Department of Philosophy (PHI)

Using Venn Diagrams to Test Arguments for Validity

An argument that consists of categorical statements can be tested for validity using a Venn diagram (provided that there are not too many different terms in the statements).

The method.Diagram the premises together, in a single diagram. Then check to see if this premise diagram contains all of the information the conclusion contains. If it does, the argument is valid. If not, the argument is invalid.

How to diagram the premises together. FIRST, construct a blank diagram with enough circles (normally two or three) for all of the independent subject and predicate terms in the entire argument. For one example, the argument 'all Fs are Gs; all Gs are Hs; therefore all Fs are Hs' has three independent terms, F, G, and H, and requires a three-circle diagram. For another, the argument 'no Fs are non-Gs; no Gs are non-Hs; therefore no Fs are non-Hs' also has only three independent terms. Although 'Gs' and 'non-Gs' are not the same term, they are not independent of one another. A diagram that has a circle representing Gs will automatically represent non-Gs by the points outside that circle.

NEXT, diagram each premise exactly as you would if you were doing it alone. If two different premises require X-entries, put in two separate X-entries. Do not join Xs entered for one statement with Xs entered for another statement, even if both entries have an X in the same cell. If one premise requires an X in a cell that must be shaded for another premise, go ahead and put both the X and the shading in that cell. Here are some examples of statements diagrammed together:

All Fs are Gs All Gs are Hs	Some Fs are Gs All Gs are Hs	Some Fs are Gs Some Gs are Hs

How to see whether the conclusion's information is in the premise diagram. Broadly speaking, there are two approaches to determining whether the premise diagram contains all of the information that the conclusion contains. One approach is direct, the other indirect. The indirect method is simpler, but I will discuss the more popular direct approach first.

The direct method.

The direct approach involves examining the premise diagram to see whether the conclusion is "in there" either explicitly or implicitly. Sometimes this is easily done, as in the case of the argument form called BARBARA:

PREMISES All Fs are Gs All Gs are Hs	CONCLUSION Hence, All Fs are Hs	"BARBARA" (valid)
		The two cells that represent Fs that are not Hs would be shaded to diagram the conclusion: both are shaded in diagramming the premises, hence the conclusion is already "in there" once the premises are diagrammed, and the argument is valid.

This approach also works nicely for other arguments consisting entirely of universal statements, as in the case of the following invalid form:

PREMISES No Fs are Gs No Gs are Hs	CONCLUSION Hence, No Fs are Hs	"BARBARA" (invalid)
		The two cells that represent Fs that are Hs would be shaded for the conclusion. Only one of them is shaded in diagramming the premises, so the conclusion is not "in the diagram" once the premises are entered, and the argument is not valid.

The direct approach also works for the form known as "DARII" (below), though a bit more reasoning is required. Although there is nothing in the premise diagram that exactly resembles the diagram entry for the conclusion, the conclusion is implicitly present:

PREMISES Some Fs are Gs All Gs are Hs	CONCLUSION Hence, Some Fs are Hs	"DARII" (valid)
		The X-entry in the premise diagram says that there exists something that is either F and G and not H, or F and G and H. The shading narrows that down to something that is F and G and H. This is one of the alternatives that fits the conclusion, so the conclusion is implicitly "in there".

As this last example shows, the direct method sometimes requires us to do some reasoning about the diagram to arrive at a verdict of valid or invalid. Here is another example:

PREMISES Some Fs are Gs Some Gs are Hs	CONCLUSION Hence, Some Fs are Hs	(invalid)
		If there is something that is F and G and not H, and something that is G and H and not F, then both premises are true. Obviously this is compatible with there being nothing that is F and H, hence with the conclusion's being false. That is obvious, isn't it?

There is one more complication that must be noted here. Technically, any argument whose premises are inconsistent (cannot all be true) is a valid argument no matter what the conclusion says. This is because an argument that can't have all true premises can't very well have all true premises AND a false conclusion. Thus any resemblance between the premise diagram and a diagram for the conclusion is coincidental and irrelevant when the premises are inconsistent. Here is an example:

PREMISES Only Fs are Gs Not all Gs are Fs	CONCLUSION Hence, Some Fs are Hs	(weird, but valid)
		The premises are inconsistent (note how the X-entry is completely shaded over). Thus, even though the premise diagram bears no resemblance to the conclusion diagram, the argument is valid (cannot have true premises and a false conclusion).

So in order to use the direct method we must first check for inconsistency in the premise diagram: only if the premises are consistent will it be necessary or illuminating to compare the premise diagram with the diagram we would draw for the conclusion.

Fortunately, it is very easy to see whether a diagram is inconsistent or not. A Venn diagram is inconsistent if, and only if, it contains an X-entry that is entirely shaded over. This could be a lone X (not connected to any others) lying in a shaded cell, or it could be an entire group of connected Xs that are all in shaded cells. If there are two or more separate X-entries in a diagram, only one needs to be shaded over in order for the diagram to be inconsistent. Here are some examples:

CONSISTENT: No X-entry is entirely shaded over.
INCONSISTENT: At least one X-entry is entirely shaded over.

Next we will turn to the indirect approach to determining whether the conclusion is "in" the premise diagram. This is an alternative to the direct approach that has occupied most of the last four pages of this text. As you will see, the indirect approach is much simpler.

The Indirect Method. Add the negation of the conclusion to the diagram for the premises. Then check the diagram for inconsistency. The argument is valid if this diagram is inconsistent. The argument is invalid if this diagram is consistent.

The rationale is simplicity itself. If the argument is valid, the conclusion is already explicitly or implicitly in the diagram for the premises, so that its negation will directly contradict part of what is in the diagram. If the argument is invalid, no such contradiction will arise.

How to add the negation of the conclusion. When you diagram the negation of any categorical statement, you wind up marking the same cells you would mark for the statement itself, but with the opposite kind of mark: connected Xs instead of shading, or shading instead of connected Xs. If the original statement says that things of a certain kind do exist, its negation says that things of that same kind do not exist, and vice versa. For example, the diagram for 'not all Fs are Hs', the negation of 'all Fs are Hs', has connected Xs in the same cells you would shade for 'all Fs are Hs', namely, the cells for Fs that are not Hs.

An even simpler way of understanding the indirect approach is to view the diagram as saying, in effect, that the argument has true premises and a false conclusion. Since this is possible if the argument is invalid, and impossible if the argument is valid, the diagram will be consistent if the argument is invalid, and inconsistent if the argument is valid.

Examples of the indirect approach, in which the premises and the negated conclusion (stated in parentheses after the argument) are diagrammed together:

All Fs are Gs All Gs are Hs Hence all Fs are Hs (Not all Fs are Hs)	Some Fs are Gs All Gs are Hs So some Fs are Hs (No Fs are Hs)	No Fs are Gs No Gs are Hs Ergo no Fs are Hs (Some Fs are Hs)	Some Fs are Gs Some Gs are Hs Thus some Fs are Hs (No Fs are Hs)
Inconsistent diagram: argument is valid.	Inconsistent diagram: argument is valid.	Consistent diagram: argument is invalid.	Consistent diagram: argument is invalid.

Notice, finally, that when we use the indirect approach, validity shows up in the same way in a diagram--as the shading over of an entire X-entry-- regardless of whether the conclusion is universal or existential, and regardless of whether the premises are consistent or inconsistent.