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Logic

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!