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In previous lessons, I discussed how circles can represent classes, how two-circle Venn diagrams can represent categorical statements and be used to test such statements for equivalence and contradiction. Of course, two-circle Venn diagrams can only represent logical relations between two classes; so, to handle reasoning involving a third class, you must add a third circle. Three-circle Venn diagrams―or "pretzels", as I like to call them―can be used to represent the logical relations between three classes, and test arguments involving three classes, notably categorical syllogisms, for validity.1
If you've studied the lessons covering the topics mentioned in the previous paragraph, or in some other way learned how to use Venn diagrams to evaluate categorical syllogisms2, you have every right to take pride but, before you get too arrogant, what would you do with the following argument?
Now, you might say, quite correctly: "That's not a syllogism! It has three premisses instead of two. It has four class terms rather than three. I don't know how to diagram four class terms on a three-circle diagram!" Yet, it's an obviously valid argument and all of the statements making it up are categorical statements, so it seems a shame not to know what to do with it.
Your first idea might be to add another circle to the Venn diagram, but you can't do that. You may remember from a previous lesson3 that adding a circle to a Venn diagram should double the number of areas that the circles divide the space into so that every logically possible subclass will be represented on the diagram. So, a Venn diagram with four circles should have sixteen areas for subclasses. Unfortunately, it's not possible to add a fourth circle to the diagram in such a way as to yield sixteen areas4.
However, it is possible to create diagrams with the right number of subdivisions for any number of class terms5, but using shapes other than, or in addition to, circles. However, the more classes involved, the harder it is to correctly draw such a diagram freehand. Moreover, though the diagramming process is an extension of that for three-circle Venn diagrams, the more complex and tangled the diagrams get, the more difficult it becomes to diagram the premisses correctly. So, if you wish to use the diagrammatic method to evaluate arguments such as the above example, it will help to make copies of the appropriate diagrams6.
There's another way to show that arguments such as the example above that are really two syllogisms in one are valid using only pretzels. Such arguments are called "polysyllogisms", meaning "many"―"poly-", as in "polytheism"―syllogisms. They're also sometimes called "sorites"7―which means "heap" in ancient Greek―because they're a heap of syllogisms.
More specifically, you can reconstruct the example as a chain of two syllogisms where each syllogism is a link in the chain. What links the two syllogisms is the missing conclusion: All sapsuckers are birds. Adding that conclusion to the argument gives:
This is clearly a sequence or chain of syllogisms where the conclusion of the first syllogism―"all sapsuckers are birds"―is a premiss in the last syllogism. Such unexpressed statements in a polysyllogism that serve as both conclusion and premiss are called "intermediate conclusions". The conclusion of the last syllogism in the chain―in this case, "all sapsuckers are animals"―is called the final conclusion to distinguish it from intermediate conclusions.
Both of the arguments making up this chain are syllogisms, so that you can test them individually for validity using pretzels. Chains of arguments are like metal chains in that they are only as strong as their weakest link. In other words, if even one link in the chain is invalid, then the entire chain breaks. In contrast, if every link in a polysyllogism is valid, then the original argument is valid. Since both of the arguments in the polysyllogism above are valid, the example is itself valid.
Unfortunately, there's a confusing subtlety here: if every sub-argument in a polysyllogism is valid then the polysyllogism as a whole is valid; but it's not the case that if some sub-argument in a polysyllogism is invalid that the polysyllogism itself must be invalid―the most that you can conclude is that it has not been shown to be valid. If you want to show that it's invalid you'll have to use a diagram, for if you correctly diagram the premisses of a polysyllogism and the resulting diagram does not show the conclusion to be true, then the polysyllogism is invalid.
Example:
The above polysyllogism should seem intuitively valid, but let's see how to show that it is by breaking it down into a chain of simple syllogisms. Since there are three premisses containing four class terms―"flickers", "woodpeckers", "birds", and "mammals"―this should be broken down into two syllogisms connected by an intermediate conclusion.
To find which premisses to combine, look for two statements with a common middle term. So, the first two premisses will not work since there are four terms between them, but the first and third premiss share the term "woodpeckers". Here's the resulting syllogism:
This syllogism is an instance of Barbara and, therefore, valid. Now, to complete the chain we take the intermediate conclusion―"all flickers are birds"―and combine it with the remaining premiss and final conclusion:
You can show that this syllogism is valid by a Venn diagram or any other technique you prefer. This shows that the original polysyllogism is valid.
Exercises: To practice proving polysyllogisms valid, use the same technique demonstrated in the above Example on the following arguments. The chains of syllogisms given in the Answers are not necessarily the only way to show the polysyllogisms valid, so yours may differ.
I.
II.
Notes:
Columnist Roger Harris offers an intriguing distinction between two types of movies: Country movies, which are not necessarily about the country, stress the old-time country virtues in an old-fashioned country way: you are never left in doubt as to who are the "good guys" or the "bad guys," and both always get their just deserts (he spells it "desserts," but no matter).1
Initially, I wrote the following sentence in a puzzle a few years ago: "Just planning what kind of pies to bake for desert is a puzzle." Certainly, figuring out what pie to make for a desert is quite a puzzle: cactus pie2, perhaps? Of course, I meant "dessert", not "desert".
While spelled almost the same, "desert" and "dessert" are pronounced quite differently, with "desert" accented on the first syllable while "dessert" is accented on the second. "Desert" is a noun that refers to an arid area of land3 whereas the noun "dessert" usually means a sweet dish eaten as the last course of a meal4. It's hard to remember which pronunciation goes with which number of esses, since there's nothing about one "s" or two that indicates which syllable should be stressed. It's enough to make you wish that English had accent marks as some other languages do, such as French.
Making matters more difficult is the verb "to desert", meaning to leave without permission, which is spelled exactly like the noun for a dry region but pronounced like "dessert". Also spelled and pronounced identically is the noun "desert", which is related to a different verb, "to deserve", and refers to a state of entitlement or what one deserves. In this sense, it is usually plural and most often found in the phrase "just deserts". When we say that someone got his or her just deserts, we mean that they received what they deserved, which is not necessarily a dessert5―in fact, it's often a bad thing.
Luckily, I recognized the misspelling in the puzzle before posting it. I suspect that misspellings of these words must be common, yet only two of my reference books mention the error6. To test whether spelling and grammar checking programs would have caught the mistake, I tried out the misspelled sentence in a few free online ones, and two out of three did indeed correct it.
I asked ChatGPT to check the following, more difficult example: "I will dessert my family, move to the dessert, subsist on just deserts, and get my just desserts." Here is how it responded:
You're close, but there are a couple of words that need tweaking for clarity:"I will desert my family, move to the desert, subsist on just desserts, and get my just desserts."
Here's why:
- Desert (verb): to abandon.
- Desert (noun): a barren, sandy area.
- Dessert (noun): a sweet course typically eaten at the end of a meal.
- Deserts (noun): refers to what one deserves, as in "getting what you deserve."
Let me know if you need further explanation!7
You're also close, ChatGPT, but you just missed the last misspelling.
Notes: