* A long lost letter by Descartes has been found; it's to Mersenne and discusses the publication of the Meditations. You can find more details at the website of Erik Jan-Bos, the discoverer.

* A macroeconomics cube of opposition. I lack sufficient grounding in a number of the positions to know how accurate it is, but it looks interesting.

* A good post by Paul Gowder on a common error about shared responsibility.

* Dog jousts rabbit. Poor rabbit.

* Burying the dead in Haiti. Burying the dead is one of the traditional corporal works of mercy.

* Christopher Tolkien's translation of, and commentary on, The Saga of King Heidrek the Wise. (PDF)

* Erasmus's Anti-Polemus.

ADDED LATER:

* James Chastek has a good post on good taste.

* Lindsay Beyerstein has a sensible post on the Austin plane crash and terrorism.

## Friday, February 26, 2010

## Thursday, February 25, 2010

### Virtue Analysis

To wrap up the virtue ethics section of my ethics course, I had the students do the following assignment:

It's interesting to see the virtues that they chose. I've received so far (there are others who are straggling in or had family emergencies and thus have not yet turned anything in):

Humor

Docility, i.e., teachableness

Faith (2x)

Hope

Charity

Nurture

Respect

Truthfulness

Temperance (2x)

Fortitude (2x)

Patience

Justice (2x)

One sees a tendency to go for the big, obvious virtues; the only cardinal virtue that doesn't get a direct hit is prudence. That's not surprising; all the students found the paper somewhat difficult to write (as it should be), and not only did we talk cardinal virtues, but Aquinas's list is right there, divided up according to the seven virtues. The papers on the theological virtues all treat them as acquired virtues, of course; the tools they had available were tools most suitable for discussing acquired virtues, and the assignment sets things up on the assumption that they would be talking about moral virtues (since somehow it didn't occur to me that they might choose the theological ones), so it's unsurprising that faith, hope, and charity all come out sounding like special kinds of acquired virtue. But, pleasantly, this is often (partially) recognized; one paper on faith notes some difficulties with classifying faith as a virtue, and they are difficulties that arise if you think of it on the model of an acquired virtue. (The student's particular solution, to make the virtue justified faith in times of difficulty and associate it with fortitude, still treats it as an acquired virtue but is an interesting attempt.) I was interested in the non-cardinals chosen; docilitas in particular surprised me, but I was interested to see nurture, humor, and respect being chosen out as noteworthy (or at least convenient for analysis). The respect paper treats it as especially associated with prudence, which is another interesting move; in fact, the long association of pietas and prudentia gives a certain plausibility to such a classification.

As you might expect, the responses are rather uneven, but some of them clearly put some thought into the matter -- thus proving my general rule in teaching that you can never go wrong giving students room to surprise you.

Pick a virtue and analyze it using ideas we've discussed in class. (Examples of questions you should ask yourself in order to do this: What are the corresponding vices of effect and defect? What actions does this virtue involve? With which cardinal virtue is it most closely associated? Are there any vices that mimic it? Are there any vices it remedies?)

Your paper should be 800-1000 words (that's approximately three to four full pages if written out on a word processor). You should try to be as concise, focused, and organized as possible in your discussion, you should use examples to show that your analysis is a good one, you should consider possible objections to your analysis, and you should cite any sources that you use. If you have difficulty thinking of a virtue, you might consider looking at the list of virtues analyzed by Thomas Aquinas to see if you can find one that interests you.

It's interesting to see the virtues that they chose. I've received so far (there are others who are straggling in or had family emergencies and thus have not yet turned anything in):

Humor

Docility, i.e., teachableness

Faith (2x)

Hope

Charity

Nurture

Respect

Truthfulness

Temperance (2x)

Fortitude (2x)

Patience

Justice (2x)

One sees a tendency to go for the big, obvious virtues; the only cardinal virtue that doesn't get a direct hit is prudence. That's not surprising; all the students found the paper somewhat difficult to write (as it should be), and not only did we talk cardinal virtues, but Aquinas's list is right there, divided up according to the seven virtues. The papers on the theological virtues all treat them as acquired virtues, of course; the tools they had available were tools most suitable for discussing acquired virtues, and the assignment sets things up on the assumption that they would be talking about moral virtues (since somehow it didn't occur to me that they might choose the theological ones), so it's unsurprising that faith, hope, and charity all come out sounding like special kinds of acquired virtue. But, pleasantly, this is often (partially) recognized; one paper on faith notes some difficulties with classifying faith as a virtue, and they are difficulties that arise if you think of it on the model of an acquired virtue. (The student's particular solution, to make the virtue justified faith in times of difficulty and associate it with fortitude, still treats it as an acquired virtue but is an interesting attempt.) I was interested in the non-cardinals chosen; docilitas in particular surprised me, but I was interested to see nurture, humor, and respect being chosen out as noteworthy (or at least convenient for analysis). The respect paper treats it as especially associated with prudence, which is another interesting move; in fact, the long association of pietas and prudentia gives a certain plausibility to such a classification.

As you might expect, the responses are rather uneven, but some of them clearly put some thought into the matter -- thus proving my general rule in teaching that you can never go wrong giving students room to surprise you.

### Kant on Philodoxers

The artist of reason, or philodox, as Socrates calls him, strives only for speculative knowledge, without looking to see how much the knowledge contributes to the final end of human reason; he gives rules for the use of reason for any sort of end one wishes. The practical philosopher, the teacher of wisdom through doctrine and example, is the real philosopher. For philosophy is the idea of a perfect wisdom, which shows us the final ends of human reason.

Immanuel Kant, Lectures on Logic. 'Artist of reason' here seems to be fairly literal; a translation that would perhaps convey Kant's point better would be 'technician of reason'.

### Strength from Sion Hill

The Twentieth Psalm

by John Davies

The Lord give ear to thee in thy distress,

And by they shield when troubles thee oppress,

And let his help come down from heaven for thee,

And strength from Sion hill imparted be;

Let him remember and accept withal

Thine offerings, and thy sacrifices all,

And of his bounty evermore fulfill

Thy heart's desire, and satisfy thy will.

But we will glory in our great God's name,

And joy in our salvation through the same,

And pray unto the Lord our God that he

The effect of all they prayers will grant to thee.

He now I know will hear, and help will bring,

With his strong hand to his anointed king;

On chariots some, on horses some rely,

But we invoke the name of God most high.

Those others are bowed down, and fall full low,

When we are risen and upright do go;

Save us O Lord of heaven, and hear us thence,

When we invoke thy name for our defence.

## Wednesday, February 24, 2010

### Mill on Art and Science

The relation in which rules of art stand to doctrines of science may be thus characterized. The art proposes to itself an end to be attained, defines the end, and hands it over to the science. The science receives it, considers it as a phenomenon or effect to be studied, and having investigated its causes and conditions, sends it back to art with a theorem of the combination of circumstances by which it could be produced. Art then examines these combinations of circumstances, and according as any of them are or are not in human power, pronounces the end attainable or not. The only one of the premises, therefore, which Art supplies, is the original major premise, which asserts that the attainment of the given end is desirable. Science then lends to Art the proposition (obtained by a series of inductions or of deductions) that the performance of certain actions will attain the end. From these premises Art concludes that the performance of these actions is desirable, and finding it also practicable, converts the theorem into a rule or precept.

John Stuart Mill, A System of Logic Book VI, Chapter XII. 'Art' here means the systematic activity of practical reason. It's interesting that he thinks of the relationship syllogistically.

## Tuesday, February 23, 2010

### Marvelous Tyrannical Power

On the day the plane crashed into Echelon, this was part of the passage in the Gorgias that we were going to discuss:

[Plato, Gorgias, Zeyl, tr. Hackett, p. 32]

The passage led to some interesting discussion on the nature of true power, and what is merely a pitiable and pathetic grasping after it, as well as the difference between a conception of power or success based on doing something because one can and sees fit to do it -- which underlies much of the rhetoric by which people try to persuade us to buy or to vote -- and a conception of power or success based on doing what is good for oneself and others. It's one of the reasons I teach the Gorgias; there's no better Platonic dialogue for discussing in a direct way things we deal with every day even in our society.

Imagine me in a crowded marketplace, with a dagger up my seleve, saying to you, "Polus, I've just go myself some marvelous tyrannical power. So, if I see fit to have any one of these people you see here put to death right on the spot, to death he'll be put. And if I see fit to have one of them have his head bashed in, bashed in it will be, right away. If I see fit to have his coat ripped apart, ripped it will be. That's how great my power in this city is!" Suppose you didn't believe me and I showed you the dagger. On seeing it, you'd be likely to say, "But Socrates, everybody could have great power that way. For this way any house you see fit might be burned down, and so might the dockyards and triremes of the Athenians, and all their ships, both public and private." But then that's not what having great power is, doing what one sees fit.

[Plato, Gorgias, Zeyl, tr. Hackett, p. 32]

The passage led to some interesting discussion on the nature of true power, and what is merely a pitiable and pathetic grasping after it, as well as the difference between a conception of power or success based on doing something because one can and sees fit to do it -- which underlies much of the rhetoric by which people try to persuade us to buy or to vote -- and a conception of power or success based on doing what is good for oneself and others. It's one of the reasons I teach the Gorgias; there's no better Platonic dialogue for discussing in a direct way things we deal with every day even in our society.

## Monday, February 22, 2010

### Contradiction Explosion

A recent xkcd:

There is a famous story about Bertrand Russell, who proved that if 1=0, he was the Pope. Add 1 to each side of the equation. This gives you the equation 2=1. The set containing Russell and the Pope has two members. But since 2=1 it has one member. And since it is the set of Rusell and the Pope that single member must be both Bertrand Russell and the Pope. Therefore Bertrand Russell is the Pope.

There are, however, many reasons why one wouldn't want a logical system in which contradictions imply everything whatsoever; for instance, one could want a logical system in which contradictions imply nothing whatsoever. Systems in which contradictions don't explode, that is, in which they don't imply everything, are called paraconsistent systems. The usual way of making a paraconsistent system in philosophical circles is to deny disjunctive syllogism; anyone who has been reading me long enough knows that I find this an exasperatingly silly move. Disjunctive syllogism is a central inference rule; eliminating it to avoid explosion is like welding shut the vault to keep the bank from being robbed. A far more rational strategy, which is more popular in computer science circles than it is in philosophy circles, is to eliminate the rules of addition and weakening; this makes sense because they are the only rules that let you introduce absolutely anything into an inference in the first place. If you take that route, you can avoid explosion and still reason like an intelligent person.

But to some extent it is a matter of notational convenience rather than serious substance; as Hartley Slater has noted, whenever you start messing with any inference rules you are also changing the kind of operations you are doing: 'or' simply does not mean the same thing, does not indicate the same connection or operation, in a system that has disjunctive syllogism when compared to a system that does not have disjunctive syllogism. But that's an argument for a paraconsistent system that keeps disjunctive syllogism, too, because we should keep the things that are most useful.

There is a famous story about Bertrand Russell, who proved that if 1=0, he was the Pope. Add 1 to each side of the equation. This gives you the equation 2=1. The set containing Russell and the Pope has two members. But since 2=1 it has one member. And since it is the set of Rusell and the Pope that single member must be both Bertrand Russell and the Pope. Therefore Bertrand Russell is the Pope.

There are, however, many reasons why one wouldn't want a logical system in which contradictions imply everything whatsoever; for instance, one could want a logical system in which contradictions imply nothing whatsoever. Systems in which contradictions don't explode, that is, in which they don't imply everything, are called paraconsistent systems. The usual way of making a paraconsistent system in philosophical circles is to deny disjunctive syllogism; anyone who has been reading me long enough knows that I find this an exasperatingly silly move. Disjunctive syllogism is a central inference rule; eliminating it to avoid explosion is like welding shut the vault to keep the bank from being robbed. A far more rational strategy, which is more popular in computer science circles than it is in philosophy circles, is to eliminate the rules of addition and weakening; this makes sense because they are the only rules that let you introduce absolutely anything into an inference in the first place. If you take that route, you can avoid explosion and still reason like an intelligent person.

But to some extent it is a matter of notational convenience rather than serious substance; as Hartley Slater has noted, whenever you start messing with any inference rules you are also changing the kind of operations you are doing: 'or' simply does not mean the same thing, does not indicate the same connection or operation, in a system that has disjunctive syllogism when compared to a system that does not have disjunctive syllogism. But that's an argument for a paraconsistent system that keeps disjunctive syllogism, too, because we should keep the things that are most useful.

### Gobekli Tepe

Newsweek has a story about Gobekli Tepe, one of archeology's big mysteries. (ht) It appears to be a temple site, but it also appears to be so old -- about 11000 years old -- that it would predate systematic agriculture in the region (and occur at the very, very beginning of the millenia that constitute Neolithic agricultural revolution). This is extraordinarily anomalous; major construction of this sort is usually thought to postdate the rise of agriculture, and to find anything remotely like it you have to go several thousand years after it. It is sufficiently strange that one worries that it has been misdated -- but it seems not to be; and assuming that it is not, it requires a rethinking of our standard Stone Age timelines. See also this article at Smithsonian magazine; and Wikipedia actually has an unusually nice article on it.

### Sommers Notation, Part VII

(Part VI)

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:

-0M+P

-0S+M

∴-0S+P

Nothing very surprising here. We can use higher numbers, of course. For instance:

-11M+P

-333S+M

∴ -344S+P

Syllogisms with particular premises are only slightly more tricky:

-11M+P

+30S+M

∴ +19S+P

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:

(1) +11M+P

(2) +15M+S

(3) -20M-M

From the first and third we get:

(4) -9M+P

From this and the second we get:

(5) +6S+P

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:

+3T

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]

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]

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:

+3T

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:

+15(+S+T)

-1(+R+(-C))

-2(+S+C)

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.

(Part VIII)

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:

-0M+P

-0S+M

∴-0S+P

Nothing very surprising here. We can use higher numbers, of course. For instance:

-11M+P

-333S+M

∴ -344S+P

Syllogisms with particular premises are only slightly more tricky:

-11M+P

+30S+M

∴ +19S+P

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:

(1) +11M+P

(2) +15M+S

(3) -20M-M

From the first and third we get:

(4) -9M+P

From this and the second we get:

(5) +6S+P

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:

+3T

_{1}+ ((G_{123}+2B_{2})+4S_{3})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}+ ((G_{123}+2B_{2})+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}+ ((G_{123}+2B_{2})+4S_{3})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:

+3T

_{1}+ ((G_{123}+2B_{2})+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:

+15(+S+T)

-1(+R+(-C))

-2(+S+C)

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.

(Part VIII)

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