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We homeschool our kids, and "The Great Courses" offers a lot of good intro college classes that are suitable for 6-12th grade as well. I'm really not into math or logic so please forgive me for the simpleness here. The class is pretty straightforward. In one of the lectures of Categorical Logic, the professor gives 5 rules to ensure a syllogism's validity (apparently to avoid looking up 256 different moods and figures)
The Rules:
1)The middle term is distributed in at least one of the premises.
2) Any term distributed in the conclusion is also distributed in the premises.
3) At least one of the premises must be affirmative.
4) If the conclusion is negative at least one premise must be negative.
5) (Hypothetical mood only) If the conclusion is particular at least one premise must be particular.

Ok, seems straight forward enough (but we're still learning). To give an example we came up with a simple argument:

  1. No American is immortal.
  2. All Idahoans are Americans.
  3. Therefore no Idahoans are immortal.

I applied all the rules to this and it checked out.

  1. American (MT) distributed in P1.
  2. Idahoans is distributed in conclusion and P2.
  3. P2 is affirmative
  4. Conclusion is negative and P2 is negative
  5. Is not applicable bc our syllogism is existential, not hypothetical, and the conclusion is not particular anyway.

    -All of the rules checked out. Fair enough.

So, one of the kids says, "it looks to me that if you change the conclusion to 'therefore ALL Idahoans are immortal' all of the rules still check out". Well, that's OBVIOUSLY not a valid argument, but it seems it passes all the rules regardless. Am I misunderstanding one of these rules, if so, which one?
I did a Venn Diagram (with minor, middle and major term) and verified that it's invalid.
Either I don't understand one of the rules, or the list provided is not complete. Any explanation would be extremely appreciated.
I'm looking like a fool in front of my kids, so thanks in advance!

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    FWIW, this kind of logic hasn't been taught much of anywhere except history of philosophy and maybe Catholic seminaries for about 150 years. Not that there is any harm in learning it, but if you think you are preparing your kids for college work with this material, you aren't. Aug 16, 2023 at 19:07
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    I agree with David G. What you are doing here is Aristotelian logic, which has been rendered obsolete by more modern logics. There is still some value in it, but only as a precursor for going on to do propositional and predicate logic. If you want an elementary introduction to formal logic, with some but not too much in the way of symbols, I suggest looking at forallx.openlogicproject.org
    – Bumble
    Aug 16, 2023 at 20:45
  • A bad solution to an equation does not imply that the equation is invalid. The five rules validate the logical structure of the proposition set, not its falsehood or truthfulness.
    – RodolfoAP
    Aug 17, 2023 at 13:28
  • None of you people answered the question. Aristotelian logic is not "obsolete" lol. I took logic in a major university (a LONG time ago), and yes they taught Aristotelian logic, as well as propositional truth functional logic. I perform truth functional logic all day long for my job (computer science). Aristotelian categorical logic is every bit as valid as any other (but I also am not as familiar). If you don't know the answer, then I'd suggest no response.
    – Godfrey
    Sep 14, 2023 at 22:32
  • This is the comments section, we're not trying to answer the question, but comment on it.
    – Bumble
    Sep 16, 2023 at 13:09

2 Answers 2

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Muchas gracias for the question. I shall skip directly to the sticking point, which is the argument below:

  1. No Americans are Immortals
  2. All Idahoans are Americans
    Ergo,
  3. All Idahoans are Immortals

I have a different set of conditions that categorical syllogisms must meet to qualify as valid and they are:

  1. Distributed Middle Term Test: The middle term must be distributed in the premises. The middle term Americans is distributed in premise 1. Check!

  2. Distributed Conclusion Test: The terms distributed in the conclusion must be distributed in the premises. The term Idahoans is distributed in the conclusion and in premise 2. Check!

  3. Equal Quantity Test: A particular statement can't be inferred from universal statements and a universal statement can't be inferred from particular statements. The conclusion is universal (All Idahoans are immortals) and there are no particular statements in the premises. Check!

  4. Equal Negatives Test: The number of negative statements in the conclusion must be equal to the number of negative statements in the premises. Here the argument fails because ... the number of negative statements in the conclusion is 0 and there's 1 negative statement in the premises, viz. premise 1. We know 1 does not equal 0. The argument, therefore, is invalid.

I have a feeling you can map the rules I have onto the rules you have to test for validity of categorical syllogisms.

Hope that helps.

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    Thanks so much for answering my question! I used a Venn Diagram (which I had already known what the outcome would be). It's clear to me that the professor of this online university course gave incomplete standards. Thanks for yours, I think they work better! My sons syllogism was OBVIOUSLY wrong, but the profs rules didn't catch it. Wish I could give you an upvote, but it won't let me bc apparently I don't have the qualifications.
    – Godfrey
    Sep 14, 2023 at 22:38
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Am I misunderstanding one of these rules, if so, which one?

The problem is the incompleteness of rule 4. Rule 4 covers only the scenario "conclusion is negative", thereby leaving the other scenario(s) undefined.

The kid's syllogism fits the undefined scenario (i.e., conclusion is affirmative). In other words, the syllogism goes beyond the scope of rule 4. The premise in rule 4 is a non-sequitur in regard to the kid's syllogism. That non-sequitur precludes a validation [under the rules as stated] of the syllogism.

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    Thanks so much for answering the question, I agree that his rule is probably incomplete. It's clearly not a valid syllogism, and seems similar in nature to negating the antecedent in regular conditional logic. Thanks for your response!
    – Godfrey
    Sep 14, 2023 at 22:44

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