Following is a dictionary definition of a sorites:
A form of argument in which a series of incomplete syllogisms is so arranged that the predicate of each premise forms the subject of the next until the subject of the first is joined with the predicate of the last in the conclusion.
Syllogisms are incomplete because only premises are present - conclusion is not. If more than two premises are given, a pair must be selected and a conclusion drawn such that combined with another premise it will yield another conclusion and so on until all premises have been employed. The last conclusion is the statement to look for.
I'll pick a few original examples from Lewis Carroll's Symbolic Logic.
- All my sons are slim.
- No child of mine is healthy who takes no exercise.
- All gluttons, who are children of mine, are fat.
- No daughter of mine takes any exercise.
As with syllogism, we first have to determine the universe of discourse - a set that encompasses all the objects mentioned in the given statements. "My children" is suitable for this example. Elements of this set may or may not have the following attributes:
- a - fat (i.e., being fat)
- b - glutton
- c - healthy
- d - sons
- e - taking exercise
Note that implicitly d' means "being a daughter" while a' denotes "slim", and so on. The four statements above are reformulated in the abstract form as
- All d are a'
- No e' are c
- All b are a
- No d' are e
We have to find a pair of premises that may lead to a conclusion. This implies that the two selected statements should include like attributes - candidates for elimination. Consider 1 and 3 with a and a' being eliminands. You may use the trilateral diagram applet to figure things out. From these two we draw a couple of conclusions:
- All d are b'
- All b are d'
Let's now combine #6 with #4. From we conclude
- No b are e
At this point, only #2 remains unused. Combined with #7, it yields
- No b are c
- No c are b
whose interpretation is
- No glutton among my children is healthy
- No healthy offspring of mine is glutton
You may want to try a few more:
- Things sold in the street are of no great value
- Nothing but rubbish can be had for a song
- Eggs of the great Auk are very valuable
- It is only what is sold in the street that is really rubbish
(Universe: "things", a - able to be had for a song, b - eggs of the great Auk, c - rubbish, d - sold in the street, e - very valuable. Conclusion: An egg of the Great Auk is not to be had for a song.)
- Remedies for bleeding, which fail to check it, are a mockery
- Tincture of Calendula is not to be despised
- Remedies, which will check the bleeding when you cut your finger, are useful
- All mock remedies for bleeding are despicable
(Universe: "remedies for bleeding", a - able to check bleeding, b - despicable, c - mockeries, d - Tincture of Calendula, e - useful when you cut your finger. Conclusion: When you cut your finger, you will find Tincture of Calendula useful.)
- No interesting poem are unpopular among people of real taste
- No modern poetry is free from affectation
- All your poems are on the subject of soap-bubbles
- No affected poetry is popular among people of real taste
- No ancient poetry is on the subject of soap-bubbles
(Universe: "poems", a - affected, b - ancient, c - interesting, d - on the subject of soap-bubbles, e - popular among people of real taste, f - written by you. Conclusion: All your poems are uninteresting.)
- Animals, that do not kick, are always unexcitable
- Donkeys have no horns
- A buffalo can always toss one over a gate
- No animals that kick are easy to swallow
- No hornless animal can toss one over a gate
- All animals are excitable, except buffalo
(Universe: "animals", a - able to toss one over a gate, b - buffaloes, c - donkeys, d - easy to swallow, e - excitable, f - horned, g - kicking. Conclusion: Donkeys are not easy to swallow.)