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How to Use Visual Models

The verbal models of argumentation presented here in sections on Conditionals and Syllogisms are one way to explain the workings of logic. Visual models, in which the interaction of the terms of a claim are represented by the intersections of circles, are another. We will be using two different visual models: two movable interacting circles (which we'll call "Veb diagrams") and three fixed intersecting circles (called "Venn diagrams"). Both Veb and Venn diagrams can be used for both conditionals and syllogisms, though some people find it best to use the two circles of Web diagrams for conditonals, where there are two terms, P and Q, and the three circles of Venn diagrams for syllogisms, where there are three terms, X, Y, and Z. Use whichever diagram works best for you in any given application.

1. Models for Four Basic Claims

Claim	Veb Diagram	Venn Diagram
All A are B. --or-- If A then B.
No A is B.
Some A are B.
Some A are not B.

Notice that a conditional argument will always begin with the first Veb or Venn diagram, while a syllogism may begin with any of them.

2. Using Veb Diagrams

Conditionals. You saw above that a conditional premise, "If A then B," can be represented by two concentric circles.

The question is, which circle is the inner one? Consider the conditional, "If one is a professor, then he or she is an educator." Should "professor" be the inner circle and "educator" be the outer circle, or vice versa? Shortening the claim to "If professor then educator" helps make it clear that every professor is an educator. But it says nothing about every educator, some of whom may be professors and some not. And a Veb diagram looks just that way:

Notice how the outer circle, "Educators," contains the inner circle, "Professors," and more. Therefore, every dot within the "Professors" circle is also within the "Educators" circle ("If professor then educator"), only some dots within the "Educators" circle are also within the "Professors" circle. Setting it up correctly is often the most difficult part of using a Veb diagram to analyze an argument.

There are four possible second premises for our conditional, "If professor then educator." They are:

But "Professor" and "Not Educator" both create valid arguments. Click on the Veb diagram above in either of these places, and draw the valid conclusion for that argument.

Syllogisms. Syllogisms can be analyzed with Veb diagrams in much the same way as conditionals are. Just remember that the second premise determines where you must click your mouse. If we redo the above example as a syllogism, for example, "If prof then educator" becomes "All profs are educators." If the second premise is "Nadine is a professor," that instructs you to click ("Nadine") in the inner circle ("Professors"), and since that circle is completely within the outer circle ("Educators"), we can conclude, "Nadine is an educator."

3. Using Venn Diagrams

Conditionals. As in Veb diagrams, the first step in using a Venn diagram for a conditional is recognizing that "If A then B" is equivalent to "All A are B." Once that is clear, conditional arguments can be analyzed with a Venn diagram in the same way syllogisms are analyzed.

Syllogisms. Consider the claim, "All professors are educators." Imagine that circle A contains all professors, and circle B contains all educators:

The claim, "All professors are educators" can be represented by the following diagram:

Here, the circle B still represents "educators," but the only area representing "professors" is now completely within that circle, because "All professors are educators" means that there are no professors who are not educators (that is, not contained in circle B). An educator could be located in either section of circle B (there is no way to tell which from this claim), but a professor must be in the area where circles A and B intersect.

Now let's add a second premise: "Nadine is a professor." If we imagine "Nadine" to be the only thing contained in circle C, and "professors" as still contained in circle A, then the claim "Nadine is a professor" can be represented by the following diagram:

Figuring out the conclusion to this argument, then, becomes the problem of which area(s) of the diagrams overlap. Remember that you have to limit the possibilities for "professors" according to the first premise, and the possibilities for "Nadine" according to the second premise. Put your mouse on the correct area, and click!

"All professors are educators."	"Nadine is a professor."	Click on the section which the intersected areas in the two preceding diagrams have in common to indicate a valid conclusion:

Conclusion: Educator

You clicked at a point (X) in the inner circle which, as the diagram shows, is also within the outer circle, and therefore belongs to the category "Educators," which is the valid conclusion.

Conclusion: Invalid.

You clicked at a point in the outer circle which might have represented either "Educators" or "Not Professors." Neither of these produce valid arguments, because neither is limited to one area of the diagram.

Conclusion: Not Professor

You clicked at a point (X) beyond the outer circle which, as the diagram shows, is also beyond the inner circle, and therefore belongs to the category "Not Professors," the valid conclusion.

Conclusion: Invalid

You clicked on the area where circle C (Nadine) is not intersected by either A or B though only the second premise ("Nadine is a professor") has C colored. The idea is to pick the area colored in both premise diagrams.

Conclusion: Invalid

You clicked on the area where circle A (Professors) is not intersected by either B or C though neither of the premises has A colored. The idea is to pick the area colored in both premise diagrams.

Conclusion: Invalid

You clicked on the area where circle B (Educators) is not intersected by either A or C though only the first premise ("All professors are educators") has B colored. The idea is to pick the area colored in both premise diagrams.

Conclusion: Invalid

You clicked on the area where circles B (Educators) and C (Nadine) overlap, but outside circle A (Professors). Yet B and C are related by their connection with A in the premises, so some part of A must be included in the correct answer.

Conclusion: Invalid

You clicked on the area where circles A (Professors) and C (Nadine) overlap, but outside circle B (Educators). Yet the conclusion must affirm "educators"; there are no negations in either of the premises.

Conclusion: Invalid

You clicked on the area where circles A (Professors) and B (Educators) overlap, but outside circle C (Nadine). Yet the conclusion must affirm "Nadine"; there are no negations in either of the premises.

Conclusion: Nadine is an educator.

Correct! Knowing that Nadine is a professor and all professors are educators, you clicked on the area where circles C (Nadine), A (Professors) and B (Educators) overlap. This allows you to conclude validly that Nadine is an educator.

OK! Now that you understand the use of visual models to analyze arguments, you can go on to exercises in one of the following areas, or return to the main menu:

• Veb Diagrams
  • Conditionals
  • Syllogisms
• Venn Diagrams
  • Conditionals
  • Syllogisms