Infinitely Many Valid Syllogisms
When we use ordinary quantification (all/no/some/some...not), we have only a small number of valid syllogisms. But these basic quantifiers are not the only possible quantifiers. This is where precise quantification comes in. Consider the following argument:
At least all but 5 M's are P
At least all but 4 S's are M
∴ At least all but 9 S's are P
This is a valid variation of a standard Barbara syllogism; Barbara alone has infinitely many valid 'precised' counterparts. Moreover, we can make use of maximal and minimal existential suppositions. For instance, if we suppose the minimum-presupposition that there are at least eleven S's, the above premises also yield the conclusion that at least 2 S's are P. If we make the maximum-presupposition that there are at most nine M's, the following is valid:
At least 7 M's are P
At least 5 S's are M
∴ At least 3 S's are P
So the natural question is: how does this all work in the context of Sommers notation?
Ordinary Quantification and Precise Quantification (Murphree Extension)
Ordinary quantification can be seen as particular cases of numerical quantification; or, perhaps more accurately, when we think of quantification in terms of some unit defined by our terms, we get numerical quantification. "Some S is P" is linked to "At least one S is P" while "Some S is not P" is linked to "At least one S is not P." It turns out that you can render universal propositions fairly easy, as well. "No S is P" is linked to "At most zero S's are P." The counterpart for "All S is P" is rendered in the slightly more complicated form, " At least all but zero S's are P". Or to put it in other terms: ordinary universal quantification assumes no exceptions. Numerical counterparts to universal quantity quantification are what have traditionally been known as exceptives. The basic pattern for A-type propositions is "All but x S's are P"; when x = 0, we have a standard A proposition. The basic patter for E-type propositions is "At most x S's are P"; when x = 0, we have a standard E proposition. The two are related, of course. "All but x S's are P" is equivalent to "At most x S's are nonP," and "At most x S's are P" is equivalent to "At least all but x S's are nonP."
With this in mind we can build a numerical term logic that uses precise quantification. The correspondences are as follows:
All but x Ss are P = -xS+P
At most x Ss are P = -xS-P
At least x Ss are P = +xS+P
At least x Ss are not P = +xS-P
It's easy to see that this is just ordinary Sommers notation with x's, and that if x=0, the first two are equivalent to ordinary Sommers notation's -S+P and -S-P, while if x=1, the second two are equivalent to ordinary Sommers notation's +S+P and +S-P.
One tricky feature of this new numerical part of the sentence is that denial of the whole sentence requires a change of numerical quantifier. The denial of -0S+P is +1S-P; and we find that it's a general truth that the denial of a universal sentence with x in its quantification results in a particular sentence with x+1 in its quantification, and the denial of a particular sentence with x in its quantification results in a universal sentence with x-1 in its quantification.
Syllogisms in this extended Sommers notation are very similar to syllogisms in ordinary Sommers notation. An ordinary Barbara syllogism would be:
Nothing very surprising here. We can use higher numbers, of course. For instance:
Syllogisms with particular premises are only slightly more tricky:
We can establish as a general rule the following principle: To achieve the strongest conclusion the premises can bear, the numerical value of the quantifier in the conclusion is equal to the sum of the numerical value of the quantifiers in the premises.
Now, one of the problems raised by these syllogisms is the failure of the backbone of any syllogistic, DDO, at least if interpreted in too wooden a way, because it puts us in situations where we are not strictly saying P of all M. Murphree proposes a generalization of DDO, in which we understand it to mean, Whatever is said of all but x Ms is said of all but x of whatever is M. However, there are still complications, because cases where DDO would give us negative numbers in our quantification are awkward, since we usually prefer to avoid negative numbers in this context. Murphree therefore also suggests an additional principle, a dictum de aliquid (DDA): Whatever is said of some Ms is said of all but the rest of the Ms.
And we can add in our above reasoning about maximum- and minimum-presuppositions. So here's an example of what you can get. Take the following syllogism:
At least eleven members are philosophy majors.
At least fifteen members are sophomores.
There are at most 20 members.
What's the conclusion? We can convert to Murphree-expanded Sommers notation:
From the first and third we get:
From this and the second we get:
At least six sophomores are philosophers. (4) follows from (1) and (3) by DDA; (5) follows from (2) and (4) by DDO. We could use DDO to get an analogue of (4); it would be the claim that at least negative nine philosophy majors are not members. This is certainly a validly derived conclusion, and might be useful in some circumstances; but with maximum-presuppositions DDA will often get us a handier result.
Relations with Numerical Sommers notation
One of the interesting things about this extension of Sommers notation is that you can dis-ambiguate some natural language claims very easily. Consider the following sentence:
Three teachers gave four students two books.
This might mean that each of three teachers gave each of four students two books each:
+3T1 + ((G123 +2B2)+4S3)
Then there were 24 books given. Or it might mean that three teachers together gave two books total to four students as a group:
+1[3T]1 + ((G123 +2B2)+1[4S]3)
Then there were only two books given. It might also mean that three teachers together gave two books total to each of the four students:
+1[3T]1 + ((G123 +2B2)+4S3)
That gives us a total of eight books given. It could also mean that each of three teachers gave two books to four students as a group:
+3T1 + ((G123 +2B2)+1[4S]3)
That's a total of six books given. So, whether it's 24, 2, 8, or 6 books given, this extension of Sommers notation can clarify. This is not at all surprising, because it offers a finer degree of discrimination in quantification. Inferences with relationals, of course, work just as one would suspect.
Ordinary Quantification and Precise Quantification (Szabolcsi-Sommers Extension)
It is possible to handle numerical quantification in another way. We can correlate “Some S is P” with “More than zero S’s are P”; we can then treat all quantity in terms of more than x. We then get the following correspondences:
-S+P corresponds to -0(+S+(-P))
-S-P corresponds to -0(+S+P)
+S+P corresponds to +0(+S+P)
+S-P corresponds to +0(+S+(-P))
“At least x S’s are P” is then representable as +x-1(+S+P); “Exactly x S’s are P’s” is represented as the conjunction “At least x S’s are P and No more than x S’s are P”.
The ordinary conditions for validity in Sommers notation remain (the terms – not counting numerical quantifiers – must algebraically add to the conclusion and the number of particulars in the premise must equal the number of particulars in the conclusion), but a third condition, Szabolcsi’s Condition (SC) is needed: The sum of the numerical quantifiers in the premises must be equal to or greater than the numerical quantifier in the conclusion.
So take the following argument:
More than 15 of the school’s students are taking the big test.
All but 1 person on my roster are in the testing center.
No more than 2 of the school’s students are in the testing center.
This then can be represented:
And the conclusion is: +12(+T+(-R)), which is to say, “More than 12 of those taking the test are not on my roster.”
This could be pursued at greater length, but we will not do so here.