# Logic exercise (evaluation of whether an argument is valid)

Interested in others thoughts on the following logic exercise (from Peter Smith's Introduction to Formal Logic, 2nd ed., available for free at https://www.logicmatters.net).

Here are the premises: "Only logicians are wise. Some philosophers are not logicians. All who love Aristotle are wise. Hence some of those who don't love Aristotle are still philosophers."

The question is whether the argument is valid. In my opinion it is not because the premises don't support any conclusion about whether "some of those who don't love Aristotle" are or are not philosophers. I think the premises would permit non-Aristotle-lovers to be philosophers, but we can't conclude that some (or any) non-Aristotle-lovers are, in fact, philosophers.

Any thoughts would be welcome.

Well, break it down a bit.

You know that everyone who is wise is a logician. You know that only the wise love Aristotle. Therefore everyone who loves Aristotle is a logician. There exists at least one philosopher who is not a logician, and therefore does not love Aristotle (because only logicians love Aristotle).

If we must get all formal:

1. To be Wise, one must be a logician (W -> L) [given]
2. To love Aristotle, one must be wise (A -> W) [given]
3. To love Aristotle, one must be a logician. (A -> L) [syllogism]
4. If one is not a logician, one does not love Aristotle. (~L -> ~A) [modus tollens, contrapositive]
5. There exists a subset of philosophers (P1) who are not logicians. (P1 -> ~L) [given]
6. These philosophers do not love Aristotle (P1 -> ~A) [syllogism] QED

The only part of the proof that isn't a syllogism or statement of the premises is (4), which is the contrapositive of (3).

To answer the question directly: YES the argument is valid. That is if the premises are true the conclusion must also be true.

I will show a formal proof system called the Aristotelian Syllogistic to show the argument is valid. Note I am not using Mathematical Logic which is a distinct logic system.

I will use some short cuts here to save some writing time. I will post a key of abbreviations I will use for all to follow along.

A = people who love Aristotle

L = people who are logicians

W = people who are wise

P = people who are philosophers

The premises written out in standard categorical form (using the letter short cuts) are as follows:

1. All W are L. [Only s are p equals the converse: All p are s; so we convert the L & W as you see in the first premise]

2. Some P are not L.

3. All A are W ----------- / therefore Some non-A are P [notice the conclusion is an I proposition not an O proposition.]

``````Derivation        Line # used           Justification / Rules used
``````
1. All A are L -------------- 1, 3 -------------- Barbara (Valid syllogism 1 figure)

2. Some P are not W ------1, 2 -------------- Baroco (Valid syllogism 2 figure)

3. All non-L are non-A ----- 4 --------------- Contraposition

4. Some P are non-L ------- 6 ----------------Obversion

5. Some P are non-A ------6,7 ---------------Darii (Valid syllogism 1 figure)

6. Some non-A are P ------- 8 ----------------Conversion
QED

• My Aristotelian logic was a bit rusty, but I've re-acquainted myself; this is wonderfully clear and thanks for taking the time to set it out so thoroughly. Jul 8, 2022 at 17:19

Here are the premises: "Only logicians are wise. Some philosophers are not logicians. All who love Aristotle are wise. Hence some of those who don't love Aristotle are still philosophers."

I think you are correct. I got the reasoning to work only by adding a premise about non-Aristotle-lovers. I could not decide what the middle term might be.

One overall problem is that the argument has three premises and a conclusion. A valid syllogism has exactly two premises. This syllogism, as written, contains the fallacy of four terms.

Next, the opening statement is actually two premises: All Logicians are Wise, and All Wise people are Logicians.

Continuing:

Some Philosophers are not Logicians.

Some who do not love Aristotle are not Wise.[unspoken. necessary for conclusion.]

Thus: Some who do not love Aristotle are Philosophers.

Semi-Q.E.D.

• The opening statement does not contain two premises. "Only logicians are wise" is not the same as the statement "ALL logicians are wise". We don't need to add a statement, because we know that W -> L, A -> W, therefore A (loving aristotle) implies that one must be a logician (A -> L). Therefore ~L -> ~A. There Exists P1 in P such that P1 -> ~L, therefore P1 -> ~A, as was to be shown. Jul 7, 2022 at 21:47
• @philosodad. Thank you. (1) You solved the three-premise problem by dividing the argument in two and making the conclusion of the first part a premise of the second. (2) I disagree that an additional statement is unneeded. I added “some who do not love Aristotle are not Wise.” You added “Therefore ~L -> ~A”. So something had to be added to make the whole thing work. (3) “Only logicians are wise” looks to me as an “if and only if” statement. It is indeed two statements about the world. Jul 7, 2022 at 23:23
• It really is not two statements. If there are 10 wise people in the world, and 20 logicians, and all 10 wise people are logicians, than the staement "Only logicians are wise" holds true, while the statement "All Logicians are wise" is false. The extra statement isn't "added" to the argument. If A -> B, ~B -> ~A. That's just the contrapositive. It isn't added, it's just a restatement of an existing proposition. Jul 7, 2022 at 23:58
• Thanks for all the input. The exercise would allow the addition of statements if such statements are logically entailed by the premises. With the addition of premises logically entailed by what's given, I do think this can be shown to be valid -- in the sense that if the premises are true, the conclusion must be true. Thanks again. Jul 8, 2022 at 0:12