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Ok, here is my attempt to find the valid syllogisms for singular-singular propositions. I have named them R2 and G2, following the standard singulars:

R2 : This S is this P

G2 : This S is not this P

So altogether there are 8 x 8 = 64 possible premises for each figure. However, since R2 and G2 are both convertible, isn't it the case that results for any one figure will apply to all? Anyway, I'll just give the results for the first figure.

Premises involving the plurals can be discounted since there will be no connection, which leaves just the following 7 (ignoring those in which the premises are the same but the order is different):

R,R2

R,G2

G,R2

G,G2

R2,R2

R2,G2

G2,G2

And here are the full syllogisms:

R This M is P

R2 This S is this M

con This S is P

R This M is P

G2 This S is not this M

con NONE

G This M is not P

R2 This S is this M

con This S is not P

G This M is not P

G2 This S is not this M

con NONE

R2 This M is this P

R2 This S is this M

con This S is this P

R2 This M is this P

G2 This S is not this M

con This S is not this P

G2 This M is not this P

G2 This S is not this M

con NONE

So there are only 4 valid syllogisms (first figure):

R R2 R

G R2 G

R2 R2 R2

R2 G2 G2

I may return to the other figures later, but for now I'm keen to continue on to part II of the book. Hopefully I haven't completely messed up!

So altogether there are 8 x 8 = 64 possible premises for each figure.

This should have been 64 possible pairs of premises, of course.

Hi MDB, well done, but you missed one valid mood.

Not so interesting are the moods you listed with a plural major term (P). To be thorough, you need to also look into moods with a plural minor term (S) (I mention a negative example higher up - No Asian city is Paris ... a positive example would be ?). Let's just focus for now on the fully singular syllogisms.

sing-sing syllogs:

Let S, M, P be three singular terms (or this S, this M, this P, as you have it) (e.g. the present capital of France, the location of the Eiffel Tower, Paris or Vienna). We have three out of four valid moods in the first figure.

M is P

S is M

S is P – valid

M is not P

S is M

S is not P – valid

(You missed this one)

M is P

S is not M

S is not P – valid

M is not P

S is not M

No conclusion

The most interesting mood here is of course the third one, because it has a negative minor premise and yet is valid, which is atypical/exceptional. For this mood alone it was worth looking into.

Let us compare this mood to plural syllogism. If the major premise was compound, with both directions universal, then the conclusion would be valid, because a 2nd fig. syllogism would actually be operative (not really the 1st fig. one).

(All M are P and) All P are M

No S is M

conclusion No S is P is valid

So that's why the sing-sing version works, really.

Regarding the other 3 figures, as you say, we just need to convert the major premise (2nd fig.) or the minor premise (3rd fig.) or switch the premises (4th fig.). Since all these propositions are fully convertible, there will be 3 valid moods out of 4 in each figure.

Something about you (optional) logician-philosopher

Avi, thanks for the feedback.

M is not P

S is M

S is not P – valid

(You missed this one)

M is P

S is not M

S is not P – valid

Aren't these essentially the same? I did have both on my list but discarded one because it's just a repeat of the other with Major/Minor premises reversed. This isn't the same as saying that in the plural syllogisms (fig. 1), EAE is the same as AEE, because the latter is invalid, but with singular-singular syllogisms both terms are the same from a distribution point of view. i.e. S is P = P is S and S is not P = P is not S. What am I missing?

Hi MDB.

If you switch 1st fig. premises you get a 4th fig. syllog. BUT the term symbols S and P must be changed.

P is M

M is not S

S is not P – valid

P is not M

M is S

S is not P – valid

As discussed in FL, and as Aristotle considered, the 4th fig. is somewhat artificial, not so natural a movement of thought as the other figures. But still, in some cases it brings an interesting conclusion not otherwise evident.

As regards the 1st fig. valid mood with a negative minor premise, as I said before it is unusual for a 1st fig. syllog. to so behave - and the explanation is given by me with reference to the 2nd. fig. in my previous post.

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Avi, with regard to chapter 15, section 3 (validations), I'm having a hard time convincing myself that pnp in the 1st figure is invalid, although I've taken on board your comments on it.

I've tried comparing it with the 3rd fig. pnp, which is valid, but it's still not clear to me how the differences make one valid and the other invalid.

e.g. an invalid mood in pnp (1st fig.) would be:

All M can be P (p)

All S must be M (n)

therefore, All S can be P (p)

Rephrasing in terms of "circumstances", this would be

In some circumstances all M is P

in all circumstances all S is M

therefore in some circumstances all S is P

I'm trying to come up with suitable concepts for the terms which makes this syllogism intuitively invalid, but so far haven't been able to. Any suggestions?

Hi MDB,

In the first figure, the 'all circumstances' of the minor premises are those applicable to the minor term S, they are not all circumstances concerning the middle term M. For this reason, there is no guarantee that the 'some circumstances^concerning M in the major premise are covered.

On the other hand in the third figure, M being the subject in both premises, you can be sure of overlap of circumstances.

Best, Avi.

Something about you (optional) logician-philosopher

Thanks Avi, for some reason I just wasn't getting it. I think I was confusing myself by reading the premises as "all S in all circumstances is what P is in some circumstances". I was looking at the quantity and not the modality. Then I drew a diagram to represent the circumstances "surrounding" the terms. Being more of a visual thinker, the invalidity then became clear.

But still, some of these modal syllogisms aren't at all obvious.

In fig. 1, if you tried converting the minor premise, you would have immediately seen that while S was distributed, M was not. All S must be M converts to Some M can be S. Remember All S means: Each S - it is distributive, not collective or collectional. So, the all circumstances are all circumstances concerning each S, which are not the same set of circumstances... The universal proposition is a mere summary formula.

Note additionally that in the past, some logicians thought that a universal potential major premise in the first fig. could yield a valid conclusion. They did not realize the issues involved. So, you are in good company!

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