| 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. |
One 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.
Your 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) and B(x)." |
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 | The 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.
Note 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)."
The 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. |
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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.
Your 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).
Thus, 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. |
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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?
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The 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 gray rectangle.
Now 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 premises. |  |
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| 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 |
| The 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 to prove. |
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| 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. |
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