it is not clear that there is a justification of the pedagogical use of Venn
diagrams. Shin is correct that VENN does not justify it.
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It is Shin’s conjecture that a confusion of syntax and semantics is at the basis of
the pedagogical use of Venn diagrams that is a major concern for the logic teacher who
wants to continue using Venn diagrams as a tool for understanding the logic and
inculcating some of its norms. As I mentioned earlier, if her conjecture is tenable, logic
teachers will have to accept that the ascriptions of perceptual intuitions they routinely
rely on are meaningless. I will now begin to make the case that Shin’s conjecture is
untenable.
Venn was the first to rely on ascriptions of perceptual intuitions for an
understanding of the Boolean logic of categorical syllogisms. He believed that his
diagrams have two interrelated interpretations. On the one hand, he believed that the
diagrammatic regions may represent classes of things as, say, VENN interprets them, but
Venn (pp. 113-140) also believed that they contain and represent compartments for
placing the classes of things otherwise represented. The interpretation of his diagrams in
terms of compartments is not entirely clear in Venn’s writing, in part because he does not
provide a definition of the term ‘compartment’. Nevertheless, Venn (p. 124) believed it
makes sense to say that a compartment represented by a diagram is empty should there be
no corresponding class of things to place in it. When there is a class of things to put in a
represented compartment, Venn (p. 168) believed it makes sense to say that the
compartment is not empty. In turn, Venn believed that talking about compartments being
empty and not empty and his diagram’s ability to represent that information about
compartments could help the logicians of his day understand and accept the class identity
relation Boole introduced for interpreting categorical proposition on an algebraic domain
of things. The way Venn stated it is, “this will set before the eye, at a glance, the whole
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import of the propositions collectively” (Venn, p. 123). Today, logic teachers have
dropped the talk about compartments, but they still expect students to see that the
diagrammatic regions are empty, or not empty, as the case may dictate.
Here is what Venn said when he introduced his insight about compartments:
The best way of introducing this question
[of how a diagrammatic scheme can be
worked so as to represent propositions] will
be to inquire a little more strictly whether
it is really classes that we thus represent, or
merely compartments into which classes may
be put (Venn, p. 119) .
This is his answer:
The most accurate answer is that our diagrammatic
sub-divisions . . . stand for compartments and not
for classes. We may doubtless regard them as
representing the latter, but if we do so we should
never fail to keep in mind the proviso, “if there
be such things in existence.” And when this
condition is insisted upon, it seems as if we express
our meaning best by saying that what our symbols
stand for are compartments which may or may not
happen to be occupied (Venn, pp. 119-120).
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Despite his failure to define ‘compartment’, ‘stand for’, and ‘to be occupied’,
Venn provided an example of the pedagogical benefit of interpreting his diagrams as
illustrations of compartments. His objective was to help his contemporaries understand
how universal affirmative and negative categorical propositions can be consistent in the
Boolean logic:
Take the two following, which in technical language are
termed ‘contrary’ the one to the other: ‘All x is y’, and
‘No x is y’. It will sound rather oddly at variance with
with ordinary associations to ask if these two propositions
are compatible with each other, that is, if they can both be
admitted simultaneously? The reply of ordinary logic
would be an emphatic negative; and this reply would be
valid enough from the predicative point of view, provided
we look no further than that. But if we take a wide, and I
should say a sounder view, we shall readily see that there is
no reason whatever against our accepting both propositions.
Look at them from the class or compartment point of view,
and we see at once that ‘All x is y’ empties out x−y, whilst
‘No x is y’ empties out xy. There is no harm in this, no
suggestion of conflict or inconsistency (Venn, p. 162).
Now according to Shin, a confusion of syntax and semantics is all there is to
Venn’s inclusive class or compartment point of view. She makes clear the basis for her
conjecture:
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From our point of view, this question [Venn uses to introduce
his insight] does not itself make sense, since the two alternatives
Venn suggests, that is, classes and compartments, do not belong
to the same category. Compartments represent classes in a diagram.
That is, a compartment with a shading is a representing fact, whereas
a class without any members is a represented fact. This passage shows
that we cannot expect Venn to perform a semantic analysis of his
diagrammatic system. We need to wait for the development of
semantics in the history of logic (Shin, 1994, p. 20).
Clearly, the basis for Shin’s conjecture is, as I said earlier, a point of view she has
adopted on representational competence. Again, the point of view is that representational
competence is a matter of following distinct rules of syntax and semantics (Shin, 1994, p.
8, & p. 185). Diagrammatic competence is a special case: the ability to follow rules of
syntax and semantic rules that involve homomorphic representation relations. This
conception of representational competence is justified, but only for its intended use. It is
justified in achieving command over the content of ascriptions of perceptual intuitions
that can appropriately occur in diagrammatic proofs.
The reason Shin’s conjecture is ultimately untenable is that there is a more
charitable point of view on diagrammatic competence according to which Venn’s insight
makes sense. I suggest that we have to use Nelson Goodman’s (1976, pp. 27-31) general
theory of symbols and his distinction between dyadic denotative representation and
representation-as to analyze Venn’s insight.
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Representation-as informs us of the intended classification of a representation. A
more detailed discussion of representation-as is in the next section. My point here is that
knowing what a representation denotes and how the representation is classified is often
required to comprehend a representation. In cases of fictive representation, all that
matters is how the representation is classified.
Representation-as and dyadic denotative representation along with the algebra of
logical classes provide a model of Venn’s insight that his diagrams contain and represent
compartments which may, or may not, be occupied by actual classes. When Venn would
say that a diagram contains and represents a compartment occupied by an actual class, we
can say that the diagrammatic region represents an actual class as a compartment; in other
words, it is a compartment-representation and it denotes an actual class. When Venn
would say that a diagram contains and represents a compartment for which there is no
actual class to occupy it, we can say that the diagrammatic region represents the null class
as a compartment; it is a compartment-representation and it denotes the null class. The
classification of diagrammatic representations in terms of compartment-representations
can be understood precisely as a classification of logical-class-representations. A
diagrammatic region may then be a logical-class-representation that denotes a logical
class as well as an actual class. The classification of representations may include emptyclass-representations and not-empty-class-representations.
If we interpret a Venn diagram as an illustration of an ideal domain of logical
classes, we can fix the factual content of the ascriptions of perceptual intuitions of the
class-illustrations by means of a mention-selective use of the class terms and the terms
‘empty’ and ‘not empty’. Their mention-selective use has a basis in the denotations of
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corresponding compound predicates. Furthermore, we can understand how a student can
use the ascriptions of perceptual intuitions to form a tentative understanding of the
identity relation on the corresponding classes of things of an intended domain as, say, we
find through the semantics of VENN.
In general, diagrammatic illustrations do help us understand things, and in the
process, we use the terms that denote the illustrated objects to say what we see in the
illustrations. It is common practice to say that we see things about people, places, and
things in pictures and to use the information content of the self-ascribed perceptual
intuitions to sort out the people, places, and things illustrated. In other words, we claim
to see what is illustrated. Let me now provide an overview of the alternative point of
view on representational competence that can make sense of ascriptions of perceptual
intuitions of this sort.
Another Point of View
The point of view I have in mind has a basis in Catherine Z. Elgin’s (Goodman &
Elgin, 1988, pp.101-120) discussion of representational competence. She maintains that
both conceptual and perceptual skills are involved in comprehending a representation, but
she does not limit representational competence to following rules of syntax and
semantics. It is Elgin’s view that comprehending a representation may involve terms that
are not specifically syntactical or semantical. Moreover, the modes of reference that
guide the terms are various. They include literal and metaphorical denotation and
mention-selection. The interpretation of a representation will accordingly depend on how
one uses terms to comprehend it, and there are no strict standards determining a uniquely
correct interpretation, according to Elgin.
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Equally relevant to my project is Elgin’s (Goodman & Elgin, 1988, p. 116)
conjecture that one can simply see what a representation represents upon mastering the
terms involved in comprehending the representation. As she puts it, the comprehension
becomes so nearly automatic that we are apt to forget that interpretation has occurred.
Following Elgin’s lead, I maintain that a Venn diagram is an effective tool for
understanding the logic of categorical syllogisms and inculcating some of the norms of
logic because it can function as a diagrammatic illustration and is open to a couple of
extra-logical interpretations. Both interpretations rest on a system of diagrammatic
classification that is not specifically syntactical or semantical. The basis for
differentiating the interpretations is in the way the teacher and student use the terms to
comprehend the diagrammatic illustration.
The terms in question are those Shin would say the contemporary logic teacher
misapplies to the Venn diagram. I explain how they can function as captions and play a
role in comprehending the diagrammatic illustration.
Theoretically speaking, captions refer to a diagrammatic illustration by means of
mention-selection. In the case of a Venn diagram, I conjecture that the student learns to
comprehend a Venn diagram by using terms (the teacher employs mention-selectively) to
refer indifferently to the diagrammatic illustration and the logical classes Venn
unwittingly designed his diagrams to illustrate as compartments. I also maintain that the
student’s inability to distinguish the mention-selective and the denotative uses of her
terms puts the teacher in position to conjecture that there is a sense in which the student
understands the logic by seeing what a Venn diagram illustrates. In addition, I claim that
the student’s somewhat unclear self-ascriptions of perceptual intuitions play a role in
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forming an understanding of the logic. I will give some examples to support these claims
in the following sections.
Diagrammatic Illustrations
Diagrammatic illustrations help us learn the denotations of terms. They are in
dictionaries, field guides, and textbooks. The following diagrammatic illustration from
Trees and Schrubs (Petrides, 1986, xxii) provides a good example:
Fig. 10
Isreal Scheffler (1979, pp. 31-36, pp. 45-49 & pp. 51-53) has developed a theory
on the pedagogical employment of captions. The basis is Nelson Goodman’s (1976)
general theory of symbols. Scheffler’s theory states that the habits governing a term’s
employment to caption an illustration guide the learning of the term’s denotation of the
objects illustrated. I use the theory to justify the pedagogical use of the Venn diagrams in
VENN.
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First, I will introduce some terminology. The expression ‘x represents a class as
empty’ has two interpretations (Goodman, 1976, pp. 27-30). On one interpretation, ‘as
empty’ modifies the noun ‘class’. In this case, the expression means that x represents the
empty class. It is a matter of dyadic denotative representation, and all that is said is that
the empty class is represented. On the other interpretation, ‘as empty’ modifies the verb
‘represents’. In this case, the expression means that a class is represented as empty. In
Goodman’s terminology, it involves representation-as. What is said is that a class is
represented and how it is represented.
An additional nuance in the meaning of ‘x represents a class as empty’ arises in
the illustrative use of a Venn diagram because the student does not need to know the
identity of any actual classes to comprehend the diagram. To model how the Venn
diagram works in this situation, I treat the diagram as an illustration of the logical classes
of an ideal domain. Since there are no such classes per se to represent (Shapiro, 1991, p.
18), I treat the Venn diagram as a fictive illustration of logical classes (Goodman, 1976.
p. 26). Saying that a Venn diagram illustrates a logical class is then a matter of
representation-as exclusively. All it tells us is the kind of illustration we have.
Although the student is not likely to be aware of the distinction between dyadic
denotative representation and representation-as, the logic teacher should have a way to
avoid confusing them. To keep in mind when she is thinking about a case of
representation-as, the teacher can say, for instance, that a region of a diagram is an
empty-class-representation. The compound term ‘empty-class-representation’ is a oneplace predicate that denotes regions of a diagram (Goodman, 1976, p. 21). Compound
24
predicates formed in this way are neither predicates of syntax (Elgin, 1983, p. 45) nor
semantic predicates of VENN.
Finally, it will be helpful to call the extension of a term its primary extension. For
any term t, the extension of a corresponding compound predicate of the form ‘trepresentation’ will then be called t’s secondary extension (Goodman, 1972, p. 227). The
terminology helps provide an extensional semantics for the teacher’s use of captions.
Strictly speaking, a caption is a token of a term employed in context to refer to an object
in its secondary extension. In the example above, the inscription of ‘Bud Scale’ refers to
a Bud-Scale-representation.
In terms of extensions, the pedagogical use of a diagrammatic illustration involves
a transfer of classification from the secondary to the primary extension of a term. Using
Scheffler’ terminology (p. 56), I call it pedagogical transfer. The term itself is the vehicle
of the transfer, not a resemblance between the objects in the primary and secondary
extensions of the term. Along with Scheffler (pp. 46-47), I conjecture that the
pedagogical transfer may go unnoticed despite there being some grasp of the general
conventions of representation involved because there is a psychological confusion
involved: a failure to draw the theoretical distinction between the two uses of the term.
The psychological confusion facilitating pedagogical transfer is not use-mention
confusion (Elgin, 1983, p. 48). Because the difference is subtle and easily missed, I will
explain it.
In cases of use-mention confusion, a symbol is confused with the symbolized.
The following sentence is an example of a type of sentence that invites the confusion:
Scale denotes a thin membranelike covering of a bud.
25
In this example, the subject term that is used should be the object mentioned. Taken
literally, the sentence is false. The following sentence has the correction that insures
against a mistaken interpretation of the sentence:
The term ‘scale’ denotes a thin membranelike covering of a bud.
When we caption an illustration, we use a token of a term correctly to refer to the
illustration. We do not confuse the illustration with the illustrated objects that the term
denotes. The prior recognition of some general conventions of illustration precludes that
confusion. Nor do we confuse the syntax of the illustration with the illustrated object, I
might add.
To place captions within Goodman’s general theory of symbols Scheffler (p. 46)
aptly calls their peculiar referential function mention-selection. Mention-selection is an
extensional mode of reference. In general, a token of a term t mention-selects the
denotation of the corresponding compound predicate of the form ‘t-representation’
(Elgin, 1983, p. 47).
The psychological confusion facilitating the pedagogical employment of a
diagrammatic illustration is accordingly a confusion of two uses of a term, the mentionselective and the denotative uses of a term. With this in mind, the teacher may
reasonably conjecture that the student uses a caption to refer indifferently to the
recognized illustration and the putative object illustrated (Scheffler, p. 46-47). It is a
reasonable conjecture since the student is not likely to have recourse to a compound
predicate that denotes just the illustration. The teacher’s strategy is then to prompt the
student to draw from her comprehension of the illustration that the teacher has selected
according to its kind (as determined by a suitable compound predicate) to help sort out
26
the primary extension of the term (functioning as a caption) on an intended domain
(Scheffler, pp. 46-49 & Elgin, 1983, pp. 47-50).
In view of the teacher’s recourse to the compound predicates, the student’s
psychological confusion is just an expedient (Scheffer, p. 47). We could teach the
denotations of terms with illustrations by using the compound predicates; however, this
would needlessly complicate the instructions. Let us now consider how Scheffler’s
theory on the pedagogical use of captions can justify the ascription of the common
perceptual intuitions of Venn’s diagrammatic illustrations in teaching students to
understand the language of the logic of categorical syllogisms.
A Rough Lesson Plan
The teacher can present the lesson on the illustrative use of Venn diagrams in
three phases of instruction. Each phase has a specific goal for the student to meet. The
goals are to understand an algebraic structure on an ideal domain, to understand the
Boolean formulations of categorical propositions on such a structure, and to understand
why a Boolean formulation of categorical syllogism is, or is not, valid on such a
structure. For each goal, there is a series of objectives: to comprehend a Venn diagram as
a fictive illustration, to ascribe perceptual intuitions to oneself on the basis of the
comprehension of the illustration, and to use their information content to form an
understanding of structural features that have analogues on the domains referred to in the
semantics of VENN. With her empathetic observations that the student has met the
objectives, the teacher can conclude each phase of instruction by conjecturing that there
is a sense in which the student sees what the diagram illustrates and has thereby met the
goal of the lesson.
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In the first phase of instruction, a Venn diagram should consist of just overlapping
closed curves enclosed in a rectangle along with some captions. Diagram D8 is an
example:
Fig. 11. D8
With the captions in place, the diagram can help the student understand the algebraic
structure of an ideal domain of four atoms. The first objective is to comprehend the
diagram as a fictive illustration of the non-empty subsets of the ideal domain: to
comprehend the diagram in terms of the captions that refer to it.
The cognitive value of comprehending the Venn diagram in terms of captions
rests on the corresponding compound predicates, and their classification of the diagram is
entirely heuristic. In the semantics of VENN, for instance, the identity relation
interpreting the Boolean formulation of a categorical proposition is set-identity on the
logical subsets of a non-empty domain. For an introduction to the logic of VENN, an
initial understanding of a Boolean formulation may however be partial and tentative
(Goodman & Elgin, 1988, pp 117-129). The initial understanding may be in terms of an
indeterminate proper congruence relation on the ideal parent algebra of a non-empty
domain in VENN (Stoll, pp. 259-266). The null class may serve as the domain of the
ideal parent algebra. At the outset, a Venn diagram need only illustrate fictional non-
28
empty subsets of the ideal parent algebra. The captioned illustration D8 may accordingly
illustrate no domain in the semantics of VENN, but it may function as a four-atomalgebraic-domain-representation. In the second phase of instruction, the student can then
learn to comprehend a more developed diagram as an illustration of an indeterminate
proper congruence relation on the fictional non-empty subsets of the ideal domain D8
illustrates.
At the outset of the lesson, the teacher should bear in mind the disparity in the
conceptual resources she and the student bring to the diagram. The teacher can base her
use of the captions on the corresponding compound predicates. The student will not have
recourse to the compound predicates and she will therefore be unaware of the theoretical
distinction between the mention-selective and the fictive denotative uses of the terms the
teacher is using as captions. The conceptual disparity does not prevent the student from
meeting the behavioral criteria for learning to comprehend the diagram as an illustration
of an ideal domain from the teacher’s use of the captions; however, the student’s
comprehension will have to be considered somewhat confused given her lack of recourse
to the compound predicates. The point of having the teacher keep the conceptual
disparity in mind is this. She can then have an empathetic observation of the student’s
observation of the diagram and take the student’s behavior as evidence that the student
has an adequate comprehension of the diagrammatic illustration.
The teacher should tell the student that the diagram’s function is to illustrate the
Boolean operations of union and intersection on the subsets of an ideal domain. The
teacher should also tell the student that the equation and the class terms accompanying
the diagram function as captions. The student’s familiarity with captioned illustrations
29
and her capacity for pattern recognition (Resnik, 1997, p. 225) enables her to learn the
captions ostensively. At a minimum, the student should learn to use the following class
terms to refer to the diagram: ‘D’, ‘M’, ‘D∩−M’, ‘D∩M’, and ‘(D∩−M) + (D∩M)’.
Once the student learns to use the captions, the teacher can conjecture that the student has
an adequate comprehension of the diagram as an illustration of the subsets of an ideal
domain.
Even with a somewhat confused comprehension of the diagrammatic illustration,
the student can use her skill at seeing that something is the case to help select information
content from her comprehension of the diagram. For instance, the teacher can ask the
student to say what she sees about D in D8. If the student says something like, “I see that
D is the union of D∩M and D∩−M,” then the teacher has evidence that the student has
“correctly” used her perceptual skill to help select some information content from her
comprehension of the illustration.
To maintain command of the perceptual idiom, ‘sees that p’, the teacher must be
careful not to lose sight of the factual content of the student’s self-ascribed perceptual
intuition. The teacher must carefully interpret the content clause the student uses.
Treating the student’s content clause as mention-selective, she can interpret the student’s
content clause literally to mean that the D-class-representation is composed of the D∩Mclass-representation and the D∩−M-class-representation. The teacher can then claim to
have surreptitiously embedded that factual content in the student’s unclear self-ascription
of a perceptual intuition of the diagram, so to speak. In saying that the student’s selfascription of a perceptual intuition is unclear, I mean that the student is using her terms in
a manner that obscures the truth condition of her self-ascription of a perceptual intuition.
30
Once again, the teacher can empathize with the student as she is in a position to
conjecture that the student is not aware of the literal interpretation of the content clause
that she uses to describe her perceptual intuition. The teacher can accordingly
acknowledge that the factual content of the student’s perceptual intuition is not clear to
the student. The reason it is not clear to the student is that she lacks the conceptual
resources—the corresponding compound predicates—to sort out the extensions of the
mention-selective and denotative uses of the terms. We can then conjecture that the
disparity in conceptual resources the teacher and student bring to the diagrams does affect
the quality of the content of their respective perceptual intuitions (the sentence following
‘that’), but not the factual content.
The next objective is to have the student draw on the content of her self-acribed
perceptual intuition (again, the sentence following ‘that’) to form an understanding of the
ideal domain as illustrated. Aware that the student is indifferent to the distinction
between dyadic denotative representation and representation-as, the teacher can ask the
student to use the captions to say what D8 shows about the class D as it is illustrated. If
the student responds that D8 shows that the class D is the union of the classes D∩M and
D∩−M, then the teacher has evidence that the student has formed an understanding of a
part of the algebraic structure of the illustrated ideal domain. Never mind that the class D
per se does not exist and that the student is not likely to be aware of the truth condition
for her implicit self-ascription of a perceptual intuition: the student has nonetheless
acquired an understanding of a fictional domain as illustrated (Elgin, 1983, p. 47). Her
understanding is a fictive belief that she has implicitly ascribed to herself as a perceptual
intuition. Moreover, the teacher is in a position to conjecture that there is a sense in
31
which the student understands the fictional domain by seeing what a Venn diagram
illustrates.
I admit that claiming empathy with the student’s implicit self-ascription of a
perceptual intuition in meeting this objective is tenuous, as I believe most would doubt
that the teacher could imagine herself confusing the belief about the class D with a
perceptual intuition about it. In defense of the claim, I suggest that claiming to see what
a Venn diagram illustrates about an ideal domain is akin to the standard practice of
claiming to see things about people, places, and things in pictures and then using the
information content to sort out the people, places, and things. In these cases, the selfascription of a perceptual intuition is a convenience. It involves a deliberate blurring of
the boundary between the ascription of belief and the ascription of perceptual intuition,
but the ascription of a perceptual intuition is a reasonable attitude to take given the ability
to confuse the mention-selective and denotative uses of the terms involved. Of course,
we never go so far as to claim our “visual achievement” is epistemic in character, and we
will retract our self-ascribed perceptual intuition if pressed to do so. Similarly, I think we
can maintain that a student’s implicit self-ascription of a perceptual intuition in response
to the question about what the diagram shows is reasonable. Given her use of the key
terms involved, the student is not likely to be aware that her implicit self-ascription of a
perception to the leading question is just a belief. Of course, we do not have to consider
her “visual achievement” epistemic in character any more than we do in our own similar
situations. Fortunately, for an introduction to the logic of categorical syllogisms, the
student’s implicit self-ascription of a perceptual intuition does not have to be epistemic in
32
character. It is merely a convenient propositional attitude the student is disposed to adopt
in forming an understanding about an ideal domain of classes.
The lack of a clear boundary between the ascription of belief and the ascription of
perception (from the student’s point of view) is what gives the teacher a pedagogical
advantage in shaping the student’s judgments of validity and invalidity. Again, the
student is unlikely to be aware of the truth conditions for her self-ascribed perception
intuitions since the teacher intends to interpret their content clauses as a mentionselective claim, in response to her first question, and a fictive claim, in response to her
second question. The truth conditions of her respective intuitions are obscured by her use
of the key terms to refer indifferently to the diagrammatic illustration and the ideal
domain as illustrated. Nevertheless, because the student has a grasp of some of the
conventions governing representation we can conjecture that she does discern that she has
distinct intuitions in response to the questions the teacher poses: the student is aware that
the sentences she uses are not paraphrases (Elgin, 1997, chapter 6).
Again, it is the teacher who can interpret the student’s sentences with sentences
that make clear their separate, albeit compatible, information content, sentences that state
the truth conditions of the student’s perceptual intuition and belief. They are,
respectively, that a D-representation is a ((D∩M) + (D∩−M))-representation and that a
D-description is a ((D∩M) + (D∩−M))-description. It is enough that the student’s verbal
behavior suggests that she is aware of a distinction in the content of her intuitions for the
teacher to conjecture rightly that she has formed the tentative understanding expected of
her. The lack of clarity in the content of the student’s perceptual intuition and her belief
works to the teacher’s pedagogical advantage in just this way. The student holds the
33
fictive belief she has formed with all the conviction of a self-ascribed perceptual
intuition.
Theoretically speaking, the transfer mechanism at work in forming the student’s
understanding of the ideal domain is pedagogical transfer. The habits guiding the
student’s initial use of the captions to refer to the diagrammatic illustration transfer to
guide the terms’ reference to the non-empty subsets of the ideal domain. The student is
not aware of a transfer at work because she does not have the devices for fixing in her
mind separate extensions for the mention-selective and denotative uses of the terms in her
content clauses. She has not yet learned to use the explicit compound predicates to
denote the ranges of mention-selections. Again, the student’s indifference to the
distinction between the mention-selective and denotative uses of her terms is what
enables the teacher to empathize with the student and to conjecture reasonably that there
is a sense in which the student understands the ideal domain by seeing what a Venn
diagram illustrates.
In the second phase of instruction, the goal is for the student to form an
understanding of the Boolean formulations of categorical propositions. The teacher can
achieve this goal through another series of objectives. For each Boolean formulation, the
first objective is to teach the student to comprehend the corresponding diagram as a
fictive illustration of an indeterminate proper congruence relation on the non-empty
subsets of an ideal domain.
For a couple of examples, let us consider again diagrams D1 (Figure 2, page 10)
and D5 (Figure 6, page 13). With the appropriate adjustments, what I say in connection
34
with D1 and D5 holds, mutatis mutandis, for the other two standard Boolean
formulations.
Again, there will be a disparity in the clarity of the teacher’s and the student’s
comprehensions of the diagrams and a difference in the quality of their respective
ascriptions of perceptual intuitions. For instance, the teacher can base her comprehension
of D5 on a further classification of some of the class-representations in D5 by classifying
the D∩H-class-representation a not-empty-class-representation. To meet the first
objective of this phase of the lesson, the teacher can use the caption ‘not empty’ to
mention-select the not-empty-class representation in D5.
The representing facts of VENN in both D1 and D5 happen to make it easy for the
student to learn to use the additional captions ostensively. In general, the shaded regions
of diagrams can be empty-class-representations and the teacher can caption them empty;
regions containing x-sequences can be not-empty-class-representations and the teacher
can caption them not empty. (I will address the classification and captioning of regions
not containing representing facts later.) Behavioral criteria are involved in determining
that the student has learned to use the captions adequately. The student demonstrates her
comprehension of the diagrammatic illustrations by pointing to the class-representations
and referring to them “correctly” with the terms ‘empty’ and ‘not empty’.
Once again, the teacher should bear in mind that the student’s comprehension of
the diagrammatic illustrations is confused. As is the case with the captions learned in the
first phase of instruction, the student lacks the resources for fixing in her mind the
distinction between the mention-selective and the denotative uses of the terms the teacher
uses to caption the illustration. Nevertheless, the student’s comprehension is adequate,
35
and the teacher can move on to the second objective by prompting the student to use her
skill at seeing that something is the case to help select information content from her
comprehension of the illustrations. The teacher simply asks the right question. For
instance, the teacher can ask, “What do you see about D∩H in D5?” If the student says
that she sees that D∩H is not empty, the teacher has evidence that the student has used
her perceptual skill “correctly” and is now in position to form an understanding of the
Boolean formulation.
To achieve an understanding of the Boolean formulation, the student need only
transfer the habits she has acquired in learning to use the captions to guide their reference
to the fictive subsets illustrated. The appropriate prompt that can affect the transfer is to
ask the student to say what the diagram shows about the class D∩H as illustrated. If the
student responds that the diagram shows that the class D∩H is not empty, then the
teacher has evidence that the student has formed an understanding of the Boolean
formulation ‘D∩H ≠ Ø’.
Here again the student’s understanding is a fictive belief that she implicitly
ascribes to herself as a perceptual intuition. Again, the teacher can conjecture that there
is a sense in which the student understands the Boolean formulation by seeing what a
Venn diagram illustrates. The student’s implicit self-ascription of a perception is
understandable. It is akin to claims of seeing things about people, places, and things in
pictures. In this case, the student’s prompt response to the leading question indicates that
she is unaware that interpretation and pedagogical transfer occur.
It is imperative that the teacher not confuse the factual content of the student’s
explicitly self-ascribed perceptual intuitions with any of the Boolean interpretations of
36
the diagram in VENN. In the case of D5, the factual content of the student’s perceptual
intuition of the illustration is that the D∩H-class-representation is a not-empty-classrepresentation. In VENN, the information content that is semantically involved in D5 is
that the class denoted by the D∩H-class-representation is not the empty set. If the
teacher loses sight of the distinction, she will lose command of the ‘sees that p’ idiom,
and it would then be difficult for the teacher to set aside the charge that she has just
confused the syntax and semantic of VENN.
The goal in the third stage of instruction is to teach the student to use a single
more complex Venn diagram to understand why a Boolean formulation of a categorical
syllogism is, or is not, valid. Let us return to the diagrams of syllogisms discussed earlier
along with their Boolean formulations serving as general captions. Let us start with D3:
Fig.12. D3
I maintain that the student can have an understanding of the validity of the Boolean
formulation of a categorical syllogism captioning the diagram from the content of her
self-ascribed perceptual intuitions of D3.
37
The student can see the class-representations in D3. Drawing on the captions she
has learned to use in the first and second phases of instruction, the student can readily
form a comprehension of the diagram with the aid of its captions. With the appropriate
series of prompts, she can use her skill at seeing that something is the case to help select
information content from her comprehension of the illustration, and she can form an
understanding of the Boolean formulation of the syllogism from that information content.
In summary, the teacher can acquire the evidence (including her own empathetic
observations) needed to conclude that there is a sense in which the student sees that the
union of D∩−M and H∩−D in D3 is empty, sees that it contains H∩−M, and sees that
H∩−M is accordingly empty. Once the student has explicitly ascribed these perceptual
intuitions to herself, she is in position to form an understanding of the validity of the
Boolean formulation. To achieve it, the student need only implicitly ascribe to herself the
perceptual intuition of seeing that the formulation is valid.
The role of the teacher is to prompt her to ascribe the perceptual intuition to
herself implicitly. The teacher need only ask the right sequence of questions. The
following will work: what does the diagram show about the union of the classes D∩−M
and H∩−D, what does the diagram show about the class H∩−M, and finally does the
diagram show that that the formulation is valid? If the student responds “correctly” to the
questions, the teacher has evidence that the student has formed an understanding of the
validity of the formulation. The teacher can conjecture that there is a sense in which the
student understands the validity of the argument formulation by seeing what the diagram
illustrates.
38
For an introduction to the logic of categorical syllogism, it does matter that the
student’s self-ascriptions of perceptual intuitions are not clear to her. The underlying
confusion is in her use of the terms and it obscures the truth conditions of her selfascribed perceptual intuitions. The student is accordingly not likely to be aware that she
does not actually have a visual proof of validity. The teacher should inform her of this.
Notwithstanding, the student can be credited with an understanding of the validity of the
Boolean formulation of a categorical syllogism. Her understanding is in the belief system
she has formed from her explicit self-ascriptions of perceptual intuitions of an illustration
of a fictive model-illustration.
I can agree that there is a sense in which Tennant (p. 305) is correct in claiming
that the Venn diagram is chimerical. I do not agree with him that the Venn diagram has
no peculiar and irreducible role to play in learning the language of the semantics of
VENN. Using the Boolean terms and the ordinary English predicates to refer
indifferently to the class-representations in the Venn diagram and the putative subsets of
the ideal domain is a basis on which the student can form a tentative understanding of an
argument formulation’s validity.
Let us now consider diagram D6 together with its corresponding Boolean
formulation serving as a general caption:
39
D∩H ≠ Ø, D∩−M = Ø so M∩H ≠ Ø
Fig. 13. D6
First, let me digress for a moment to identify problems with the accounts that are
currently available to justify the pedagogical use of diagrams like D6. Remember, Shin’s
work demonstrates that a student cannot grasp the validity of the argument formulation
with a perceptual inference.
One account calls the applications of ‘empty’ and ‘not empty’ to diagrammatic
regions metaphorical applications (Tidman & Kahane, 1999, p. 302). It does not pass a
close examination. It is an easy mistake to make since both metaphorical and pedagogical
uses of terms involve a transfer of habits guiding the use of the terms (Scheffler, p. 48).
Here is the problem. Metaphors have an impact on conceptual development by
working on the philosophical fringe of a theory (Quine, 1981, p. 188). They serve to
limn a new order when the old idioms begin to fail. The new order a metaphor brings to
a domain is transferred from another domain where the terms have literal denotations
(Goodman, 1976, p. 74-85). By the time the new order is accepted, the novelty that is
characteristic of metaphor has worn off: the terms acquire new literal denotations and
40
convey literal truths. A new model for a formulation of an existing theory has emerged.
A metaphor’s usefulness in conceptual development is accordingly brief. Either it
reorganizes a domain or we discard it.
In view of this process, the problem with calling the applications of ‘empty’ and
‘not empty’ to Venn diagrams metaphorical is that the practice persists without the terms
denoting regions of the diagrams. In other words, no Venn diagram of a valid categorical
syllogism will ever literally exemplify a model in the semantic of VENN.
Another standard account uses the idiom ‘read off the content of a conclusion
from the diagram of the premises’ (Klenk, p. 381). Referring back to D6, for instance,
the teacher is to inform the student that an x-sequence in a non-atomic-region of a
diagram “means” that there is something in one of the regions, but not both. This
exclusive disjunctive information is literally about the diagram, and it mirrors the
exclusive disjunctive information about the sets of an actual domain referred to in VENN.
Concomitantly, a shaded region would have to “mean” that the region shaded literally has
nothing in it. With this vague extra-logical literal diagrammatic information in mind, the
student is to use an intuitive version of disjunctive syllogism to infer that there must be
something in the M∩H region of D6. Then, from this inferred information about the
M∩H region, the student is to infer analogically (and thereby understand) that there must
be something in the set denoted by the M∩H region.
The problem here is that there is nothing to suggest that the student’s perceptual
intuition has played a role. In other words, there is nothing of a visual nature to speak of
in this process of “reading off” the conclusion from the diagram of the premises. First,
there is no perceptually obvious way to represent exclusive disjunctive information even
41
if that information is literally about a diagram in VENN. Second, the student’s intuitive
execution of disjunctive syllogism does not make it reasonable to conjecture that the
student sees what the diagram illustrates. Therefore, the student might as well be using
an intuitive version of disjunctive syllogism to infer the content of the conclusion directly
from the intended interpretation of the diagram in VENN without considering the vague
extra-logical interpretation of the representing facts on the diagram.
The advantage of the pedagogical theory I am using is that it justifies a teacher’s
belief that a student’s explicit self-ascription of a perceptual intuition plays a role in
understanding the validity of categorical syllogisms even in cases where particular
premises are involved. It also provides the teacher sufficient command of the perceptual
idiom to help the student learn to use ‘not empty’ to refer to representations like the
M∩H-class-representation in D6. Once this use of the caption is learned, the third phase
of the learning process proceeds as smoothly as in cases involving only universal
premises.
It will also be helpful to take a moment for a more detailed discussion of the
lesson on captioning class-representations. The teacher should of course weave the
lesson on captions through the three phases of instruction. The lesson on ‘empty’ and
‘not empty’ should begin in the second phase of instruction. In the second phase of
instruction, the captions refer to atomic-class-representations. Among the atomic-classrepresentations, a one-to-one correlation exists between the representing facts in the
class-representations and their classification as empty- and not-empty-classrepresentations, and the teacher can rely on the correlation to help the student learn to
caption the class-representations “correctly”. The teacher can point out a class-
42
representation that is shaded and say that it is empty (mention-selectively speaking). Just
as easily, the teacher can point out a class-representation that contains an x-sequence and
say that it is not empty (mention-selectively speaking).
In the third phase of instruction, learning to use the captions to refer to classrepresentations is complicated because the teacher must now caption non-atomic-classrepresentations, and for them there is no one-to-one correlation between the occurrences
of x-sequences and the not-empty-class-representations. In Venn, some not-empty-classrepresentations are not marked with an x-sequence. The student can still learn to use the
captions to refer to the unmarked not-empty-class-representations ostensively, that is, by
direct conditioning. As it happens in VENN, some of the not-empty-classrepresentations that do not contain x-sequences overlap empty-class-representations.
Since it should be clear how we can teach the use of ‘empty’ to refer to empty-classrepresentations, I will focus on teaching the use of ‘not empty’ to refer to not-emptyclass-representations.
The student will encounter not-empty-class-representations that do not contain xsequences in diagrams of Boolean formulations of syllogisms that contain particular
premises. It is best to diagram the particular premise first. Let us consider diagram D9
along with its general caption:
43
Fig. 14. D9
Any non-atomic class-representation that contains an x-sequence is a not-emptyclass-representation, as a whole, and teaching a student to “caption” it not empty can
proceed simply by pointing out the representing fact of VENN. The teacher should note,
however, that the component atomic-class-representations of D9 are neither not empty
nor empty (mention-selectively speaking). As it happens in VENN, neither atomic
component contains an x-sequence. On the other hand, the teacher can point out that any
class-representation that is in part not empty (mention-selectively speaking) is (as a
whole) not empty (mention-selectively speaking). D9 is, as a whole, not empty (mentionselectively speaking).
Before moving on to the joint diagram of the premises, D6, the teacher should be
confident that the student has learned to use the captions ‘empty’ and ‘not empty’ to refer
to class-representation in which there are representing facts. The teacher need not define
or use the corresponding compound predicates that provide the basis for the mention-
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selective use of the terms as captions. The objective is to inculcate the practice of using
the captions to refer to the class-representation, not to justify the practice to the student.
Let us now look more closely at D6 together with its general caption (Figure 12, page
40).
Again, what makes learning to use the captions difficult in cases like D6 is that
there is not a one-to-one correlation between the representing facts of VENN and the
empty- and not-empty-class-representations. Nevertheless, in view of the broader context
of D6, the student can learn to point to H∩M in D6 and say, “not empty” by direct
conditioning. This is because not-empty-class-representations occur in the diagrams of
VENN with discernable regularity despite the lack of a one-to-one correlation between
them and the representing fact of containing an x-sequence in VENN.
Having said that, it may now seem puzzling that not-empty-class-representations
can occur in the diagrams of VENN without their own corresponding representing facts,
x-sequences to which one can point. First, having an x-sequence in a region is the only
way a diagram can represent a corresponding non-empty class in VENN explicitly.
Second, even though the represented fact (that the set corresponding to the H∩M region
is not empty) is involved in the information content of D6 according to VENN, we cannot
use that fact for the meaning of the compound predicate. Finally, the classification
cannot be justified by claiming that the diagram is also an illustration of an indefinite
proper congruence relation on logical classes of an ideal domain. The classification of
diagrammatic regions in terms of empty- and not-empty-class-representations is,
theoretically speaking, prior to and the basis for comprehending the illustration.
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I suggest that a source of the puzzle is a desire for the meaning of the compound
predicate ‘not-empty-class-representation’. The fact is, however, that to learn to use the
captions adequately the student needs no prior data to work with, and this frees me from
having to posit meanings or ideas for the student to have in mind to meet the initial
objectives in the three phases of instruction. The student’s comprehension of the
diagrammatic illustrations is ultimately evident from the teacher’s empathetic observation
of the student’s overt behavior.
If we must have something more to say to maintain our subjective sense of
purpose—that we are doing something more than shaping a students verbal behavior—
we can take a wider point of view and say that we are helping the student form an
adequate understanding of the logic of categorical syllogisms. Comprehending Venn
diagrams as illustrations is not an end in itself. As I mentioned earlier, the cognitive
value of the extra-logical classification of diagrammatic regions in terms of the
compound predicates is entirely heuristic. The goal is not to comprehend a fictive
illustration of an ideal domain. The goal is for the student to form an initial, partial
understanding of the assessments of VENN. The understanding is tentative. Later, when
the student has grasped the semantics of VENN, she can dispense with treating the
diagrams as illustrations altogether. Until then, she can rely on her partial understanding
to acquire the conviction that the logic gets it right. She will have taken a major step in
her education: the inculcation of the norms of the logic. The puzzle is resolved if one
does not try to make any more of the lesson in comprehending the diagrammatic
illustration than a heuristic exercise to aid the student’s formation of a logical intuition.
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So once again, with the right set of prompts, the student can use her skill at seeing
that something is the case to help select information content from her comprehension of
the diagrammatic illustration. The teacher’s evidence that the student has used her
perceptual skill “correctly” consists of her empathetic observations and the student’s
claims “I see that D∩−M is empty,” “I see that H∩D is not empty,” and “I see that H∩M
is not empty.”
With some additional prompts, the student can ultimately tell the teacher what the
diagram shows regarding the validity of the Boolean formulation. The teacher need only
conjecture that the habits the student has learned for using the captions are guiding their
use in referring to the subsets of the ideal domain illustrated. The teacher can then
conclude that the student has achieved an understanding of the validity of the Boolean
formulation of a categorical syllogism. The teacher can basis this judgment on her
conjecture that there is a sense in which the student understands that the Boolean
formulation is valid by seeing what the diagram illustrates.
Conclusion
There is a sense in which diagrammatic illustrations enable us to see what they
illustrate. This is why we use them instead of descriptions. They are especially helpful
in introducing students to technical vocabularies. The pedagogical theory I have
employed in this paper states that captions can aid the learning of language. I maintain
that treating Venn diagrams as captioned diagrammatic illustrations can aid the learning
of the Boolean logic of categorical syllogisms. I have used the theory to justify the
pedagogical use of the diagrams of VENN alongside their logical use in an introduction
to logic. It assures the logic teacher there is a sense in which Venn diagrams enable
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students to claim to see that the Boolean formulations of categorical syllogisms are valid
even when there is no visual proof.
Talk of ascriptions of de dicto perceptual intuitions has played a major role in my
discussion of both the logical and the pedagogical use of Venn diagrams. I have claimed
that the student’s explicit self-ascribed perceptual intuitions involved in the pedagogical
use of the diagrams have factual content and that it is distinct from the factual content of
the perceptual intuitions appropriately involved in the logical use of the diagrams. I have
also claimed that the ascriptions of perceptual intuitions of the diagrams differ in quality
for the teacher and the student, and that some of them have a common factual content. I
have explained the difference in quality to be the result of the student’s lack of recourse
to the compound predicates that form the basis of the teacher’s use of captions.
I admit to using the intensional idiom of de dicto perceptual intuition only to
systematize a student’s demonstrated ability to judge correctly the validity and invalidity
of Boolean formulations of categorical syllogisms from diagrams like D3 and D6 without
the aid of the VENN rules for transforming diagrams or a clear and distinct understanding
of the semantics of VENN. I will acknowledge that the perceptual intuitions associated
with the behavior may ultimately come down to neural events, but I am under no delusion
that I have provided any clarity about the neural events. Nor have I speculated how the
diagrams might look to the teacher and the student. By recourse to the extensions of the
compound predicates, I have described a factual content for the explicit ascriptions of
perceptual intuitions that can ground the pedagogical use of Venn diagrams, and I have
thereby avoided any reference to the mysterious realm of introspection. Moreover,
nothing I have said entails that we can have perceptions of classes or sets.
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I do maintain that both VENN and a pedagogical theory along the lines I have put
forward are required to secure enough benefits from teaching Venn diagrams to justify
their use in an introduction to logic. Let me list them.
First, the teacher can introduce students to proofs of validity without teaching
them a formal deductive system: students can rely on their self-ascribed perceptual
intuitions of diagrammatic syntax to make perceptual inferences that can prove the
validity of some categorical syllogisms. This will work for only one third of the valid
forms of categorical syllogisms. Second, the teacher can use the diagrams as illustrations
to help students understand the logic without teaching them a model-theoretic semantics:
students can rely on their self-ascribed perceptual intuitions of the diagrammatic
illustrations to form a tentative understanding of the validity and invalidity of the
syllogistic forms. Third, students get a complete lesson on the norms of the modern logic
of categorical syllogism. The tentative understanding the diagrammatic illustrations
provide can convince the student that the logic’s evaluations of categorical syllogisms are
right in every case, even when perceptual inferences are not possible and the student’s
pre-formal “logical intuition” is either incorrect or absent.
On balance, some of the recent theoretical developments in the logical use of
perceptual intuition can improve the logic curriculum. VENN can help the logic teacher
reform the standard practice of blurring the logical and pedagogical roles of Venn
diagrams in the logic curriculum. The pedagogical use of Venn diagrams in the
curriculum does however remain intact, and it should continue. Here is what Venn (p,
138) said about his diagrams in this regard:
That letters, or some such symbols, are the truly appropriate
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instruments of calculation must of course be maintained. . . .
But I shall make frequent appeal to diagrams also; both for
purposes of mere illustration, and because they will occasionally
afford much briefer modes of proof.
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