Wednesday, November 17, 2010

A logic problem considered

This morning, the Maverick Philosopher posed a logic problem, and I decided to go all out and make a Venn Diagram...and why not, it's Wednesday?

The Argument
1. A necessary truth is true.
2. Whatever is true is possibly true.
3. Whatever is possibly true could be false.
4. A necessary truth could be false.

The Problem
There appears to be two ways to interpret "possibly true" in this argument.  Here are the two ways:

Possibly true - not impossible (the entire area of the left circle on the chart)
Possibly true' - not impossible and not necessarily true (the darkly shaded middle area).  I believe the term "contingent" would apply here...not necessarily false and not necessarily true.  It could be either true or false in some possible worlds.  This, I believe, is the everyday use of the term.

Possibly True
We can see from the chart that premise 3 is painfully false if we interpret possible in the first sense.  The left circle clearly has areas that don't overlap the right circle, so therefore there are possible truths which couldn't be false (again, think of the definition of necessary truth).  I didn't label necessary truth, but it is area of the left circle that isn't shaded dark.

Possibly True'
On the other hand, if we use the other meaning of possible ("not impossible and not necessary"), then premise 3 is true but now premise 2 is false, since "whatever is true" is not the same as the darkly shaded middle.  Clearly "whatever is true" isn't confined to contingent truth as shown below.  Of course, on this account of "possibly true" what should we label the left circle on the diagram?  I don't know either, which is why I made one diagram and labeled the conjunction as Contingent.

Can this argument be considered fallacious on the account of equivocation?  After all, there were two possible uses of the term "possibly true."  I believe that it can, though I am more inclined to simply say that premise 3 is false.  Perhaps the type of argument will dictate which approach is the preferred response.  Many times (in more complex arguments) we may not be able to show that a premise is false, but we can show that the argument doesn't work...the beauty of critical thinking if you ask me.

Let's assume the arguer is guilty of equivocation, I'm not 100% certain whether to categorize it as formal or informal.  Usually equivocation is labeled informal, though I'm not exactly sure why.  Perhaps because it involves the meaning of language and not merely an incorrect form?  Actually Wikipedia claims that it is both informal and formal, but who really takes their opinion seriously.

Equivocation Fallacy: Formal, Informal or Both?
1. Fans make a lot of noise
2. My wife uses a fan
3. Therefore, my wife makes a lot of noise

Now, if we take the first premise to be referring to rowdy sports fanatics, then clearly there is a problem with the argument.  But what exactly is the problem?  It is true that sports fanatics make a lot of noise, and it's true that my wife uses a fan.  The problem is those premises are irrelevant to the truth of the conclusion.  They offer no support for it.  I have a hard time believing this fallacy occurs very often in actual arguments.  Can any anyone think of some real examples, especially ones that aren't syllogistic?  In addition to being equivocal, it seems the previous example is an invalid syllogism with four terms (another fallacy).

Here's a modus ponens rendition:
1. If you catch a bug, you will need to take a sick day
2. You caught a bug [in a jar]
3. Therefore, you will need to take a sick day

The argument doesn't work, even though premise 2 might be true in the sense that you caught a bug in a jar.

So wouldn't this be a formal fallacy?  Technically not, since we had to establish the alternate meaning before we could determine that it was invalid.  In other words, we couldn't show the argument to be invalid by merely analyzing the form.  Only after we have distinguished the ambiguity, can show the form to be invalid:

If a, then b
therefore, b

So equivocations can only be shown invalid once you notate the ambiguous term separately.  That's interesting (to me at least), because originally I assumed that equivocations were only formally invalid as syllogisms.  Then my little modus ponens example came to mind.

Am I missing something?  Feel free to correct me in the comments box.

Dr.Vallicella showed that equivocation (in deductive arguments) does indeed induce a formal fallacy.


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