Exclusive help

Unit 4: Arguments in Normal English in my Introductory Logic text is a difficult section, primarily because of the ambiguities within English. But if we want to be able to apply the tools for analyzing syllogisms to everyday arguments, it is essential that we understand it.

One of the more difficult parts of this difficult section ideals with translating exclusive statements into categorical form. Exclusives are statements that exclude all or part of the predicate of the subject, statements that use words like only, unless, except. Let me give some suggestions that may help.

Exclusives using “only”

Consider the statement “Only women are mothers.” This statement is true. It does not mean “All women are mothers,” which is false, but rather “All mothers are women.” This leads us to this basic rule:

Only S is P = All P is S

Thus, “Only jet airplanes are supersonic” translates into “All supersonic airplanes are jets.”

“Nothing but” or “nobody but” acts the same way. The statement “Nothing but the best is good enough for my son” is best translated “All things good enough for my son are the best things.”

Exclusives using “unless”

Consider the statement “Our eyes always see unless we are unconscious.” This means that if we are not unconscious, then our eyes are seeing, or better “All times we are not unconscious are times our eyes see.” This leads us to this rule:

P unless Q = All non-Q is P

So the statement “He wins the game unless he makes a mistake” is translated “All games in which he makes no mistake are games he wins.”

If the “unless” starts the sentence, it works something like this:

Unless P, Q = All non-P is Q

So “Unless you try you are guaranteed to fail” is translated “All times you do not try are times you are guaranteed to fail.”

Exclusives using “except”

Consider this statement: “Everyone is a sinner except Jesus.” This statement clearly implies or contains this statement: “All people who are not Jesus are sinners.” But it also implies “Jesus is not a sinner,” or in categorical form, “No Jesus is a sinner.” This leads us to this complex rule:

All S is P except Q = All non-Q S is P

All S is P except Q ≈ No Q is P

This second rule is an approximately equal to, because it may not always be the case. Consider the statement “All creatures died except those on Noah’s ark.” This implies “All creatures not on Noah’s ark were creatures that died.” But it does not really imply that “No creatures on Noah’s ark were creatures that died,” because a creature (such as one of the many beetles) could have died in the ark. So with this second rule, you simply need to use common sense to see if it applies.

The above rules will be modified if negative. For example, consider this translation sequence:

“No one can come to me unless the Father draws him” (John 6:44) =

No people not drawn by the Father are people who can come to Jesus =

All people who can come to Jesus are people drawn by the Father.

Following the above rules, we can even turn this back into another exclusive:

Only people who are drawn by the Father are people who can come to Jesus.

Exclusive statements are common in English, so the logic student should study them assiduously in order to analyze them as used in arguments. But it does take extra mental effort to translate them into categorical form correctly.

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