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Chapter 3: Logical Relations

I was wondering how you would define the relation of "no relation", meaning Independent propositions. Could this be defined as a relation between P and Q such that P neither implies nor is implied by Q?

Something about you (optional) student

Hi Greg.

No - "no logical relation" is a denial of not only of implications either way but also of incompatibility and exhaustiveness. That is, in Table 3.1 there would be four question marks.

However, this is only true at this level of consideration, because even with four question marks there may at a deeper level (i.e. more conditionally, dependent on other factors) a logical relation. This is dealt with in detail in The Logic of Causation.

So perhaps we should speak here of no "direct" logical relation.

Something about you (optional) Logician, philosopher

Thanks Avi. Yes, it seems obvious now you point it out; if two propositions are independent then they can't at the same time be mutually exclusive, or exhaustive.

Something about you (optional) student

But Greg, note well that not-implying, not-implied, compatible and inexhaustive are all logical relations of sorts, just weaker ones.

Let me add, just to be clear, if all four of these are true of two propositions, then you have the nearest thing to "no relation" possible at this level of consideration. In truth, as I said before, there may be some still weaker (conditional) relation at a deeper level.

P.S. you will find more about the possible combinations of logical relations in the chapter about Oppositions.

Something about you (optional) Logician, philosopher

Thanks Avi, noted!

I have just finished reading Chapter 10 and am about to start Modal Categoricals, but regarding Actual Categoricals, do they not include syllogisms involving identities? e.g.:

Socrates was the first western philosopher.

The first western philosopher lived in Greece.

So, Socrates lived in Greece.

Something about you (optional) student

Avi, I was referring to propositions in which both subject and predicate are singulars, like "this A is this B" etc. In section 5.3 you say that R and G are effectively distributive, but "this A is this B" is not a component of an R form (because the predicate is not a category, but a singular).

Perhaps I'm making a difficulty where none exists. It just seemed to me that these forms are syllogisms and yet they're not mentioned.

Something about you (optional) student

Hi Greg. You are quite right to raise the question.

First, let me clarify that when I said that this is a normal first figure syllogism, I meant that it does not matter that the minor premise "Socrates was the first western philosopher" is convertible to "The first western philosopher was Socrates", because the latter converse does not affect the inference of the conclusion you gave. As far as this syllogism is concerned, it is redundant information (as we have seen together in another example).

"The first western philosopher was Socrates" could be used in a third figure syllogism yielding (arguably) the same conclusion. But in that syllogism, the converse "Socrates was the first western philosopher" would be redundant. So, in fact there is not involvement of "identities" here as you suggested. These are ordinary arguments. Note that the fact that this proposition is singular is not the issue here, really; the same could be said with regard to a two-way general proposition like "all humans are rational animals and all rational animals are humans".

Secondly, notice that the middle term is an abstraction "the first western philosopher" - this is effectively a class with only one member rather than a singular term of the type "Socrates". (Incidentally, Socrates was not the first Western philosopher by far - but I assume you knew that.) Of course, we could have a proposition like "Socrates is (the same guy as the one we know as) Socco" - so your question still stands. For Aristotle, the middle term had to be an abstraction, at least in the first figure; but I would agree with you that a singular term is in practice occasionally found the middle term as your example suggests. On the other hand, the third figure allows for a singular middle term (see the moods RRI and GRO in FL chapter 9). So we could argue that your argument works because it is really a third figure syllogism rather than a first figure one.

One more thing parenthetically. I recommend when discussing logic to always place the major premise first and the minor premise second, with the conclusion last - even though in ordinary discourse the order is not important. In this way, you show you always know which proposition is which. How to know? The minor premise is the one with the subject of the conclusion, and the major premise is the one with the predicate of the conclusion.

Something about you (optional) Logician, philosopher

Greg - upon reflection, I'd like to add the following to my preceding comment.

Although the first figure syllogism with a singular minor term (as subject) and a singular middle term is no different from one with a plural middle term, the same is not true of the third figure. In the latter case, the conclusion is not merely particular (as is normally the case for this figure), but more specifically singular. This means that the third figure syllogism with a singular middle term and a singular minor term (as predicate) is indeed special.

So your criticism was correct. I do not remember, frankly, if I dealt with this special case anywhere (do let me know if you come across this issue in FL). If I did not, I ought to have for the sake of completeness. Thank you for your comment.

One other reflection is that the concept "the first western philosopher" is not definitely singular. There may be many simultaneous first western philosophers, in which case of course Socrates (or whoever) would be "among the first". We can only use the term "the first" if we know or believe there is only one first. This confirms my contention that your term "the first western philosopher" is best viewed as a one-member class rather than as a singular term in the usual sense.

Also note, some people claim that Aristotle did not recognize singular syllogism. Here however is an example drawn from his Prior Analytics 2:27 "Wise men [i.e. at least some of them] are good, since Pittacus is not only good but wise". This says: Pittacus is good and Pittacus is wise, therefore [at least some] wise men are good (3rd fig. singular middle term - square brackets mine). This is a normal case, not a special one, note.

Something about you (optional) Logician, philosopher

Hi Avi,

Well I seem to have opened a can of worms here, but actually I think my original criticism (although it wasn't really intended as a criticism) was misplaced, because I was only thinking about the kind of arguments which involve identities between singulars, such as "Mt Everest is the highest mountain", or "Bacon is Shakespeare". But if these are the only kinds of propositions in an argument, it isn't really a categorical syllogism, and so shouldn't be included in part 1 of FL.

Arguments of the form:

a = b

b = c

so, a = c

involve only simple substitution of like for like, so perhaps on this basis can be excluded from the traditional syllogism?

Something about you (optional) student

Hi Greg. There is no can of worms. Every criticism is welcome and may bring to the fore issues that need dealing with, as was the case here. I'm grateful. I almost always learn something from criticism. Always be critical!

That said, I think I have answered your objection correctly earlier.

You bring up now the issue of substitutive syllogism. Here I refer you to FL 19.1 for some general comments. However, as regards your formula - a=b and b=c, so a=c, what you have in mind seems to be more an equation of symbols (words, labels) than a chain of thoughts.

When dealing with the latter, it is wise to refer to syllogisms, and in such case you must proceed with respect to the combinations of syllogism involved (if any) and acknowledge redundant information (if any). If you learn to avoid short cuts, your reasoning powers (accuracy) will increase exponentially.

Something about you (optional) Logician, philosopher

Greg: I return to your formula, a = b, b = c, so, a = c.

I want to make clear to you that this formula is only possible in conceptual logic, as against mathematics, by means of two syllogism. It is not a primary argument, but a derivative compound argument.

What does X=Y mean? If it is singular, e.g. John and Johnny are one and the same person, it means X is Y and Y is X, two predicative propositions. Similarly, if it is plural (general), e.g. all humans are rational animals and vice-versa, it means all X are Y and all Y are X.

Thus, your formula is really constituted of two syllogism, and does not constitute a separate form of reasoning. These are:

b is c and a is b, therefore a is c

and

b is a and c is b, therefore c is a

With predicative propositions, you cannot just convert at will. You must follow the rules of conversion for these forms, and use two propositions to express a two-way relation.

Something about you (optional) Logician, philosopher

Avi,

Ok then, I won't hold back from criticism!

What I like about your work is that it broadens the scope of logic; most books on the subject don't cover anything more than the propositional and predicate calculus ( I'm not a fan of the latter because of the excessively technical symbolism involved ), but then in my work ( computer programming and circuit design ) I haven't had the need for anything more advanced, although I've long had an interest in philosophy. I'm particularly looking forward to the section on factorial induction, which seems to be an integrated system with a definite plan and purpose, but I've a way to go before getting there!

Avi

If you learn to avoid short cuts, your reasoning powers (accuracy) will increase exponentially.

Interesting. I would have thought that short cuts would HELP your reasoning powers.

This is the first time I've been exposed to the "traditional" Aristotelian syllogism in such detail, and I must admit I'm finding it pretty hard going. Not because of any conceptual difficulties, but because of the large amount of data involved. All the tables, figures, moods, relationships etc. Do you have any advice on how best to assimilate it all? I was hoping you might have a few short cuts!

There are no exercises or problems in the book, but I suppose I can make up my own. I think logic is like maths in that you can't learn it merely by passively reading about it; you have to actively solve problems.

Something about you (optional) student

Hi Avi,

In reply to your last post, I've taken on board your comments. The reason I used "=" in the argument was because the word "is" is ambiguous. It seemed to me that the "=" better reflected the meaning of an identity, whereas the usual meaning of "is" in a proposition is predication.

Something about you (optional) student

Hi Greg.

About my last post concerning the "=" - I should have added at the end of it that, of course, as you imply, if X and Y are both singular, then X is Y is convertible to Y is X and vice-versa (whereas if they are universals, such conversion is not possible and both propositions must be given).

Regarding your latest comments, you are right to look forward to the chapters on inductive logic. They are a mind opener as to the way we actually reason - very far really from the deductive method that logic is usually associated with. Once induction is understood, many philosophical problems just melt away.

About short cuts - you don't want them. There is no need to memorize stuff. What is important is to understand, and to do that you must look at the details and not just skim over them. But once you have understood through scrutiny of the details, you are free to forget, because when you need the information you will be able to generate it by yourself.

Something about you (optional) Logician, philosopher

Hi Avi,

Thanks for the advice. Are there any dependencies between the parts of FL? I'm tempted to skip ahead to read the sections on factorial induction (which seems to depend only on knowledge of modal categoricals), or do you recommend that I work linearly through the book?

Something about you (optional) student

H Greg. The only part you could skip is Class Logic (Va).

You need parts I-V and Vb to understand part VI. The latter, on Factorial Induction, is the culmination of the former parts and depends on them for its full understanding.

Take your time.

Best regards,

Avi

Something about you (optional) Logician, philosopher

Hi Avi, I'm a little way into reading Future Logic and have a query. On page 37, under table 6.1 you say:

"The seven definite oppositional relations are mutually exclusive (i.e. contrary, to be exact), but one of the seven must hold."

But according to the table, contrary relations allow nonP + nonQ (i.e. all the relations can be false together), but if that's the case, doesn't that contradict the proposition that "one of the seven must hold"?

Or perhaps my logic is faulty.

Hi JayDee, in reply to your question.

When two propositions are contrary it means that they cannot be both true, but may be both false.

In the case of seven (or any set of three or more) propositions, we can well say that more than one of them cannot be true, but still at least one of them must be true (i.e. they cannot be all false).

That 'only one of them can be true' does not exclude that 'one of them must be true'.

This holds, notice, even if you take any two at random and say that they are contrary to each other, i.e. that the pair may be both false.

Something about you (optional) logician-phil.osopher

Thanks Avi, I guess I'm still a bit confused. If the relations (propositions) are mutually exclusive and exhaustive I understand that to mean that for any pair of propositions, one and only one of the relations *must* be true. Am I wrong?

By "exhaustive" I mean as fits the definition as given on page 25 and in table 3.1. Here, nonP + nonQ is logically impossible, but in table 6.1 it isn't (although "contrariety" isn't the same as "exhaustive").

It seems to me that the relations (taken as a set) are contradictory and not contrary because they are mutually exclusive, meaning if one is true then the others must be false, and also since one must be true then they cannot all be false. I.e., a "no" under columns P+Q and nonP+nonQ (which meets the criteria of contradictory).

I understand your point that 'only one of them can be true' does not exclude that 'one of them must be true', but isn't the issue rather that they cannot all be false?

The logic of oppositions is best understood by matrix analysis.

In the case of two propositions, the matrix is:

1) P + Q

2) P + notQ

3) notP + Q

4) notP + notQ

These are the 4 general, prima facie, logical possibilities for two propositions.

If the propositions are shown to be contradictory, it means that the alternatives 1 and 4 are impossible, while the alternatives 2 and 3 are possible. If the propositions are contrary, then the alternative 1 is impossible, but the other three are possible. If the propositions are subcontrary, then the alternative 4 is impossible but the otherr three are possible. And so on.

Now when you deal with three or more (e.g. seven) propositions, obviously the matrix is going to be accordingly larger, with more alternatives, and more ways they might be combined. For instance, with three propositions we have:

P + Q + R

P + Q + not R

P + notQ + R

notP + Q + R

and so on... till notP + notQ + notR (this is given in full in the book somewhere).

That's how we know we have exhausted the logical possibilities - by listing them all.

Something about you (optional) logician-phil.osopher

Thanks Avi, I guess I'm still a bit confused. If the relations (propositions) are mutually exclusive and exhaustive I understand that to mean that for any pair of propositions, one and only one of the relations *must* be true. Am I wrong?

NOT WRONG. THIS IS THE DEFINITION OF CONTRADICTION (NOT CONTRARIETY).

By "exhaustive" I mean as fits the definition as given on page 25 and in table 3.1. Here, nonP + nonQ is logically impossible, but in table 6.1 it isn't (although "contrariety" isn't the same as "exhaustive").

YES, EXHAUSTIVE MEANS THE NEGATION OF ALL THE PROPOSITIONS IS IMPOSSIBLE. MUTUALLY EXCLUSIVE MEANS THE AFFIRMATION OF ALL THE PROPOSITIONS IS IMPOSSIBLE.

It seems to me that the relations (taken as a set) are contradictory and not contrary because they are mutually exclusive, meaning if one is true then the others must be false, and also since one must be true then they cannot all be false. I.e., a "no" under columns P+Q and nonP+nonQ (which meets the criteria of contradictory).

THIS WORKS WITH TWO PROPOSITIONS, WITH THREE OR MORE PROPOSITIONS THERE ARE MORE POSSIBILITIES, AS IS CLEAR WITH REFERENCE TO MATRICES.

I understand your point that 'only one of them can be true' does not exclude that 'one of them must be true', but isn't the issue rather that they cannot all be false?

ONE MUST BE TRUE MEANS THE SAME EXACTLY AS THEY CANNOT ALL BE FALSE. IN SOME CASES, ONLY ONE CAN BE TRUE. IN OTHERS, MORE THAN ONE MIGHT BE ALLOWED TO BE TRUE (UP TO ALL BUT ONE).

Something about you (optional) logician-phil.osopher

Ok, thanks Avi, sorry about the confusion.

Do these relations hold for ALL the forms of proposition which are covered in the book? (categorical, modal, conditional, etc).

I notice that the unconnectedness relation has no column in which a pair of propositions is not possible. Would it be true to say that in general, the "strength" of a relation is correlated (positively) with the number of impossible combinations? So the strongest relations are implicance and contradiction, each having two "no's", and the weakest is unconnectedness, which has none.

Hi JD,

Do these relations hold for ALL the forms of proposition which are covered in the book? (categorical, modal, conditional, etc).

THE LISTED OPPOSITIONS (CONTRADICTION, CONTRARIETY, ETC.) ARE OBVIOUSLY APPLICABLE TO ANY PAIR OF PROPOSITIONS OF ANY SORT, IN THIS BOOK OR ANY OTHER. THEY ARE JUST EXTENSIONS OF THE THREE LAWS OF THOUGHT.

I notice that the unconnectedness relation has no column in which a pair of propositions is not possible. Would it be true to say that in general, the "strength" of a relation is correlated (positively) with the number of impossible combinations? So the strongest relations are implicance and contradiction, each having two "no's", and the weakest is unconnectedness, which has none.

YES, THAT IS CORRECT. THE MORE COMBINATIONS ARE IMPOSSIBLE, OBVIOUSLY THE STRONGER THE 'OPPOSITION' RELATION BETWEEN THEM.

JD, keep in mind that the named oppositions refer to PAIRS of propositions. When dealing with sets of 3, 4, 5, 6, etc. propositions, the different possibilities of 'impossible' and 'possible' rows are ALSO 'oppositions', except that we have not given them any specific names because that would require a never-ending vocabulary! Instead we resort to descriptive sentences like "one of them must be true" or "all but one of them must be false" etc. All this, to repeat, is merely application of the laws of thought.

Something about you (optional) logician-phil.osopher

Avi

Hi JD,

JD, keep in mind that the named oppositions refer to PAIRS of propositions. When dealing with sets of 3, 4, 5, 6, etc. propositions, the different possibilities of 'impossible' and 'possible' rows are ALSO 'oppositions', except that we have not given them any specific names because that would require a never-ending vocabulary!

JD, keep in mind that the named oppositions refer to PAIRS of propositions. When dealing with sets of 3, 4, 5, 6, etc. propositions, the different possibilities of 'impossible' and 'possible' rows are ALSO 'oppositions', except that we have not given them any specific names because that would require a never-ending vocabulary!

Thanks Avi. Noted!

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