In Part I, I gave a rough-and-ready characterization of the basics of SETL. In Part II, I looked at some basic issues. We can handle categorical assertions; singular terms; relations; singular identities; propositions about propositions; and propositions about the domain of discourse. In this post I will look at two issues that are more complicated, namely, modality and irreducible identity between variables. (As noted previously, sources and relevant readings will follow in a later post.)
(1) Modality. Basic SETL has no modalities (necessity, possibility); and the extension for handling modalities is still incomplete. It appears that SETL can actually handle certain forms of modality fairly easily, with only a small extension. De re modality turns out to be nothing other than ordinary term-logic arguments with modalized terms. (Indeed, this is what distinguishes 'de re modality' from 'de dicto modality', at least using the terms in this sense: de re modalities modalize the terms of the proposition, whereas de dicto modalities modalize the proposition itself. Thus we could symbolize "All S is possibly P" as (-S+◊P) and add rules of inference relevant to such statements. 'De dicto modality' is considerably trickier. In any case, I will look at extension to modality a little more closely in a later post.
(2) Identity between variables. William Purdy has argued, with considerable force, that while SETL can handle just about anything modern predicate logic can, there is one thing that the latter appears to handle more easily: identity in cases where both sides of the identity are variables (in the predicate logic). SETL can easily handle identity in cases where at least one side is not a variable. Sommers and Englebretsen often talk as if this ended the matter; SETL can handle identity. But Purdy pointed out that they are always speaking of what I called above 'singular' identity; and there is one form of identity in modern predicate logic that turns out to be fairly important that is not 'singular' identity -- the case already noted, where both sides of the identity would be variables in the predicate logic. He rigorously argued that PCS, a formal language like modern predicate logic in many ways, but like SETL (and unlike modern predicate logic) able to put a singular term in the predicate, and having the theory of identity associated with SETL, turns out to be equivalent to a subset of modern predicate logic, but fails to handle well-formed formula involving identity between two variables (when those formula are not reducible in predicate logic to a form not involving variable identity).
The argument is interesting and important. And, as it's quite advanced, I can't be certain I've adequately understood. I'm not sure it is completely adequate, however. One thing that appears to be missing from PCS that is clearly found in SETL is what I've called category nominalization. Now, since every variable has an associated domain of discourse (namely, the domain of discourse for propositions with that variable), at least some identity between variables should be expressible in SETL. So, for instance, if (x)(y)(x=y) is the identity in question, what is naturally relevant to the truth of this identity is the domain of discourse; the identity basically says, for every member in the domain relevant to x and every member of the domain relevant to y, x is identical to y. Which yields:
±/TERM1/±/TERM2/
Where /TERM1/ is the nominalization of the domain associated with x and /TERM2/ is the nominalization of the domain associated with y. This, and equivalent propositions, are expressible in SETL. So at least some variable identity is expressible, because nominalized categories or domains can do the work of variables -- and, indeed, this is not surprising, because you can't have variables without domains; it would be like having variables that are incapable of having values, which is to say, variables that are not variables at all. And when you translate variables directly from predicate logic to SETL, e.g., (x)(Px), you translate using domain nominalization, namely,
-/P/+P
/P/ is the domain associated with x in (x)(Px), assuming that this one sentence is the only relevant sentence. Any weirdness up to this point is due to the fact that you can be entirely arbitrary about the domain associated with a variable; which means that for a lot of domains in which variable identity would be important there would be no ready-made terms for handling the identity (we'd have to neologize). In fact, Purdy notes that the limitation of PCS is that it can't handle unnamed elements in the domain of discourse. But there seems to be no reason why nominalized domains or categories can't cover unnamed elements (indeed, it would seem that they must).
Thus Purdy seems to be right that the theory of identity associated with SETL doesn't on its own allow for irreducible variable identities. But SETL does allow for category nominalization, which does appear to allow SETL to handle unnamed elements; and given this, it looks like at least some irreducible variable identities could be handled in SETL. In any case, this is an issue that needs further investigation.
So, having looked at some of these key issues, we can know proceed to discussion of inference in SETL; which we will begin in the next post.
Saturday, August 26, 2006
Sommers-Englebretsen Term Logic, Part II
In Part I, I discussed some of the basics of SETL. In this post, I want to look at how SETL handles certain tricky issues that are important for the proper handling of propositions in logic. (There will be a later post with additional readings and sources.)
Singular Terms
It's easy enough to see how SETL handles universal quantity, like 'All dogs are canines' or 'No dogs are felines':
-D+C
-D-F
It's also easy enough to see how SETL handles particular quantity, like 'Some dogs are tame' or 'Some dogs are not housebroken':
+D+T
+D-H
But what if we have a singular term, like, 'Fido is a dog'? In SETL this is handled fairly easily. For singular terms, the distinction between P-opposition and C-opposition turns out not to be significant; and every singular subject can be treated as having a 'wild quantity', because they are indifferent to whether you treat them as universal or as particular. Thus 'Fido is a dog' would be symbolized as:
±F+D
In an argument you can treat the singular proposition as universal or particular, as you need; the only tricky thing is that you sometimes need to keep track of what you are doing with it. In any case, SETL has no problem with singular terms.
Relations
When people give a reason for rejecting traditional term logic in favor of modern predicate logic, one of the reasons at the top of the list is that traditional term logic can't handle relational propositions. As it happens, SETL can handle relational propositions by allowing complex predicate terms. Take the proposition, "All sophists take money from some fools". The basic format of this proposition is:
-S+P
But the P term is a complex term consisting of other terms. So we can expand the predicate in this way:
-S+(T+M+F)
Then we can do all sorts of things with this. For instance, suppose we add to it the proposition, "All money is gold." The conclusion is:
-S+(T+G+F) [All sophists take gold from some fools]
Sometimes it is useful to use subscripts, when the direction of the relation is important. So, we could symbolize this proposition as:
-S1+(T123+G2+F3)
This, however, is just a convenience to help us keep track of what the terms mean in complex relational predicates. Subscripts can do a little more than this, for which see below.
On this basis we can translate any relational you could want. Here are some examples and their translations.
Richard loves Richard. ±R +(L±R)
Every boy loves every girl. -B1+(L12-G2)
Every boy loves some girl. -B1+(L12+G2)
Some boy loves every girl. +B1+(L12-G2)
Some boy loves some girl. +B1+(L12+G2)
No boy loves every girl. -B1-(L12-G2)
Every boy sends a rose to some girl. -B1+(S123+R2+G3)
Some girl was sent a rose by every boy. +G3+(S123+R2+B1)
Note that the last two are equivalent, which is precisely the result you should get. There are more complicated predicates that can't be handled so easily, for instance,
Some girls who think that all love is easy are unhappy.
To do this one must introduce propositional nominalization, which we will get to below. But even without this we can do a lot, as we will see in a later post.
Singular Pronouns
Suppose we have a sentence like: "Some boy kissed some girl and she clobbered him."
The first conjunct is easy: +B1+(K12+G2). Given this, we can represent the whole sentence as:
(+B1+(K12+G2))+(±2+(C21±1)
But the use of the subscripts in this way is just a matter of convenience -- to show that we are dealing with pronouns. Singular pronouns are just singular terms, and are treated as such.
(Singular) Identity and Existence
SETL can also handle singular identity very easily. Because singular terms are indifferent to quantity and can be qualified, we can handle such an identity very easily. 'Socrates is Socrates' becomes:
±S±S
Thus there is no need to bring in any special way of handling identity in order to handle singular identity statements. (Identity between variables is more difficult, and we'll look at that below.)
Just as identity is handled by normal predication in SETL, so, too, are existential statements: existence is a predicate in SETL.
Propositional Nominalization
How would we handle propositions about propositions? The natural way is to treat them as complex terms. Consider the sentence we noted above:
Some girls who think that all love is easy are unhappy.
If we use [p] to indicate the proposition, "All love is easy", we get the following rendering:
(+G+T+[p])-H)
But because [p] is a complex term, we can treat it as one, keeping it in square brackets to indicate that it is a nominalized proposition:
+G+T+[-L+E])-H)
In nominalizing, we have embedded one sentence in another by treating it as a term. This turns out to be a means of doing quite a few things that are rather fun. Most important of these is that we can handle propositional logic in our term logic. Consider the following sentences in propositional logic and their categorical SETL forms:
(If p then q) = -[p]+[q]
(p and q) = +[p]+[q]
(If p then if q then r) = -[p]+[-[q]+[r]]
This can be extended. SETL, unlike predicate logic, does not presuppose propositional logic. Likewie, we can handle metapropositions easily:
[p] is false
It is not the case that [p]
And so forth.
Category (Domain) Nominalization
In propositional logic every use of a variable technically requires a domain of discourse (sometimes called a universe of discourse. This is obvious when one consider Lewis Carroll syllogisms. ends up being of this general structure
/domain of discourse for propositions with x/
If it is P, it is Q (where 'it' refers back to the things in the domain of discourse)
In common usage, people don't worry about domain of discourse much; but technically you can't have a variable without it being a variable capable of ranging over a domain of some sort, so it's always there, and necessarily so.
In SETL, as with any term logic, domain of discourse is much less important, but it still can be defined for every sentence, and usefully so, because every statement is true if and only if it denotes its domain of discourse. In SETL every term has a domain (or, if you prefer, category), and the domain of discourse for any sentence is the intersection of the domains (or, if you prefer, categories) of all its terms. Thus
We symbolize the domain (or, if you prefer, category) indicated by the term D with /D/. This is what I am calling 'category nominalization' or 'domain nominalization'. /D/ consists of everything that is D or nonD, where D and nonD are both taken to be part of a category. Thus if D is 'red', then /red/ consists of everything that can be truly characterized by 'red' or 'nonred', where the latter is understood in such a way that it only applies to things falling in the same category as red things. Thus, blue things might fall under this category, but not the number two or the pain in my left hand, because these are a different category (we can meaningfully say of them that they are not the sort of thing that could be either red or nonred). Now, we can use this to handle a particular type of proposition:
Everything is P
Something is P.
In these the subject is the nominalized domain, so they are respectively translated as:
-/P/+P
+/P/+P
Or in other words, every member of the category or domain associated with P is P; some member of the category or domain associated with P is P. (Of course, things get more complicated if we need to use a larger domain of which P is only part; e.g., if there are lots of sentences, and we need to say that everything in the domain of all the sentences is P. But it works the same way.)
In the next post I will look briefly at some loose ends that are not covered by the above points.
Singular Terms
It's easy enough to see how SETL handles universal quantity, like 'All dogs are canines' or 'No dogs are felines':
-D+C
-D-F
It's also easy enough to see how SETL handles particular quantity, like 'Some dogs are tame' or 'Some dogs are not housebroken':
+D+T
+D-H
But what if we have a singular term, like, 'Fido is a dog'? In SETL this is handled fairly easily. For singular terms, the distinction between P-opposition and C-opposition turns out not to be significant; and every singular subject can be treated as having a 'wild quantity', because they are indifferent to whether you treat them as universal or as particular. Thus 'Fido is a dog' would be symbolized as:
±F+D
In an argument you can treat the singular proposition as universal or particular, as you need; the only tricky thing is that you sometimes need to keep track of what you are doing with it. In any case, SETL has no problem with singular terms.
Relations
When people give a reason for rejecting traditional term logic in favor of modern predicate logic, one of the reasons at the top of the list is that traditional term logic can't handle relational propositions. As it happens, SETL can handle relational propositions by allowing complex predicate terms. Take the proposition, "All sophists take money from some fools". The basic format of this proposition is:
-S+P
But the P term is a complex term consisting of other terms. So we can expand the predicate in this way:
-S+(T+M+F)
Then we can do all sorts of things with this. For instance, suppose we add to it the proposition, "All money is gold." The conclusion is:
-S+(T+G+F) [All sophists take gold from some fools]
Sometimes it is useful to use subscripts, when the direction of the relation is important. So, we could symbolize this proposition as:
-S1+(T123+G2+F3)
This, however, is just a convenience to help us keep track of what the terms mean in complex relational predicates. Subscripts can do a little more than this, for which see below.
On this basis we can translate any relational you could want. Here are some examples and their translations.
Richard loves Richard. ±R +(L±R)
Every boy loves every girl. -B1+(L12-G2)
Every boy loves some girl. -B1+(L12+G2)
Some boy loves every girl. +B1+(L12-G2)
Some boy loves some girl. +B1+(L12+G2)
No boy loves every girl. -B1-(L12-G2)
Every boy sends a rose to some girl. -B1+(S123+R2+G3)
Some girl was sent a rose by every boy. +G3+(S123+R2+B1)
Note that the last two are equivalent, which is precisely the result you should get. There are more complicated predicates that can't be handled so easily, for instance,
Some girls who think that all love is easy are unhappy.
To do this one must introduce propositional nominalization, which we will get to below. But even without this we can do a lot, as we will see in a later post.
Singular Pronouns
Suppose we have a sentence like: "Some boy kissed some girl and she clobbered him."
The first conjunct is easy: +B1+(K12+G2). Given this, we can represent the whole sentence as:
(+B1+(K12+G2))+(±2+(C21±1)
But the use of the subscripts in this way is just a matter of convenience -- to show that we are dealing with pronouns. Singular pronouns are just singular terms, and are treated as such.
(Singular) Identity and Existence
SETL can also handle singular identity very easily. Because singular terms are indifferent to quantity and can be qualified, we can handle such an identity very easily. 'Socrates is Socrates' becomes:
±S±S
Thus there is no need to bring in any special way of handling identity in order to handle singular identity statements. (Identity between variables is more difficult, and we'll look at that below.)
Just as identity is handled by normal predication in SETL, so, too, are existential statements: existence is a predicate in SETL.
Propositional Nominalization
How would we handle propositions about propositions? The natural way is to treat them as complex terms. Consider the sentence we noted above:
Some girls who think that all love is easy are unhappy.
If we use [p] to indicate the proposition, "All love is easy", we get the following rendering:
(+G+T+[p])-H)
But because [p] is a complex term, we can treat it as one, keeping it in square brackets to indicate that it is a nominalized proposition:
+G+T+[-L+E])-H)
In nominalizing, we have embedded one sentence in another by treating it as a term. This turns out to be a means of doing quite a few things that are rather fun. Most important of these is that we can handle propositional logic in our term logic. Consider the following sentences in propositional logic and their categorical SETL forms:
(If p then q) = -[p]+[q]
(p and q) = +[p]+[q]
(If p then if q then r) = -[p]+[-[q]+[r]]
This can be extended. SETL, unlike predicate logic, does not presuppose propositional logic. Likewie, we can handle metapropositions easily:
[p] is false
It is not the case that [p]
And so forth.
Category (Domain) Nominalization
In propositional logic every use of a variable technically requires a domain of discourse (sometimes called a universe of discourse. This is obvious when one consider Lewis Carroll syllogisms. ends up being of this general structure
/domain of discourse for propositions with x/
If it is P, it is Q (where 'it' refers back to the things in the domain of discourse)
In common usage, people don't worry about domain of discourse much; but technically you can't have a variable without it being a variable capable of ranging over a domain of some sort, so it's always there, and necessarily so.
In SETL, as with any term logic, domain of discourse is much less important, but it still can be defined for every sentence, and usefully so, because every statement is true if and only if it denotes its domain of discourse. In SETL every term has a domain (or, if you prefer, category), and the domain of discourse for any sentence is the intersection of the domains (or, if you prefer, categories) of all its terms. Thus
We symbolize the domain (or, if you prefer, category) indicated by the term D with /D/. This is what I am calling 'category nominalization' or 'domain nominalization'. /D/ consists of everything that is D or nonD, where D and nonD are both taken to be part of a category. Thus if D is 'red', then /red/ consists of everything that can be truly characterized by 'red' or 'nonred', where the latter is understood in such a way that it only applies to things falling in the same category as red things. Thus, blue things might fall under this category, but not the number two or the pain in my left hand, because these are a different category (we can meaningfully say of them that they are not the sort of thing that could be either red or nonred). Now, we can use this to handle a particular type of proposition:
Everything is P
Something is P.
In these the subject is the nominalized domain, so they are respectively translated as:
-/P/+P
+/P/+P
Or in other words, every member of the category or domain associated with P is P; some member of the category or domain associated with P is P. (Of course, things get more complicated if we need to use a larger domain of which P is only part; e.g., if there are lots of sentences, and we need to say that everything in the domain of all the sentences is P. But it works the same way.)
In the next post I will look briefly at some loose ends that are not covered by the above points.
Sommers-Englebretsen Term Logic, Part I
The purpose of this post and its sequels is to provide a basic introduction to elements in the Sommers-Englebretsen Term-Functor Logic, also called TFL, or, as I will tend to call it, SETL. This is a way of handling propositions that has a great many advantages, being more closely allied to natural language than modern predicate logic, and being surprisingly flexible and powerful, given how simple it is.
Three Kinds of Opposition
The most natural place to start if you want to understand how SETL works is to look at what sorts of opposition can be logically relevant. SETL starts from the basic idea that every categorical assertion is the affirmation or denial of a simple or complex predicate of all or some of a subject. When you look at assertions in this way, we can see that they admit of three basic kinds of oppositions
1. Opposition of Quality. Every term has either a positive or a negative term quality. 'Red' would be an example of a term with positive term quality; 'Nonred' would be an example of a term with negative term quality. Likewise, every predicate has either a positive or a negative predicate quality. 'Is red' has a positive predicate quality; 'Isn't red' has a negative predicate quality. One of the features of SETL is that there is no significant distinction between term quality and predicate quality; 'S is non-P' is not significantly different from 'S isn't P'. Changing the quality does make an important difference to a logical argument, so this opposition, which I will (following Englebretsen) call C-opposition, because it is the foundation of logically contrary propositions.
2. Opposition of Quantity. Every predicate is predicated of some or all of a subject. This sort of opposition, which will be called Q-opposition, is an opposition between a universal subject and a particular subject. So 'Some S is P' is Q-opposed to 'All S is P'.
3. Predicative Opposition. Every predicate is affirmed or denied of its subject. This is the third opposition, which we will call P-opposition. 'It is not the case that S is P' is P-opposed to '(It is the case that) S is P'.
The upshot is as follows.
(a) Every categorical assertion has a subject (S) and a predicate (P).
(b) Every term, independently of its role in the assertion, has a mark of C-opposition.
(c) Every P as a complete term has a mark of C-opposition and as a predicate has a mark of P-opposition.
(d) Every S is a term with a mark of Q-opposition.
Given these four basics, which I will not argue for here, we can develop the basic format of SETL.
Plus and Minus
We have three oppositions. Sommers's great idea was to take these oppositions and note them down as plus and minus in a subject-predicate proposition. So we have (on the basis of (a) above)
S...P
as our assertion. However, we know from (b) that every term has its own mark of C-opposition. Thus:
(±S)...(±P)
We know from (d) that every subject has its mark of Q-opposition. Thus:
±(±S)...(±P)
And since each predicate may be itself a complex term, it has as predicate another C-opposition mark (c). Thus:
±(±S)±(±P)
And we know from (c) as well that every predicate, as predicate, has a P-opposition mark, which we can symbolize, putting the P-opposition mark over the whole predication (and thus at the beginning) as:
±(±(±S)±(±P))
Of course, this is just a general format. Let's take a basic assertion: All S is P. This can be symbolized by:
+(-(+S)+(+P))
S and P are both of positive quality (thus their positive C-opposition signs); P is affirmed of S (thus its positive P-opposition sign); and S is of universal quantity (thus its negative Q-opposition sign). This is pretty intuitive, except, perhaps, for the reason why the universal quantity is given a minus and the particular quantity is given a plus. The reason for this is (if you want the crude and read version) is that if we do it this way the whole thing works. More technically, however, we make the universal minus and the particular plus in order to preserve the contraposition of the A categorical (All S is P) and the conversion of the I categorical (Some S is P). That is, we want these two equivalences:
+(-(+S)+(+P)) = + (-(-P)+(-S)) [i.e., All S is P is equivalent by contraposition to All nonP is not nonS]
+(+(+S)+(+P)) = +(+(+P)+(+S)) [i.e., Some S is P is equivalent by conversion to Some P is S]
It's easy to recognize these equivalences if we give the universal a minus and the particular a plus. But that's the only tricky thing about this basic format: + and - simply indicate an opposition, and '-' in particular shouldn't be confused with negation.
In the above format, all we've marked are the terms and their oppositions. Which opposition is relevant is entirely a matter of where it is positioned in the assertion, so we don't have to worry about distinguishing them in any other way; + and - will do for everything. And that's where it gets neat. The really neat stuff we'll get to later. For now, we'll note just one neat feature that this way of symbolizing yields us. If we treat +'s as we usually treat +'s (e.g., in math), we can contract a string of plus signs. Thus,
+(+(+S)+(+P))
can be written as
+S+P
without any loss of logical function. Likewise, we can treat -'s in a complementary way, such that two minuses together become a plus, and a minus and plus contract to a minus. Thus
-(+S)-(-P)
can be written as
-S+P without any loss of logical function. They are always logically equivalent, although in their expanded forms they may look different. We can then give a simplifed form to all the basic Aristotelian categoricals:
A (All S is P) -S+P
E (No S is P) -S-P
I (Some S is P) +S+P
O (Some S is not P) +S-P
But, given that we can do all Aristotelian syllogisms. A syllogism works when it can be formulated as a true equation and both sides are similar. Take the famous Baraba (AAA) syllogism:
All S is M
All M is P
Therefore, All S is P.
This has the equation:
(-S+M) + (-M+P) = -S+P
Just treat it as you would treat it if it were an algebra equation. You'll see that the left side is indeed equal to the right. All we have to do in order to be certain that it is a valid syllogism is to make sure that the two sides are similar. The two sides are said to be similar if (a) they have the same extremes (i.e., terms that are not arithmetically eliminable); and (b) they have the same quantity (the conjunction including a particular always being particular). In the Barbara case, the sides are clearly similar. Therefore it is valid. We can even handle 'weakened syllogisms' (syllogisms with universal premises that have particular conclusions) if we assume that they have the hidden premise +S+S (which, as we'll see, is a tautology and can be introduced at will). Thus Camestrop (AEO) would be:
(-S-M) + (-P+M) + (+S+S) = +S-P
This is all quite cool. But I'm partly getting ahead of myself here. SETL is more powerful than I've suggested so far, and we need to introduce a few additional tools if we are to see this and handle all the kinds of argument SETL is capable of handling. So in the next post on this subject we'll look at how SETL handles various key issues (singular terms, relations, identities, meta-propositions, existence). And then we'll handle arguments.
Three Kinds of Opposition
The most natural place to start if you want to understand how SETL works is to look at what sorts of opposition can be logically relevant. SETL starts from the basic idea that every categorical assertion is the affirmation or denial of a simple or complex predicate of all or some of a subject. When you look at assertions in this way, we can see that they admit of three basic kinds of oppositions
1. Opposition of Quality. Every term has either a positive or a negative term quality. 'Red' would be an example of a term with positive term quality; 'Nonred' would be an example of a term with negative term quality. Likewise, every predicate has either a positive or a negative predicate quality. 'Is red' has a positive predicate quality; 'Isn't red' has a negative predicate quality. One of the features of SETL is that there is no significant distinction between term quality and predicate quality; 'S is non-P' is not significantly different from 'S isn't P'. Changing the quality does make an important difference to a logical argument, so this opposition, which I will (following Englebretsen) call C-opposition, because it is the foundation of logically contrary propositions.
2. Opposition of Quantity. Every predicate is predicated of some or all of a subject. This sort of opposition, which will be called Q-opposition, is an opposition between a universal subject and a particular subject. So 'Some S is P' is Q-opposed to 'All S is P'.
3. Predicative Opposition. Every predicate is affirmed or denied of its subject. This is the third opposition, which we will call P-opposition. 'It is not the case that S is P' is P-opposed to '(It is the case that) S is P'.
The upshot is as follows.
(a) Every categorical assertion has a subject (S) and a predicate (P).
(b) Every term, independently of its role in the assertion, has a mark of C-opposition.
(c) Every P as a complete term has a mark of C-opposition and as a predicate has a mark of P-opposition.
(d) Every S is a term with a mark of Q-opposition.
Given these four basics, which I will not argue for here, we can develop the basic format of SETL.
Plus and Minus
We have three oppositions. Sommers's great idea was to take these oppositions and note them down as plus and minus in a subject-predicate proposition. So we have (on the basis of (a) above)
S...P
as our assertion. However, we know from (b) that every term has its own mark of C-opposition. Thus:
(±S)...(±P)
We know from (d) that every subject has its mark of Q-opposition. Thus:
±(±S)...(±P)
And since each predicate may be itself a complex term, it has as predicate another C-opposition mark (c). Thus:
±(±S)±(±P)
And we know from (c) as well that every predicate, as predicate, has a P-opposition mark, which we can symbolize, putting the P-opposition mark over the whole predication (and thus at the beginning) as:
±(±(±S)±(±P))
Of course, this is just a general format. Let's take a basic assertion: All S is P. This can be symbolized by:
+(-(+S)+(+P))
S and P are both of positive quality (thus their positive C-opposition signs); P is affirmed of S (thus its positive P-opposition sign); and S is of universal quantity (thus its negative Q-opposition sign). This is pretty intuitive, except, perhaps, for the reason why the universal quantity is given a minus and the particular quantity is given a plus. The reason for this is (if you want the crude and read version) is that if we do it this way the whole thing works. More technically, however, we make the universal minus and the particular plus in order to preserve the contraposition of the A categorical (All S is P) and the conversion of the I categorical (Some S is P). That is, we want these two equivalences:
+(-(+S)+(+P)) = + (-(-P)+(-S)) [i.e., All S is P is equivalent by contraposition to All nonP is not nonS]
+(+(+S)+(+P)) = +(+(+P)+(+S)) [i.e., Some S is P is equivalent by conversion to Some P is S]
It's easy to recognize these equivalences if we give the universal a minus and the particular a plus. But that's the only tricky thing about this basic format: + and - simply indicate an opposition, and '-' in particular shouldn't be confused with negation.
In the above format, all we've marked are the terms and their oppositions. Which opposition is relevant is entirely a matter of where it is positioned in the assertion, so we don't have to worry about distinguishing them in any other way; + and - will do for everything. And that's where it gets neat. The really neat stuff we'll get to later. For now, we'll note just one neat feature that this way of symbolizing yields us. If we treat +'s as we usually treat +'s (e.g., in math), we can contract a string of plus signs. Thus,
+(+(+S)+(+P))
can be written as
+S+P
without any loss of logical function. Likewise, we can treat -'s in a complementary way, such that two minuses together become a plus, and a minus and plus contract to a minus. Thus
-(+S)-(-P)
can be written as
-S+P without any loss of logical function. They are always logically equivalent, although in their expanded forms they may look different. We can then give a simplifed form to all the basic Aristotelian categoricals:
A (All S is P) -S+P
E (No S is P) -S-P
I (Some S is P) +S+P
O (Some S is not P) +S-P
But, given that we can do all Aristotelian syllogisms. A syllogism works when it can be formulated as a true equation and both sides are similar. Take the famous Baraba (AAA) syllogism:
All S is M
All M is P
Therefore, All S is P.
This has the equation:
(-S+M) + (-M+P) = -S+P
Just treat it as you would treat it if it were an algebra equation. You'll see that the left side is indeed equal to the right. All we have to do in order to be certain that it is a valid syllogism is to make sure that the two sides are similar. The two sides are said to be similar if (a) they have the same extremes (i.e., terms that are not arithmetically eliminable); and (b) they have the same quantity (the conjunction including a particular always being particular). In the Barbara case, the sides are clearly similar. Therefore it is valid. We can even handle 'weakened syllogisms' (syllogisms with universal premises that have particular conclusions) if we assume that they have the hidden premise +S+S (which, as we'll see, is a tautology and can be introduced at will). Thus Camestrop (AEO) would be:
(-S-M) + (-P+M) + (+S+S) = +S-P
This is all quite cool. But I'm partly getting ahead of myself here. SETL is more powerful than I've suggested so far, and we need to introduce a few additional tools if we are to see this and handle all the kinds of argument SETL is capable of handling. So in the next post on this subject we'll look at how SETL handles various key issues (singular terms, relations, identities, meta-propositions, existence). And then we'll handle arguments.
Five Points of Calvinism
Strict Calvinism is a position that for most people is more easily caricatured than understood. This is unfortunate, because whatever its weaknesses may be, it has been carefully thought out. So I thought I'd say something about a common misunderstanding of Calvinism, one which involves a false view of what the 'Five Points' or 'TULIP' aspect of Calvinism really involves.
To understand the Five Points properly, you have to understand that the Five Points (Total Depravity, Unconditional Election, Limited Atonement, Irresistible Grace, and Perseverance of the Saints) are not the heart of Calvinism. They are not the most important doctrines for Calvinists generally, nor are they especially central to the Calvinist way of life. They may be in particular cases; but not generally.
The reason is that the Five Points were not formulated in order to sum up the Calvinist view of the world, but to sum up how Calvinists were distinguished from the followers of Arminius. While 'Arminian' tends to be used loosely, genuine Arminians (those in the Remonstrant tradition) actually share a lot of common ground with Calvinists. They are close cousins. However, they are also often in disputes with each other over issues related to atonement and free will; and it's necessary to have a clear way to distinguish the two. Enter the Five Points: the Five Points are things strict Calvinists agree on that Arminians don't. They are important in the sense that they are important for distinguishing Calvinists from Arminians; and since Calvinists very often have to distinguish themselves from Arminians, they come up a lot. But this does not mean that they are the most important Calvinist doctrines; nor does it mean that all Calvinists will regard all of the Five Points as being of equal importance. It certainly doesn't mean that Calvinists go to church each Sunday and discuss nothing but Total Depravity and Limited Atonement.
In fact, the Calvinists are in a sense not the ones responsible for the Five Points. The Five Points are nothing other than a summary of the response by the Synod of Dordt to the 'Remonstrance' of the Arminians (although probably not exclusively those articles), arranged on a point-by-point basis to correspond to those five articles. If you're a Calvinist and have difficulty believing that the Arminians are among your closest cousins, I recommend you read the five points of the Remonstrance and try to identify precisely where you differ from the strict Arminian view. They definitely differ; but the difference is subtle. The two groups can say almost identical things but mean them rather differently. This is because Calvinists and Arminians are not diametrical opposites; the Synod of Dordt didn't go through the Arminian list and simply contradict each point (they were better theologians than that). The Five Points of Calvinism and the Five Articles of Remonstrance overlap; they distinguish by not coinciding. So the Five Points are just a summary of the Calvinist response to the Arminians, on issues determined by the Arminians themselves; they are not a Calvinist summary of the core of Calvinism, but a Calvinist summary of Calvinist doctrine on points chosen by the Arminians (who were protesting the Belgic Confession on only a small handful of points, however important those points may be).
The Five Points, in other words, are not the heart of Calvinism, but an outer perimeter for it. Think of the Calvinist neighborhood as a group of houses. In one house we have the Calvinists in the strict and proper sense, and around this house we have other houses that are in many ways similar but in important ways not the same -- Amyraldians ('Four Point Calvinists', who accept all the Five Points except Limited Atonement), Arminians, and the like. The Five Points are the five posts that hold up the fence that distinguishes the Calvinist lot from the others. Now, that fence turns out to have a good deal of importance, because people keep trying to knock it down. So Calvinists end up arguing with their neighbors about the fence a lot. But no Calvinists in their right mind would hold that the fence is where they eat and sleep. It is not the heart and hearth and home of a healthy Calvinist life; and, however important the fence may be, obsessing about the fence to the detriment of the house is a case of bad priorities. The heart and hearth and home of Calvinism is elsewhere -- in the Trinity, and in Chalcedonian Christology, and in the centrality of Scripture, and so forth. (It's worthwhile on this point to read the Belgic Confession, which, because it was written in order to identify things they would not compromise on even in the face of persecution, is much closer to being a summary of the heart of Calvinism; then compare it with the impoverished view you'd get if you thought that TULIP were the heart.)
That's would I would think is the most important thing to keep in mind in order not to misunderstand Calvinism and Reformed traditions. I'm not Calvinist, though; one thing that would be neat, I think, if it is possible, is for Calvinist bloggers, like Jeremy or Rebecca or David Wayne or any others to say something about what they think is the single most important thing to keep in mind in order to avoid collapsing into a mere caricature of Calvinism or Reformed life.
To understand the Five Points properly, you have to understand that the Five Points (Total Depravity, Unconditional Election, Limited Atonement, Irresistible Grace, and Perseverance of the Saints) are not the heart of Calvinism. They are not the most important doctrines for Calvinists generally, nor are they especially central to the Calvinist way of life. They may be in particular cases; but not generally.
The reason is that the Five Points were not formulated in order to sum up the Calvinist view of the world, but to sum up how Calvinists were distinguished from the followers of Arminius. While 'Arminian' tends to be used loosely, genuine Arminians (those in the Remonstrant tradition) actually share a lot of common ground with Calvinists. They are close cousins. However, they are also often in disputes with each other over issues related to atonement and free will; and it's necessary to have a clear way to distinguish the two. Enter the Five Points: the Five Points are things strict Calvinists agree on that Arminians don't. They are important in the sense that they are important for distinguishing Calvinists from Arminians; and since Calvinists very often have to distinguish themselves from Arminians, they come up a lot. But this does not mean that they are the most important Calvinist doctrines; nor does it mean that all Calvinists will regard all of the Five Points as being of equal importance. It certainly doesn't mean that Calvinists go to church each Sunday and discuss nothing but Total Depravity and Limited Atonement.
In fact, the Calvinists are in a sense not the ones responsible for the Five Points. The Five Points are nothing other than a summary of the response by the Synod of Dordt to the 'Remonstrance' of the Arminians (although probably not exclusively those articles), arranged on a point-by-point basis to correspond to those five articles. If you're a Calvinist and have difficulty believing that the Arminians are among your closest cousins, I recommend you read the five points of the Remonstrance and try to identify precisely where you differ from the strict Arminian view. They definitely differ; but the difference is subtle. The two groups can say almost identical things but mean them rather differently. This is because Calvinists and Arminians are not diametrical opposites; the Synod of Dordt didn't go through the Arminian list and simply contradict each point (they were better theologians than that). The Five Points of Calvinism and the Five Articles of Remonstrance overlap; they distinguish by not coinciding. So the Five Points are just a summary of the Calvinist response to the Arminians, on issues determined by the Arminians themselves; they are not a Calvinist summary of the core of Calvinism, but a Calvinist summary of Calvinist doctrine on points chosen by the Arminians (who were protesting the Belgic Confession on only a small handful of points, however important those points may be).
The Five Points, in other words, are not the heart of Calvinism, but an outer perimeter for it. Think of the Calvinist neighborhood as a group of houses. In one house we have the Calvinists in the strict and proper sense, and around this house we have other houses that are in many ways similar but in important ways not the same -- Amyraldians ('Four Point Calvinists', who accept all the Five Points except Limited Atonement), Arminians, and the like. The Five Points are the five posts that hold up the fence that distinguishes the Calvinist lot from the others. Now, that fence turns out to have a good deal of importance, because people keep trying to knock it down. So Calvinists end up arguing with their neighbors about the fence a lot. But no Calvinists in their right mind would hold that the fence is where they eat and sleep. It is not the heart and hearth and home of a healthy Calvinist life; and, however important the fence may be, obsessing about the fence to the detriment of the house is a case of bad priorities. The heart and hearth and home of Calvinism is elsewhere -- in the Trinity, and in Chalcedonian Christology, and in the centrality of Scripture, and so forth. (It's worthwhile on this point to read the Belgic Confession, which, because it was written in order to identify things they would not compromise on even in the face of persecution, is much closer to being a summary of the heart of Calvinism; then compare it with the impoverished view you'd get if you thought that TULIP were the heart.)
That's would I would think is the most important thing to keep in mind in order not to misunderstand Calvinism and Reformed traditions. I'm not Calvinist, though; one thing that would be neat, I think, if it is possible, is for Calvinist bloggers, like Jeremy or Rebecca or David Wayne or any others to say something about what they think is the single most important thing to keep in mind in order to avoid collapsing into a mere caricature of Calvinism or Reformed life.
Friday, August 25, 2006
The New Atlantis
The new issue of The New Atlantis is up (h/t: prosthesis).
Shop Class as Soulcraft discusses the value of manual crafts. It's quite good and well worth reading.
The Self-Portrait of a Scientist discusses scientific memoirs. My favorite of those mentioned is Planck's, which should be required reading for everyone.
Shop Class as Soulcraft discusses the value of manual crafts. It's quite good and well worth reading.
The Self-Portrait of a Scientist discusses scientific memoirs. My favorite of those mentioned is Planck's, which should be required reading for everyone.
Thursday, August 24, 2006
SG-1
As one might expect, a Save Stargate SG-1 campaign has started. It has the great merit, however, of being different from most such campaigns. One of the major differences is that both Stargate Productions and MGM(who own the relevant rights) have indicated that they are strongly supportive of continuing SG-1 in some form. MGM in particular has found SG-1 to be a very lucrative investment; unlike SciFi and other networks, which only receive income from SG-1 through advertising, MGM benefits from network licensing, overseas distribution, DVD sales, game sales, and other merchandise, and the show has been very good to them on all such accounts, so they are naturally interested in continuing with it if there is any feasible way to do so. It's just unclear at present whether continuing with it will mean continuing in the same format (another season) or shifting formats (e.g., TV movies or feature films). So the movement is not a protest movement (as save-the-show campaigns are) but a support movement -- to support the studio in its effort to find some definite way to continue SG-1 beyond Season 10, whether that continuation be in a new season on another channel, a mini-series, a feature film, or what have you. In essence, it's not trying to change minds so much as trying to make sure that plans already in the air don't fizzle out.
In any case, the web headquarters are here, for those interested in more information about it.
[UPDATE: According to this, SciFi is using its contract to block MGM from giving a new SG-1 season to another (U.S.) channel. But there are still the other possibilities.]
In any case, the web headquarters are here, for those interested in more information about it.
[UPDATE: According to this, SciFi is using its contract to block MGM from giving a new SG-1 season to another (U.S.) channel. But there are still the other possibilities.]
A Poem Draft
Dhruvasimha
Beyond the first awareness is the seed,
a source untouched by any craving need,
a spark forever steadfast in its light,
constant in reflection and in fight,
where thinker is but thought, and doer deed.
Sacred text in hand, the lion waits;
teaching is the path through golden gates
that reach to other realms and then
the abyss of light beyond all human ken.
One right question every answer dissipates.
A lion for reflection on the plains
of deep delusion, in the falling rains
looks out on golden grasses and the sky.
The golden eyes outlooking wait to die.
When self is overcome, no self remains;
thoughts beyond all craving know no pain.
Beyond the first awareness is the seed,
a source untouched by any craving need,
a spark forever steadfast in its light,
constant in reflection and in fight,
where thinker is but thought, and doer deed.
Sacred text in hand, the lion waits;
teaching is the path through golden gates
that reach to other realms and then
the abyss of light beyond all human ken.
One right question every answer dissipates.
A lion for reflection on the plains
of deep delusion, in the falling rains
looks out on golden grasses and the sky.
The golden eyes outlooking wait to die.
When self is overcome, no self remains;
thoughts beyond all craving know no pain.
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