Saturday, February 13, 2010

Sommers Notation, Part IV

(Part III)

Conjunctive and Disjunctive Predicates

Something we have not directly dealt with before is how to handle something like this:

Every dog is both a mammal and a vertebrate.

In this we are predicating a conjunction of attributes (mammal and vertebrate) to every dog. It turns out that conjunctive predicates are easily handled in Sommers notation. We could translate the above proposition along the following lines:


Recognizing this allows us to identify some obvious inference rules for conjunctive predicates. For instance, we can obviously have a Rule of Conjunctive Predicate Simplification (CPSimp):

±X+(+Y+Z) ∴±X+Y or ±X+Z

And associative shift works here, as well; from

-D+(+M+V) = Every dog is a mammal and a vertebrate (or, equally, Every dog is a mammal that is a vertebrate)

we can conclude

-(+D+M)+V = Every dog that is a mammal is a vertebrate

We could have more if we choose to elaborate them. But I want to ask the next major question, which is how to handle disjunctive predicates, as in the following:

Every dog is either a mammal or a vertebrate.

We can think this through and use what we know of how to handle conjunctive predicates to handle this new kind of predicate. If every dog is either a mammal or vertebrate, what is definitely ruled out is the case where all the dogs are both non-mammals and non-vertebrates. Thus we can take disjunctive predicates to deny this conjunctive predicate:


We could develop rules for handling disjunctive predicates, as well; but we will not do so here.


In propositional logic every use of a variable technically requires a domain of discourse (often called a universe of discourse). In Sommers notation, 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 Sommers notation 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.

For convenience, I will symbolize the domain itself (or, if you prefer, category) with a bolded term, like D, and indicate what falls within that domain with /D/. This latter 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 belong to the domain or universe indicated by D. Thus if we take the ordinary term 'red', then red is the domain that consists of red and nonred things (we could just as easily call it nonred or red/nonred), i.e., which consists of all and only those things which could intelligibly (even if wrongly) be characterized by ‘red’ and ‘nonred’; 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). Similarly, /red/ indicates the actual members of this domain. We would use red to talk about the domain of red and nonred things itself; we would use /red/ to talk about what is included or excluded from it. 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:


Or in other words, every member of the category or domain P is P; some member of the category or domain P is P.

It is by combining domains with conjunctive and disjunctive predicates that we are able to handle noncategorical logic: propositional logic derives from an analysis of predication. To see this we need to learn how to nominalize propositions and therefore use whole propositions as terms.

Propositional Nominalization

In order to get to noncategorical propositions, the question we want to ask ourselves is: How can we handle propositions about propositions? 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" (or perhaps more accurately, to indicate what we mean when we say, "(such) that all love is easy"), we get the following rendering:


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.


In nominalizing, we have embedded one sentence in another by treating it as a term. This allows us to do a number of things.

(Part V)

A Beauty in Its Doubt

A Sea-Side Walk
by Elizabeth Barrett Browning

We walked beside the sea,
After a day which perished silently
Of its own glory---like the Princess weird
Who, combating the Genius, scorched and seared,
Uttered with burning breath, 'Ho! victory!'
And sank adown, an heap of ashes pale;
So runs the Arab tale.

The sky above us showed
An universal and unmoving cloud,
On which, the cliffs permitted us to see
Only the outline of their majesty,
As master-minds, when gazed at by the crowd!
And, shining with a gloom, the water grey
Swang in its moon-taught way.

Nor moon nor stars were out.
They did not dare to tread so soon about,
Though trembling, in the footsteps of the sun.
The light was neither night's nor day's, but one
Which, life-like, had a beauty in its doubt;
And Silence's impassioned breathings round
Seemed wandering into sound.

O solemn-beating heart
Of nature! I have knowledge that thou art
Bound unto man's by cords he cannot sever---
And, what time they are slackened by him ever,
So to attest his own supernal part,
Still runneth thy vibration fast and strong,
The slackened cord along.

For though we never spoke
Of the grey water and the shaded rock,---
Dark wave and stone, unconsciously, were fused
Into the plaintive speaking that we used,
Of absent friends and memories unforsook;
And, had we seen each other's face, we had
Seen haply, each was sad.

Thursday, February 11, 2010

Drink, Pilgrim, Here; Here Rest!

Inscription for a Fount
by Samuel Taylor Coleridge

This Sycamore, oft musical with bees,--
Such tents the Patriarchs loved! O long unharmed
May all its ag├ęd boughs o'er-canopy
The small round basin, which this jutting stone
Keeps pure from falling leaves! Long may the Spring,
Quietly as a sleeping infant's breath,
Send up cold waters to the traveller
With soft and even pulse! Nor ever cease
Yon tiny cone of sand its soundless dance,
Which at the bottom, like a Fairy's Page,
As merry and no taller, dances still,
Nor wrinkles the smooth surface of the Fount.
Here Twilight is and Coolness: here is moss,
A soft seat, and a deep and ample shade.
Thou may'st toil far and find no second tree.
Drink, Pilgrim, here; Here rest! and if thy heart
Be innocent, here too shalt thou refresh
Thy spirit, listening to some gentle sound,
Or passing gale or hum of murmuring bees!

Wednesday, February 10, 2010

Sommers Notation, Part III

(Part II)

Now we get to the good stuff: actual inferences.


Loosely following Aristotle, we will call a perfect inference any inference that does not require a proof of its validity (because it is clearly valid); one in need of a proof of validity is imperfect. All basic rules of inference are patterns for perfect inferences. For our purposes we will not count as perfect inferences anything with more than two premises, however obviously valid it may be. Thus there are three types of perfect inference: two-premise perfect inferences, one-premise perfect inferences, and zero-premise perfect inferences. It may seem odd to talk about a 'zero-premise' inference; but what is meant is that you don't need a premise to introduce it. Thus all tautologies are zero-premise inferences; they can be introduced whenever you please.

We know that there are two necessarily contradictory types of proposition:

All S aren't S.
Some S aren't S.

Denying these, we get:

All S are S.
Some S are S.

These are tautologies; as will be "All S that are M are S" and "Some S that are M are S". From this we can formulate our first rule of inference, the Rule of Identity (Id):

∴ ±(±S±M)+ (±S)

where M may be empty (i.e., discardable). (In this rule, as in all the others we will use, ± means 'either plus or minus', and the assumption is that you will be consistent in which you choose.) Basically, the rule is that we may introduce a universal or particular tautology whenever we wish, without regard for prior premises. You may add –S+S or +S+S whenever you require. This will be the only zero-premise perfect inference we require.

Immediate Inference

A single-premise perfect inference is also called an immediate inference. The most obvious rule for such inferences is the Rule of Self Inference (SI):

±S±P ∴ ±S±P

Another rule of immediate inference has to do with universal affirmations; it is the Rule of Contraposition (Contrap), which I will break into two parts purely for the sake of clarity:

-(+S)+(+P) ∴ -(-P)+(-S)
-(-S)+(-P) ∴ -(+P)+(+S)

Basically this says that you can infer "All nonP is nonS" from "All S is P", and vice versa.

The third rule of inference has to do with particular affirmations; it is the Rule of Conversion (Con):

+(±S)+(±P) ∴ +(±P)+(±S)

That is,if you have +S+P, you can always conclude +P+S, and if you have –S-P, you can always conclude –P-S So we have rules governing affirmations. But we need rules for doing things with denials. The easiest way to do this is to formulate a rule for converting denials into affirmations; then the rules we have already will apply for all our needs. The Aristotelian claim that any denial is equivalent to an affirmation is very plausible. So we can formulate a Rule of Internal Negation Distribution (IN), which again I will split into two parts for clarity:

±S-(+P) ∴ ±S+(-P)
±S-(-P) ∴ ±S+(+P)

Related to this, we have a Rule of Double Negation (DN) to simplify and expand expressions; any (+T) term can expand to (-(-T)) and any (-(-T)) can simplify to (+T).

A useful additional rule is the Rule of Commutation (Com), which basically says that for any terms we can rewrite (+P+Q) as (+Q+P). If we want to, we can throw in a Rule of Association (Assoc) to handle groupings:

+S+(+Q+P) ∴ +(+S+Q)+P

It's also possible to have a Rule of Relational Simplification (Simp), to handle inferences like "John smiles at Mary; therefore John smiles." Thus we could things like:

±S+(+Q+P) ∴ ±S+Q

One last, and very useful, rule of immediate inference is the Rule of Composition (Comp):

-S+P ∴ -(+S+Q)+(+P+Q)

where Q is an adjective term (whether monadic or relational doesn't matter). This allows you to handle inferences like "All circles are figures; therefore all drawers of circles are drawers of figures" and "All men are mortal; therefore all rational men are rational mortals."

Mediate Inferences

With immediate inferences in hand we are ready to tackle syllogistic, which in term logic covers all mediate inferences (two premises or more). To do this we need to add just one basic rule, the extremely important dictum de omni et nullo (DDO), which can be put in the same form as our other rules to constitute a Rule of Mediate Inference (MI):

-M+P, ±S+M ∴ ±S+P

The basic point is this: whatever is affirmed of all of something is likewise affirmed of what that something is affirmed of.

Given this, we are doing swell. We can show that the syllogism Darapti is valid by the following proof:

1. -M+P premise
2. -M+S premise
3. +M+M Id
4. +M+P MI from 1,3
5. +P+M Conv, from 4
6. +P+S MI from 2,5
7. +S+P Conv from 6

Thus every Darapti syllogism, i.e., every mediate inference of the form "All M is P; All M is S; therefore some S is P" is valid.

It's easy enough to identify the necessary and sufficient conditions for validity in Sommers notation. They are:

(1) A universal conclusion is only validly drawn from universal premises.
(2) A particular conclusion is only validly drawn from a set of premises that contains one and only one particular premise.
(3) The premises in a valid inference sum up to the conclusion.

So this is invalid:

∴ -M+P

(2) is irrelevant; it meets (1), but the sum of the premises is -M+P+S. This is also invalid:

∴ +S+P

(1) is irrelevant; it meets (3), but it tries to draw a particular conclusion from two particular conclusions.

Relational Arguments

Consider the following inference:

Some atheist mocks every prayer.
Every Shiite recites a prayer.
∴ Every Shiite recites what some atheist mocks.

In Sommers notation the premises will be:

1. +A1+(M12-P2)
2. -S3+(R32+P2)

The middle term here is P2. Thus (pretty much ignoring the parentheses, which I only put in here to make it easier to translate without accidental shift of meaning) by MI we can conclude:

3. +A1+(M12-S3+R32)

This is clearly a valid syllogism; the first condition for validity is irrelevant, and it meets both the second and the third. And taking this conclusion, by Com we get:

4. -S3+(R32+A1+M12)

Relational Simplification Arguments

Consider the following argument: John smiles at Laura; therefore, John smiles. This is easily handled by Sommers:

1. ±J+(+S±L) premise
2. (±J+S)±L Assoc from 1
3. ±J+S Simp from 2

Similarly, for "John smiles at some person; therefore, John smiles":

1. ±J+(+S+P) premise
2. (±J+S)+P Assoc from 1
3. ±J+S Simp from 2

Associative Shift

Consider the argument:

Someone Thomas studied was Aristotle.
Therefore, Thomas studied Aristotle.

Compare it to:

Someone Thomas studied was the disciple of Aristotle.
Therefore, Thomas studied the disciple of Aristotle.

These are easily handled in Sommers notation by association. The first argument becomes:

1. (±T1+S12)±A2 premise
2. ±T1+(S12±A2) Assoc from 1

And the second:

1. (±T1+S12)+(+D±A2) premise
2. ±T1+(S12+(+D±A2)) Assoc from 1

The form of argument is very simple and straightforward in both cases.

Passive Transformation

Consider the argument, "Paris loves Helen; therefore Helen is loved by Paris." In Sommers notation:

1. ±P1+(L12±H2) premise
2. (±P1+L12)±H2 Assoc from 1
3. ±H2+(±P1+L12) Comm from 2
4. ±H2+(L12±P1) Comm from 3

So far we have considered only categorical propositions. Sommers notation can also handle noncategorical propositions. Before we get to them, however, it will be helpful to look at two other features of Sommers notation: its handling of conjunctive and disjunctive predicates and its handling of domains of discourse.

(Part IV)

Dangle and Hook

I was in a conversation today with someone who tried to enlist me for Amway -- or Quixtar, as the North American portion of the company is now called. He is a genuinely nice guy, very likable, and he threw a decent sales pitch, but from a rhetorical point of view he should have led off with the Exciting Products rather than the Multi-Level Marketing; that immediately raises specters of pyramid schemes, and had the effect not of exciting me but amusing me. Is there really anyone these days who thinks that even legitimate MLM's (Avon, Quixtar, and the like) are a reliable way to make more than pocket change? Perhaps only the salesmen. But perhaps in this economic climate people are more likely to make such leaps out of hope for something better; these are times when even pocket change is tempting for many people.

Also (although he could not have known this, despite knowing that I taught philosophy), he should not have put so much emphasis on the dangling lure of money and nice cars and houses; I had just been reading up on Boethius's Consolation of Philosophy (Relihan's The Prisoner's Philosophy in particular, on which I will probably eventually have a post) and therefore had in my mind very firmly that all-important distinction between the fractured false goods of fortune and true goods. Yes, I am paid relatively little for what I do, but absolutely speaking I need relatively little. Plus, as God and the Devil know, the only material goods that are truly significant temptations for me are books. But despite the fact that I will certainly run into him again and, not being able just to flake, will now have to come up with a gentle way to let down a tenacious salesman who is a genuinely nice guy and is mostly just trying to make a few dollars (he had a spinal injury that limits his job options -- he can't do any sort of lifting and has only a high school education), and despite the fact that I would much rather have spent my lunch working on the design for a hybrid course I'm developing, the whole discussion was really quite worth it: when I teach Boethius later this term I will have a real-life example of the Consolation of Philosophy as a defense against folly.

Had I known I would have had a conversation like this, though, I would have brought a multi-level marketing bingo card. I would not have had Bingo, but I only missed a definition of insanity. As I said, the man in question is a nice guy, but the funniest thing in the whole discussion was that at an early point I thought, "If I tell him I want to do independent research, he's going to scramble to try to make sure that I don't pay attention to critical websites." And when I mentioned later that I'm not the sort of person who makes decisions without research, he gave me some websites but also immediately launched into a spiel about why I shouldn't pay attention to websites by people who 'did absolutely nothing, or thought they were doing it but weren't and so became disappointed when it's really a matter of building an asset over time'. (To his credit, he didn't try, as most MLM people do, to suggest that it worked without effort.) While he would never have admitted it, of course, I could tell from his eyes he was disappointed that I didn't just jump right into the mix.

He did make me want to buy Believe: The Movie, though.

ADDED LATER: Speaking of which, I love these previews:

Edith Stein on Reading John of the Cross

"Pure love" for our holy Father John of the Cross means loving God for his own sake, with a heart that is free form all attachment to anything created: to itself and to other creatures, but also to all consolatins and the like which God can grant the soul, to all particular forms of devotion, etc.; with a hear tthat wants nothing more than that God's will be done, that allows itself to be led by God without any resistance. What one can do oneself to attain this goal is treated in detail in the Ascent of Mount Carmel. How God purifies the soul, in the Dark Night. The result, in the Living Flame and the Spiritual Canticle. (Basically, the whole way is to be found in each of the volumes, but each time one or the other of the stages is predominant.)

Edith Stein, Self Portrait in Letters, 1916-1942, Koeppel, tr., ICS Publications (Washington, DC: 1993) Letter 311 (to Sr. Agnella Stadtmüller), p. 318.

Tuesday, February 09, 2010

Random Musing on Politics

In The Mind of the Maker, Dorothy Sayers suggests that every work of art or craft should be seen as exhibiting three aspects, analogous to the Father, the Son, and the Holy Spirit in Christian theology. Elsewhere she refers to these three aspects as the Idea, the Energy, and the Power. The Idea is what is unfolded in the work of art; the Energy is what we would often call the execution; and the Power is roughly its ability to initiate a response. In principle every artist aims to find the perfect proportion and balance among these three. In practice no one succeeds: everyone's triangle is scalene, sometimes slightly, sometimes severely.

Vico regarded government as a factibile, something made, and I think one can, as a rough but interesting exercise, think of governments as exhibiting these three. And as with works of art, every triangle is at least slightly scalene, and sometimes severely so.

I was somehow put in mind of this by a side comment in Arsen's post on archaism and futurism:

The French Revolution serves as a classical and large-scale example of a real try at futurism. Its henchmen even changed the names of the months on the calendar. The current Tea Party movement serves as a minor example of an archaist reaction to the initiatives of the Obama administration. That administration is wrongly seen as futurist in its intentions. Alas, it’s merely rational—but unfortunately also inept. Its futurist coloration is a very pale shade of pink and derives from the illusion that change can be made almost entirely by magical PR gestures intended to influence public opinion.

This does seem to be a besetting sin -- an artistic heresy, to use Sayers's phrase -- of the Obama administration, one that was actually foreshadowed, I think, by some of Obama's approach to campaigning. The administration takes an Idea, often a very interesting Idea worth developing, but then goes directly for the Power, as if the Idea itself would overwhelm all opposition by a kind of intrinsic magic. This was much the way Obama campaigned, too: ideas robed in a cloud of BOMFOG and glitter.

One is tempted to contrast this with the Bush administration, with all its emphasis on energy and forcefulness and decision and changing the world, and say that that administration was all Energy and no Idea. But this is not right, I think, and would come from accepting too uncritically the administration's own view of itself. In fact, the Bush administration was pretty thoroughly inept, as well: it, too, was an Idea-ridden administration (Spread Democracy! Go to Mars!) that expected Ideas magically to glide directly into Power without the difficult, careful, intricate work -- the actual act of administration -- required to make it happen; it just happens that one of the Ideas with which the Bush administration was enamored was the Idea of what Hamilton called an "energetic magistrate". But that Idea, too, was an Idea that they simply expected to overwhelm with its own intrinsic force; the attempt actually to put it into practice was slipshod, confused, and inept. Their Idea of Energy had no Energy.

I suspect that you might actually find this to be a common problem among American administrations. The degree to which it is a problem no doubt varies from administration to administration -- no government is scalene in exactly the same way. But American-style campaigning, especially at the Presidential level, gives almost no room for making a choice of the President that takes Energy into account. We do not test out administrative competence on the campaign trail; it is not part of our run-up to the election that each candidate is required to spend a week as Acting President so we can see how they handle the day-to-day details. At most we look at their record, and even there we don't look at the day-to-day but at the big things they did. Nobody runs on a campaign platform of quiet and tidy management that slowly and carefully refines and improves what we have; they run on some big idea which they insist will change the way we do things forever. After all, what would be stirring about someone who said that his goal as President would be to improve government accounting practices, reorganize the executive branch to improve efficiency, and to initiate a more systematic ethics review system in order to reduce corruption? It often seems like the last time that this sort of thing was a campaign issue was in the struggle over the spoils system, and like the last time that a President made a name doing this sort of thing was Chester A. Arthur, who put an end to all sorts of corruption by pushing forward a regular civil service -- and that, beneficial as it was, still only got people's attention by being a big, flashy idea.

So perhaps democracies and republics are plagued by the problem of Energy-deficiency. That is indeed very close to how some theorists of democratic government have seen them. Montesquieu's solution (found encapsulated for American's in George Washington's Farewell address) was that this sort of problem could only be remedied by virtue among the people. So perhaps we are doomed, and our good intentions, piling up without the care and craft to put them to good effect, are paving the road into our inevitable future.

Sommers Notation, Part II

(Part I)

Singular Terms

It's easy enough to see how Sommers notation handles universal quantity, like 'All dogs are canines' or 'No dogs are felines':


It's also easy enough to see how Sommers notation handles particular quantity, like 'Some dogs are tame' or 'Some dogs are not housebroken':


But what if we have a singular term, like, 'Fido is a dog'? Every singular subject can be treated as having a 'wild quantity', because as far as the term itself goes it doesn’t matter whether you treat it as universal or as particular. Thus 'Fido is a dog' would be symbolized as:


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. Sometimes, to help keep track, it’s useful to mark a term with a star to indicate that it is singular, e.g.,


It would be possible to handle singular terms without this convention, however; one could treat singular propositions as always universal, but as implicitly paired with corresponding particular propositions. This would, in effect, be treating every singular proposition as a noncategorical proposition, but for practical purposes it would amount to the same thing.


Take the proposition, "All sophists take money from some fools". This is called a relational proposition because it has a term that relates other terms to each other. The basic format of this proposition is:


But the P term is a complex term consisting of other terms, and sometimes these terms play a role in inference. What can we do? Wecan expand the predicate in this way:


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:


The 1, 2, 3, indicates that the action of the term, T, is going from S to F through G.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, since we can use them as pronouns, for which see below. But for the most part we don’t need them to do so.)

On this basis we can translate any relational proposition you could want. Here are some examples and their translations.

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 logically equivalent, which is precisely the result you should get. There are more complicated terms 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 later. But even without this we can do a lot.


Suppose we have a sentence like: "Some boy kissed some girl and she clobbered him."

The first conjunct is easy: +B1+(K12+G2). The second conjunct has a pronoun, however. How will we handle this? Given this, we can represent the whole sentence as:


Singular pronouns are just singular terms, and are treated as such. Nonsingular pronouns act like normal terms. But the use of the subscripts in this way is just a matter of convenience, to keep track of the fact that we are dealing with pronouns.

(Singular) Identity and Existence

Because singular terms are indifferent to quantity and can be qualified, we can handle identities between singular terms very easily. 'Socrates is Socrates' becomes:

±S+S or

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 will not discuss it here.)

Just as identity is handled by normal predication in Sommers notation, so, too, are existential statements: existence is a predicate in Sommers notation. If I say, “Socrates exists,” I can represent it as:


There are other ways to handle this, as well, but we won’t get into them here.

Having looked at some of these we will get into actual arguments, starting with some simple ones. Then we will look at how Sommers can handle whole propositions as terms.

(Part III)

Monday, February 08, 2010

Fallacies and Invalid Argument Forms

At, someone asked the question:

How do you prove that a certain logical fallacy is a fallacy indeed? Are there "fallacies" about which there is a controversy if it is a fallacy or not? And if in the future, a new fallacy will be discovered, what will be the outline of the proof that one will have to use to prove that it exists? (Just an application of the first question.)

Nicholas Smith provides an answer that I think is dubious:

From the point of view of deductive logic, your question is very easily answered: a fallacy is an argument form in which the premises may all be true, but the conclusion false. To prove this, one provides what is called a "counterexample," which is simply a substitution instance that has the above characteristics.

We need to distinguish between two kinds of deductive systems. In a monotonic system, which is deductive in the strict sense, something like this answer is at least plausible, but in a nonmonotonic system it certainly would not. I suspect that Smith would classify nonmonotonic systems under 'inductive logic'; this would not be unheard of, although it is misleading given that nonmonotonic systems generally have nothing to do with induction, or at least no more to do with it than monotonic systems do.

But more than this, even with a monotonic system, the plausible answer is not right if we are not merely operating within the system but applying it to actual arguments. For to apply formal systems to actual arguments we must allow for implicit premises and enthymemes, and once this is recognized there turns out to be no good way to answer the first question. In particular cases you can show that there is no way to salvage an argument with implicit premises that would not be either unreasonable or provably wrong. But Smith's argument conflates 'invalid argument form' with 'fallacy' and this is untenable as a practical matter. Even a formal fallacy as straightforward as the one Smith gives (affirming the consequent) can be salvaged in particular applications with implicit premises, e.g., premises that combined with the other premises make the relation between p and q to be one of mutual implication (equivalence). Counterexamples are not generally useful for analyzing enthymemes; and nobody commits a fallacy simply by not stating all of the premises and assumptions of the argument. At least, if we identified fallacies with formally invalid argument forms, we make the label 'fallacy' completely useless in practice. What counts as invalid is relative to the formal system we use; whenever we apply formal systems to actual arguments we have to allow ourselves so much room for the implicit and assumed that merely identifying the explicit form as invalid tells us virtually nothing about whether it is a good argument or not. (As I always tell my students, it is very, very useful to know that an argument is valid. It is at best only somewhat useful to know that it is invalid.) We can prove that particular arguments are fallacies; there appear, however, to be no generally applicable methods for doing this, because an invalid argument form that is fallacious in one context may not be in another because the second context is, so to speak, 'rigged' so that the invalid argument form, despite not being generally truth-preserving, is so in contexts like that one.

A straightforward example of this is found with fallacies of composition. An argument form like this is both formally invalid as it stands (the explicit premise does not require the conclusion) and admits of many, many counterexamples:

Each part of the wall is red; therefore the wall is red.

But we all know that under particular conditions this type of inference is not only good but certain. But pinning down these conditions is always extraordinarily tricky; what you are really trying to do is to prove something by division of possibilities, and as Aristotle pointed out long ago, even when arguments by division are certainly right they are not rigorously demonstrative.

My own view is that in the strictest sense you can't have a fallacy without an application -- that is, no argument form is fallacious as such. Rather, there are only fallacious and non-fallacious applications of argument forms. There are, of course, argument forms that are especially subject to abuse because they are not usually reliable but look superficially like argument forms that are -- the fallacy of affirming the consequent is a good example -- and we can, by a sort of metonymy, call these fallacies because their non-fallacious applications are rare enough that we can usually neglect them. But this is a metonymy; and a 'fallacy' in this sense may admit of rare cases in which an argument of exactly that form would be a perfectly good argument, and thus by definition not a fallacy at all.

Moreover, one has only to look at disputes raised by paraconsistent logical systems to see that there are problems with the counterexample approach even if the above points are set aside. Disjunctive syllogism, for instance, is valid in some formal systems and invalid in others; what counts as a counterexample to it in one system will not count as one in another. We are left with the question of whether disjunctive syllogism is really best modeled as strictly valid, or as valid only under limited conditions, like the inferences from composition; and the method of counterexamples is necessarily useless for this question.

Thus I would suggest the proper answers to the questions are, in order:

Except in a formal system there is no general method to prove that a logical fallacy is a fallacy indeed.

There are indeed argument forms whose status as fallacious is open to dispute; indeed, even disjunctive syllogism and modus ponens, which are standard inference forms, have been questioned in ways that would imply that at least some of their applications are fallacious.

It may genuinely be possible to prove that the new fallacy really is a fallacy, but without a general method for doing so, it is impossible to forecast how one would do it.

Craig Ferguson on Moral Philosophers

Jonathan Dancy gets a mention on Craig Ferguson's Late Late Show (ht):

Sommers Notation, Part I

For my students I'm putting together a new guide for using Sommers-Englebretsen Term-Functor Logic, also known as Sommers-Englebretsen Term Logic, and which I will call here, for simplicity, Sommers notation. I thought I would put up a draft version.

Parts of a Categorical Proposition

If we want to talk about categorical propositions, it’s helpful to break them down into six parts which may be briefly described as follows:

(1) Subject Term: This is what the proposition is directly about.
(2) Predicate Term: This is what is being said about the subject term.
(3) Universe (or domain) of discourse: This is something the subject and predicate term share that allows them to be compared in the proposition.
(4) Quantity of subject: how much of the subject the predicate covers.
(5) Quality of predication: how the predicate term applies to the subject term.
(6) Judgment: how the whole proposition is being put forward.

There is much more that could be said about each of these things, both separately and in their relations, but this will do for our purposes. Because the universe of discourse is already implicit in the subject term and the predicate term, we don’t usually need to include it specifically in a notation, although (as we shall see) it can be useful to have a way to do it if we need it. But, in general, we want a notation that allows us to keep track of all these parts.

Sommers notation starts from the basic idea that every categorical proposition is the affirmation or denial of a simple or complex predicate of all or some of a subject. When you look at categorical propositions in this way, we can see that they admit of three basic kinds of oppositions.

1. Term Valence. 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.

2. Opposition of 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 Sommers notation is that there is no practically significant distinction between term quality and predicate quality; 'S is non-P' is not significantly different from 'S isn't P'. But when starting out is useful to treat them as distinct.)

3. Opposition of Quantity. Every predicate is predicated of at least some or definitely all of a subject. This sort of opposition is an opposition between a universal subject and a particular subject. So 'Some S is P' is opposite in quantity to 'All S is P'.

3. Opposition of Judgment. Every proposition is put forward as true or false, that is, it is asserted or denied. 'It is not the case that S is P' is opposed in judgment to '(It is the case that) S is P'.

The upshot is as follows.

(a) Every categorical proposition has a subject (S) and a predicate (P).
(b) Every term, independently of its role in the proposition, has a mark indicating term valence.
(c) Every P as a complete term has a mark indicating opposition of quality.
(d) Every S is a term with a mark indicating opposition of quantity.
(e) The entire proposition linking S and P is itself a term with a mark indicating opposition of judgment.

Given these four basics, which I will not argue for here, we can develop the basic format of Sommers notation.

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)


as our assertion. However, we know from (b) that every term has its own valence. Thus:


We know from (d) that we want every subject to have its mark of quantity. Thus:


And since each predicate may be itself a complex term, it has as predicate a mark of quality, which allows us to take (c) into account. Thus:


And we know from (e) that every proposition has a judgment that (so to speak)links the predicate and subject together so that the proposition either asserts something or denies something. This we can symbolize, putting the mark of judgment at the beginning, as:


Of course, this is just a general format. Let's take a basic assertion: All S is P. This can be symbolized by:


S and P are both of positive quality, so the sign linked to each term is positive. P is affirmed of S and so there is a plus sign linking (+S) and (+P); S is of universal quantity, which we indicate with the minus in front of (+S); and the whole proposition is being asserted as true, so we have a plus sign in front of the whole proposition. 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. If you want a more technical answer, 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., Every S is P is equivalent by contraposition to Every nonP is 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. Another way to put the same point in technical terms would be to say that the minus for quantity tracks what the traditional theory of the syllogism calls distribution. But quantity's the only tricky thing about this basic format: + and - simply indicate an opposition, and '-' in particular shouldn't be confused with negation, which it only sometimes has.

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. 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


can be written as


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


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 Barbara (AAA-1) 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-2) would be:

(-S-M) + (-P+M) + (+S+S) = +S-P

It is useful to summarize this in a procedure that will help us to determine validity in any categorical syllogism using Sommers notation:

(1) Check to see if the argument is algebraically acceptable – that the premises add up to the conclusion.
(2) Check to see if the argument is regular. There are two, and only two, ways an argument can be regular. Either it has only universal propositions, or, if it has a particular proposition, it has one (and only one) particular proposition in the premises and a particular conclusion.
(3) To handle weakened syllogisms, if an argument fails at (1) and (2), check to see if it can pass if we add (+S+S) to the premises, where S is the subject term of the conclusion,

That is, a syllogism is valid if its premises add up to the conclusion and it is regular, either in itself or with the premise ‘Some S is S’ is added.

This is all quite cool. But I'm partly getting ahead of myself here. Sommers notation 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 Sommers notation is capable of handling. So now we'll go on to look at how Sommers notation handles various key issues (singular terms, relations, identities, meta-propositions, existence). Then we'll handle arguments.

(Part II)

Sunday, February 07, 2010

Three Poem Drafts


Lady love, my darling dear,
my honeyed life, my angel's tear,
my doggerel verse will light the way
along your path the live-long day,
and though its speech is trite and worn
it shall upon bright wings be borne;
for each cliché was turned to such
by being truth used overmuch,
and (though these words are tired speech)
when used of you, the truth they teach.


The lilting light on its lyre
makes melodies of mercy
and I, strange as it seems,
am the music, am the light's song,
a Pythagorean harmony.


The first creating one
in infinity dwells,
for the created infinite
participates the first,
and so also simple goods,
life, light, beauty,
that cause all things that have these,
for as the first cause of all
must be infinite itself
(all else follows from it)
so then life, light, beauty,
these it must then be,
and from it life and light
and beauty too
commingle and flow down
to cause mind to be.

The Sight of a Bear or a Fox

Father Stephen Freeman has a good quotation from Elder Paisios:

Elder Paisios said: Often we see a person and we say a couple spiritual words to him and he converts. 
Later we say, “Ah, I saved someone.” I believe that the person who has the disposition and goodness 
within him, if he doesn’t convert from what we say, would convert from the sight of a bear or a fox or from anything else. Let 
us beware of false evangelization.

(Paisios (also known as Eznepidis) was a twentieth-century monk at Athos, who spent much of his life at the Panagouda hermitage.)