Euler Diagrams and Quantified
| Euler diagrams, inaccurately referred
to as Venn diagrams in your text, were an informal way to convey
the ideas that are involved in what was subsequently known as
quantification. They are not the best way in which to
think about quantified statements, but the psychological literature
discussed in your text has used them in the study of what they
term categorical syllogisms.
kind of quantification is called universal quantification
and the other existential. The idea behind universal quantification
is that some statements are universally true. For example,
"All triangles have three sides." Consequently, we
don't want to have to write down this statement for each individual
... particularly, if, as is the case with triangles, there are
an infinite number of such individuals. Note that the semantics
for quantified statements is still True or False.
text provides two alternative Euler diagrams for the statement
"All A are B," and claims that this indicates that
the statement is ambiguous. The two alternative Euler diagrams
provided in your text for this universally quantified are shown
in the figure to the right. Beneath each is the quantified expression
that corresponds to the diagram. For example, the one on the
left states that: "for all x whenever A(x) then B(x)"
whereas the one on the right states that: "for all x A(x)
next figure shown on the left illustrates the Euler diagrams
that your text provides for what your text terms a particular
affirmative; namely "Some A are B." This corresponds
to what we referred to above as existential quantification. The
idea here is that there is at least one individual about whom
a statement is true. For example, "Some triangles are equilateral
triangles." Again, we are not being explicit about how many
there are, only that there is at least one.
that there are four Euler diagrams that correspond to what your
text refers to as the Particular Affirmative, namely the statement
"Some A are B." The corresponding existentially quantified
expression is shown at the bottom of the figure and reads: "there
is some x, such that A(x) and B(x)."
next figure on the lower left shows the diagram that corresponds
to what your text refers to as the "Universal Negative,"
namely, the statement "No A are B." Again, the quantified
statements that correspond to this case are shown at the bottom.
The two statements shown are equivalent.
And finally the above figure on the right
shows the diagram that corresponds to what your text refers to
as the "Particular Negative," namely, the statement
"Some A are not B." Again, the quantified statement
that correspond to this case is shown at the bottom.
text refers to these statements; "All A are B, "Some
A are B, "No A are B, "Some A are not B," as 'categorical
propositions.' And your text states on page 118 that "most
of the propositions are ambiguous." By this the author means
that there is more than one Euler diagram that can be associated
with the "categorical proposition." But this presumes
two things. First, that these "categorical propositions"
represent an appropriate syntax for logical expressions; and
second, that Euler Diagrams represent an appropriate semantic
model for these propositions. Neither of these presumptions are
made in modern logic (and, as far as I know, may never have been
made by anyone other than some psychologists who used these syllogisms
in their research).
in making sense of this chapter, it is useful to distinguish
between an English statement, e.g., All A are B; the logical
expression(s) that correspond to the English statement; and the
semantic model(s) that correspond to the logical expression.
Logics are typically constructed so that there is no ambiguity
in the technical sense between a logical expression and the semantic
model that corresponds to the expression.
Now, if we assume that humans
when reading such 'categorical propositions' map them directly
into the set of Euler diagrams and reason in these diagrams,
then the "ambiguity" of this mapping may explain some
of the difficulties that people may have when reasoning
in these "categorical syllogisms."
To illustrate this, we consider
the categorical syllogism:
All B are A
No C are B.
Are Some A not C?
figure to the right illustrates the mapping of the premises into
Euler diagrams (the top three diagrams enclosed in light gray
boxes); and the composition of these possibilities to yield the
four alternatives shown one level down and enclosed by a darker
in order to infer from the premises that "Some A are not
C" this statement must be true in all possible semantic
models of the premises. In this case, there are four such models,
and if you check them visually you can see that the conclusion
is true in each model. Therefore the conclusion follows from
| The figure on the right below shows the
same problem. Here the premises and the conclusion are shown
on the right and expressed in the syntax of first order predicate
logic (FOL). The proof that the conclusion follows from the premises
is shown on the right of the figure. The top line simply restates
the premises as a conjunctive statement. And, since we are not
asked to prove a universal statement, we have replaced the universal
quantification with a single constant, K
next shaded rectangle shows the premises of the rule of inference
known as modus ponens. These premises are directly obtained from
line 1 of the proof. The next lightly shaded rectangle contains
the conclusion of applying modus ponens, namely A(K). The next
line asserts the conjunction of A(K) and premise 2 from line
1. Then since we have proved it for some K, we can substitute
the variable x for K and existentially quantify over the expression.
This is shown in the next line. The last line simply eliminates
the conjunct B(x) from the previous line in order to arrive at
a form that is identical to the conclusion that we were asked
|This proof is an
example of a proof that is carried out in the syntax. Note that
it is a rather simple and straightforward proof.