Proposal for a Course in Which Students Invent Science

Part One: Multiplayer Mode

Every class begins with problem solving.  The question posed in the most recent assignment is written on the board.  The students, as a class, must solve it.  No one is allowed to give an actual answer to the question until a quorum has agreed on the best way to solve the problem, then executed the plan and interpreted the results.  The teacher can participate in the problem solving by posing leading questions, encouraging certain directions of thought, and suggesting that they try using tools they already have, but for the most part the students run this part of the show.  There are two main goals here: to develop their scientific toolboxes, and to encounter the inherent bugs of human minds (cognitive biases) so they can learn to recognize and patch them, thus solving problems more efficiently in the future.  Questions early in the course will emphasize revealing biases, and the later problems will emphasize empirical methods of inquiry and testing.  Overall, we’re working toward inventing something like Bayes theorem or another broad philosophy of science.

Part Two: The Meta-quest

After the question is answered, the lecture begins.  The teacher recaps everything that just happened, pointing out which things worked and which didn’t.  The methods and biases involved are given short and simple names the students can remember, like “testable hypothesis” and “availability heuristic”.  This serves as an outline for the lecture.  Lecture materials for the next several weeks should be assembled beforehand to allow for maximum flexibility in presentation order.  The lecture explicitly covers only those methods discovered by the students, showing how they’ve been used historically and how they’ve improved overtime.  (There is a little leeway here for closely related methods that are particularly difficult to discover in a class setting.)  When biases are identified, the lecture includes descriptions of studies and/or anecdotes evidencing or pinpointing the bias, a discussion (perhaps with class participation) of why we tend to think in that particular way, how we can notice when it presents an immediate danger to reasoning, and how to cope when it does.

Part Three: Personal Quests

An assignment is given at the end of every class: the students learn what question they’ll be answering the next day, and must come up with a plan for finding the answer.  These competing methods will duke it out in class debate the next day.  They must also propose problems whose solutions could be found by methods learned in class, which can be hypothetical or drawn from their lives or stories they’ve heard.

Part Four: Leveling Up

Tests will be given periodically, but their frequency will depend on how much has been discovered how quickly.  They will include simple questions about the material covered in the lectures, and a problem that can be solved only by using several if not all of the tools acquired since the last test.

Part Five: Winning the Game

The final will be cumulative.  There will be an in-class portion that is similar to the basic question and answer portions of previous tests.  The take-home portion of the test will have two parts.  The students have a choice on the first part.  They can either choose to answer one particularly difficult question, or they can answer three easier questions.  They must write an essay explaining how they went about solving the problem and why.  For the second part of the test, they must propose and defend a definition of Science.

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What I'd really like to see in the comments here is a brainstorming session in which we generate a whole bunch of useful project ideas for a class like this.  In particular I'd like to focus on things geared toward 8th graders, but other thoughts are also welcome.

In Defense Of Semantics, Or: Until You Can Say What You Mean, You Cannot Mean What You Say

Semantics matters.  In a debate, it is all that matters.  What matters when you’re talking with someone is what you mean by what you say, and what the other person takes you to mean by it.  That’s what communication is.  If I could just project my thoughts into your head all at once, my choice of words and the order in which I arrange them wouldn’t matter.  Words wouldn’t matter.  But I’m not a dolphin, so I have to use language if I want to communicate.

Language is a social behavior in which symbols such as sounds or gestures are agreed by participants to denote entities in the world.  The symbols are arranged according to structural guidelines of temporal progression to denote larger concepts in an organized way.  Thus, an image of a concept is projected from the mind of the speaker into the world for listeners to observe and replicate in their own minds.  This process is called communication.

The success of the project of language depends on two things: syntax and semantics.  Syntax, the rules by which symbols are organized to denote more complex concepts, is ultimately a servant of semantics.  It allows for the communication of thoughts far deeper and more intricate than the mere vocabluary.  But it has no purpose whatsoever in the absence of semantics.

In natural language, “semantics” is a set of correspondences, some between symbols and the things they denote, and others among entire sentences.  For instance, the relationship between the sentences “math is exciting and challenging” and “math is challenging” is one of semantic entailment, because the meaning of one entails the meaning of the other.  Suppose I formulate the following sentence and speak it aloud: “Oma cabeca djorglesnuff.”  Even if you know all the rules of the syntax I’m employing, my utterance will be completely useless as communication until I explain somehow that by “oma” I mean “cats”, by “cabeca” i mean “eat”, and by “djorglesnuff” I mean “mice”.  Only then can you understand what I’m trying to say, and respond with something equally meaningful that moves the discussion forward.  You can identify my declarative sentence as a specific claim.  “My oma,” you might say to me, “does not cabecca djorglesnuff.  I think you’re wrong to say they do.”

And here we’re at a point where the two of us might start “arguing semantics”, because the next thing I say is, “I didn’t mean that all oma cabecca djorglesnuff.  I only meant that some oma cabecca djorglesnuff.”  “Ah,” you say to me, “then you’re correct, but you should have specified that when you were explaining what you meant by ‘oma cabecca djorglesnuff’ in the first place.”  And you are perfectly right to call me out on that.

Why?  Because the sentence “some cats eat mice” entails the truth of different sentences than does “all cats eat mice”, and if I didn’t provide you with the tools to determine which sentences my utterances entail, then my words haven’t sufficiently meant and I’ve done a poor job of communicating.  

Consider the following conversation.

A: God exists.

B: No he doesn’t.

A: Yes he does, and I can support my claim.  Behold!

A holds up a spoon.

B: What does that have to do with God?

A: It is God.  See?  It exists.  God exists.

B: You think that God is a spoon?

A: Well... yeah.  That’s what I meant by “God”.  You’re not going to argue mere semantics with me are you?

B: You bet your boots I am.

The above two cases are perfectly legitimate grounds for substantial semantic disputes.  In both cases, one party has done a poor job of communicating, and the other rightly asked for more careful formulations of what is to be projected through language.  In the first case, the failure was a matter of ambiguity.  There were multiple propositions the speaker might have intended to convey, the distinction between the possible propositions was significant, and thus the misunderstanding was not the fault of the listener.  What the speaker actually said did mean something, but it didn’t mean as much as it should have.  What it meant was not precise enough for the purposes of the discussion at hand.  He did not mean what he said, because he did not say what he meant.

In the second case, the speaker meant by his words something outside of the standard, agreed-upon set of entities that might be denoted by them.  The reason the word “God” is mostly useless in discussions with people who are used to attending to fine conceptual distinctions is that the standard set of notions to which God might be taken to refer is very large and poorly defined; not only is there ambiguity, there is vagueness.  But in the case of a spoon, using the term “God” causes more confusion than usually comes with even that word.  The speaker did not mean what he said, because he did not say what he meant.  If A were to say “God exists,” and B were to say, “I think so too” but take “God” to mean “a porcupine”, the listener would also be making the same kind of mistake.

So you see, the more accurate and careful we are with our language, the more intricate, interesting, and useful will be our communications, and the more worthwhile will be our debates.  Clarity matters.  Precision matters.  Sensitivity to semantic distinctions is a valuable skill, as is the diligence to attend to them.  When philosophers argue semantics with you, the purpose is neither to annoy you nor to show off.  It’s to actually get somewhere with the conversation.  If we’re asking for precision and clarity, we’re doing so because we excel at identifying problems that derive from a lack of these, problems that would lead to frustrating, tiresome spirals of self-perpetuating confusion.  

There are important things to learn from people who shatter your semantic endeavors and ask you to rebuild them from the shards.  Developing the patience to face down the linguistic challenges of philosophers will lead you to wield language that is sharp and strong as the edge of a samurai blade.  And if you choose instead to dismiss such attempts at careful communication as tiresome nitpicking, do not meddle in the affairs of philosophers, nor seek what they have sought.  

If you’re too lazy to say what you mean, how are you ever to mean what you say?  And why should I believe that you do?

Who does Kant think he is?

Kant lays out his theory of apperception in “Of the Deduction of the Pure Conceptions of the Understanding”.  He does so in his usual style, which, as is widely known, includes a great deal of obfuscation, repetition, and tedious verbosity.  

A diligent and patient reader with some philosophy background and a good translation will discover, eventually, that the vast majority of the notions he means to convey in The Critique are remarkably simplistic and powerful.  I find it most unfortunate that his style has made the volume inaccessible to nearly everyone.

Below, I have attempted to render, paragraph by paragraph, Kant’s passages on apperception accessible to anyone who can handle the New York Times.  This task became much more difficult toward the end of the section on apperception, but I think it’s an important project.  There is certainly room for improvement in places, but I envision a collaborative effort to convert all of the Critique to simple English, so I intend for this to be more of a seed than a final proclamation on how best to convey these concepts.

My simple English paraphrases appear italicized, and my further explanations follow in regular font.  The paragraph numbered 1.1 corresponds to the first paragraph of the first topic in section II of chapter II, 3.2 to the second paragraph of the third topic, etc.  Without further delay, I give you the first installment of “Kant in Simple English”.

1.1

To combine a bunch of separate experiences into a single concept, there must be something outside of the experiences performing the combination.

When Kant writes “manifold content”, he’s referring to the stream of information constituting experience over time.  When we form concepts, we combine pieces of information gathered over some period of time.  It may be a short period, as when we listen to a sentence that takes two seconds to speak; or it may be a very long period, as when we come to understand what “Italian food”, as a whole, tastes like by ordering several different Italian dishes over the course of a few months or years.  Each experience is separated by time from all the other experiences.  There is nothing about the taste of garlic bread on October third that suggests it should have something to do with the smell of alfredo sauce on February second.  The relationship among experiences is not contained in the experiences themselves.  Instead, it is imposed on them artificially by something outside the experiences—by  us.

Kant calls the ability to gather diverse experiences into larger concepts “understanding”, and the gathering itself he calls “synthesis”.  The understanding can gather (or synthesize) not only experiences, but also concepts.  We might form the concept “three” by gathering together many experiences in which we sensed that number of sounds or sights, but we might also form the more abstract concept “number” by gathering together the concepts of one, two, three, four, etc.  All synthesis, writes Kant, is an act of the understanding.

1.2

“Combination” involves three things: 1)the concept of a bunch of separate things, 2) the concept of the action of gathering all those things together, and 3)the concept of unity—that the result of this process is a single whole.  (I don’t mean the category I’ve been calling “unity”; the concept of a category requires the concept of combination, so that would be circular.  This kind of unity is something more abstract still.)

Since the concept of “unity” must exist for there to be combination (or “conjunction”) in the first place, unity can’t come from combination itself.  The whole-ness of unified things must be a product of something beyond combination.

2.1

Inasmuch as my sensations are specifically my sensations, they presuppose the thing I call “me”; thus the act of sensing and the ability to sense also presuppose me.  Self-awareness cannot arise entirely from any of these things, because it is awareness of something presupposed by all of them.

All of my sensations and such presuppose exactly the same me.  

The only thing that my experiences have in common is that there is a subject who experiences all of them.  That doesn’t mean the experiencer is somehow embedded in its experiences—it is no part of the taste of garlic bread—but instead that there is a relationship that holds between the subject and its experiences.  When I reflect on this fact, I perceive unity of the subject across all of the experiences; it’s the same subject every time.  Moreover, it’s the same subject experiencing the reflection on the subject of experiences!  I call that unitary subject “me” (Kant calls it “apperception”).  

2.2

It’s interesting that there is sameness of the me presupposed by all those different sensations; the me who experiences sensations participates in of lots of different acts of experiencing, and there’s not anything about any one of those acts that ties it directly to the others.  Self-awareness is needed to combine all of these actions into a single concept of “me”.  So, for me to exist as an identity among experiences, I must be self-aware.

My awareness of myself is why my experiences can all be gathered together under the heading “my experiences”.  Without the unity in the combination described by that heading, there would be nothing tying all those experiences together; there might as well be a different experiencer for each experience.  They are, after all, each separated by time.  Since I am exactly the thing that experiences all of my experiences, the lack of unity would preclude my existence.  Unity, you’ll recall, can’t be a product simply of combination, so the combination of all those subjects does not itself account for selfhood.  There must be something above the experience/experiencer relation that makes the combination across times possible.  The self can’t exist without self-awareness (apperception).

2.3

Remember all that stuff I just said about unity and self-awareness?  Good.  It was important.

This paragraph is mostly repetition and emphasis of the concepts in the previous two paragraphs.

3.1

Earlier in this book, I talked about some other conditions presupposed by experience.  Specifically, I said that the kind of thing we usually count as “experience” can’t exist without Space and Time.  Those are both forms of what I called “sensibility”.

Self-awareness is another condition for experience just like Space and Time, but it involves the mind (or “understanding”) instead of sensibility.

Here, Kant is drawing a parallel between these passages on apperception and those on the transcendental aesthetic from earlier on.  He is also enumerating the conditions he’s so far determined are necessary for experience: space, time, and self-awareness.

3.2

A judgment about the relationship of a group of sensations to the particular object that caused those sensations is called a cognition.  The grouping of various sensations must be done by the self I’ve been talking about, because grouping is a kind of unity, and unity comes from self-awareness.  The relationship between sensations and objects is the objective validity of cognitions, so a self is required for there to be objective truth, cognitions, or anything else that goes on in minds (or “the understanding”).

When I see an aardvark in front of me and I judge it to be true that, “The thing in front of me is an aardvark,” what I’m really claiming is, “The object that caused the sensations that I grouped together into the concept ‘aardvark’ really is an aardvark, objectively speaking.”  It’s a claim about the relationship between the concept formed by my understanding and the object outside of my understanding (specifically in sensibility).  Because concepts are formed through combinations, combinations require unity, and unity is a product of self-awareness, the whole notion of objective truth would be meaningless without selfhood.

3.3

That the self is the same self across all experiences is a prerequisite for there being a mind.

This is already clear from 3.2 and earlier paragraphs.  Unity is a product of self-awareness; since unity is a product of self-awareness, everything that depends on unity also depends on the self.  The main jobs of a mind (or “the understanding”) are to form concepts and make judgments about them.  To make judgments about concepts, you first need concepts.  To form concepts, you first need to combine sensations.  Combination requires unity.  Unity requires self-awareness.  Therefore, the mind depends on self-awareness.

3.4

Read 2.2 again.  I really like what I said there.

3.5

I’m not saying that there’s a particular act of combining that must happen for there to be a mind; only that the mind must be able to combine things, and that the unity of the self is what makes that possible.

We wouldn’t even be able to conceive of a mind without self-awareness to ground it.

4

Objective unity of selfhood is different from subjective unity of selfhood.  Objective unity is the gathering up of all the different bits of self scattered across experiences into a whole.  Subjective unity is the ability of the little bits of self to be gathered up like that.  It’s necessary that the self is objectively unified, but it just so happens that it’s also subjectively unified.

Objective unity can exist if and only if there is subjective unity.  Objective unity is one of the prerequisites for experience, and that is true regardless of whether or not there really is experience.  That there is experience, however, is not logically necessary.

Absurdist Arithmetic

Earlier today, I was explaining to friends how to use polish hand magic (which I've referred to as "absurdist arithmetic" ever since multiplying 3 by 7) to multiply numbers smaller than five and greater than ten.  One of them observed, "It would be an interesting exercise to prove this by induction for all natural numbers."  That got me thinking; it obviously works for negative numbers, and there's no reason it shouldn't work for numbers other than integers.  He replied, "I doubt whether one can have pi fingers."

You can totally have pi fingers.  You just have to be willing to carry from the left as well as from the right.  Keeps you agile.  The problem is merely practical.   Furthermore, there’s no reason to prove it by induction on natural numbers.  It's easier to prove without induction for ALL the numbers.

1.      Let  ↑ = λ n.n-5 [This amounts to "number of fingers up" for any n whatsoever, no matter how many fingers you think you have.]

2.      Let  ↓ = λ m.10-m ["number of fingers down"]

3.      Fix x and y, arbitrary numbers (don't care which sort).

4.      ↓(x)*↓(y) = (10-x)*(10-y) [by def. of ↓]

5.      (10-x)*(10-y)=100-10y-10x+xy [because algebra, or so I've heard]

6.      ↑(x)+↑(y) = (x-5)+(y-5) [by def. of ↑]

7.      10 * ((x-5)+(y-5)) = 10x+10y-100 [also because algebra]

8.      Thus, 10 * (↑(x)+↑(y)) + (↓(x)*↓(y)) = xy [bippity boppity boo]

9.      Therefore, for all z and w, 10(sum of fingers up) plus (product of fingers down) equals the product of z and w. [existential generalization]

Pi times 8, by hand:

Left hand: -(5-pi)+3=pi-2=1.14ish. Put this in the 10's place.

Right hand: (10-pi) * 2 =13.72ish.

Left hand                                      Right hand

1.14                                              13.72

                                                     Cary ten over to the left.

2.14                                              3.72

Carry 0.14 over to the right.

2                                                   3.72+1.4

2                                                   5.2

Pi*8 is 25.2ish.

Wonder what it's like to have imaginary imaginary fingers.  i*8???

What Buddhism Might Be

Perhaps this will sound familiar to you.

"John: Personally, I kinda think *all* religions are bogus.

Mary: In general I agree with you.  I'm definitely an atheist and I think the whole idea of "church" is disturbing.  But Buddhism isn't actually a *religion*; it's more of a *philosophy*.

John: Dude, all that bowing, celestial Buddhas, rosary beads, meditation, nirvana, those crazy hats the Tibetans wear... that's totally religion!  It might have some philosophy mixed in, but overall it's a religion."

If you've spent any time around philosophy or religious studies departments, it almost certainly does sound familiar.  As I spend most of my time in philosophy, much in religious studies, a bit in IU's secular student alliance, I am tired of listening to people have this debate.  Usually one person has read one pop Buddhism book and has made up his mind one way or the other, while the other person is even more oblivious.  It's just tiring.

If you yourself do serious work in areas you or others call "Buddhism", you are probably also familiar with this:

"Other person: Interesting paper!  Zen is awesome!  Are you Buddhist?

You: ...um... *god damn it I hate when people do that* ...well..."

The following is a list of the candidates for the intention of the word "Buddhism".  I implore you to consider them.

1)The set of practices that actually lead to enlightenment.
2)The set of practices the historical Buddha claimed lead to enlightenment.
3) The set of practices people calling themselves "Buddhists" believe actually lead to enlightenment.
4) The set of practices people calling themselves "Buddhists" believe the historical Buddha claimed lead to enlightenment.
5)The list of dogmas to which people calling themselves "Buddhists" ascribe.
6) The list of dogmas to which the historical Buddha actually ascribed.
7) The list of dogmas to which people calling themselves "Buddhists" believe the historical Buddha ascribed.
*Note: The list of dogmas ascribing to which leads to enlightenment falls under 1.  Similarly for those Buddhists believe lead to it etc.*
8) The particular way about which people calling themselves "Buddhists" do philosophy; the methods upon which they agree are legitimate forms of argumentation (their unique brand of analysis, logic, etc.)
9) The particular set of texts whose content grounds number 8.
10) The set of texts written by actual buddhas and bodhisattvas.
11) The set of texts people calling themselves "Buddhists" believe were written by buddhas and bodhisattvas.
12) The truth to which actual buddhas actually awaken.
13) The set of truths to which people calling themselves "Buddhists" believe buddhas awaken (that is, if x believes p and y believes q and x and y are Buddhists, then p and q are elements of "Buddhism".)
14) Actual enlightenment itself, whatever it be.
15) The set of things people calling themselves "Buddhists" believe enlightenment to be.
16)The set of practices people calling themselves "Buddhist" call "Buddhist practices" (regardless of whether they also believe these to  be salvifically efficacious).
17)The set of dogmas people calling themselves "Buddhist" call "Buddhist dogmas" (regardless of whether said people actually ascribe to said dogmas).
18) Some combination of the above.

Whether or not "Buddhism" counts as a "religion", and whether or not it counts as a "philosophy", depends on which of the above you take "Buddhism" to be.  If you take it to be one of, or a combination of, the things that focus on salvation, then "Buddhism" is indeed a religion.  If you take it to be only the philosophy things, it's a philosophy.  If you take it to be only the text things, then it's a literary movement.  If you take it to be 17 or 18, it's an anthropological phenomenon whose name I don't know.

It's false that there is necessarily no fact of the matter about whether Buddhism is a philosophy, a religion, a combination of those, or something else entirely.  There may well be a fact of the matter.  But it depends on which of the available options counts as "Buddhism".  Matsumoto Shiro, for instance, takes Buddhism to be something very specific: the doctrines of dependent origination and emptiness.  This falls under category 12: the truth to which actual Buddhas actually awaken.  If you do not yourself believe that there's such a thing as "enlightenment" yet agree with Matsumoto that these two doctrines constitute Buddhism, then you think that Buddhism is (an extremely limited version of) number 13.

I think that one of the above constitutes genuine Buddhism, but I'm not going to tell you which one because it is not my point.  My point is that the "religion vs philosophy" debate over Buddhism is only substantive if you're willing to do the work of making these sorts of distinctions.  That means careful thought, study of the terms involved, and not tolerating from yourself or your debate partner near perfect ignorance of the world beyond Western pop culture.

In other news, I am in fact willing to answer the question, "Are you Buddhist?" but only if you're willing to demonstrate to me that you have a reasonable understanding of what it is you're asking.

Ray Jackendoff and Mathematical Semantics

I went to the discussion with Jackendoff for the philosophy department today and asked him about the semantics of mathematical statements. He said that semantic mentalism does indeed commit us to a rejection of mathematical realism; "2+2=4" has no mind-independent semantic content whatsoever. He also said that this fact makes him very uncomfortable, but he thinks that just means we need to look into it further, not that we need to reject semantic mentalism. I agree with him on that. But shortly afterward I realized another implication that would probably make him even more uneasy.

Semantic mentalism doesn't only preclude mathematical realism. It precludes classical mathematics entirely.

If mathematical statements have no mind-independent semantic content whatsoever, there cannot exist in that content elements relying on mind-independent existence. Total continuous functions over the real numbers rely on lawless infinite decimal expansions (irrational numbers with no rules governing the calculation of each subsequent digit). If we are to have functions over the real numbers at all and we're not going to assume that such expansions exist mind-independently, we're forced to accept the continuity principle: for every function whose source is the rational numbers, there exists a natural number m such that for any two real numbers x and y, if x and y are identical up to the m'th digit then if (x,n) is an element of the function then (y,n) is an element of the function. Without either omniscient minds or mind-independent lawless free-choice sequences, there can be no complete total functions over the real numbers. And without those we have to get rid of lots of classical math.

Obviously intuitionism doesn't preclude realism (as our resident intuitionist realist is so fond of repeating), but phenomenology is my favorite motivation for intuitionism so far. (Just don't let McCarty catch wind of that.)

So now the question becomes: is Jackendoff committed to a conceptual semantics founded on mentalism strongly enough not only to reject mathematical realism but to adopt intuitionism or some weaker version of constructivism? Does it make less sense to him to say that sentences refer to the mind-independent world than to reject the law of the excluded middle? Or double negation elimination? Or the equivilance of a negative general and a particular negative? Or Church's thesis?

He talks like the whole point of all of this is to have a semantics that makes good intuitive sense and is in line with how people actually think, but it's immediately apparent to anyone who's tested those forbidding waters that intuititionist math is deeply unintuitive. If you're not used to thinking about this stuff (or even if you are, really), trying to get your brain around something like, "it's false that 'a thing is either true or it isn't'" is incredibly difficult. It's about as cozy in human cognition as the most mind-shattering koan. This is not good news, I think, for conceptual semantics.

Squib on adverbial “at all”

As a general rule, adverbial “at all” can be used only when all of the following hold:

• The sentence is about degree, location, number, or type.
• The sentence is negative.
• If the sentence is about location, number, or type, its logical form is particularly quantified.  Quantification is irrelevant to sentences about degree.

For instance...

I don’t like mushrooms at all. [aboutness: degree, charge: negative]
There are no dogs in heaven at all. [aboutness: location, charge: negative, quantification: particular]
There are no coins in my purse at all. [aboutness: number, charge: negative, quantification: particular]
*There are coins in my purse at all. [aboutness: number, charge: positive, quantification: particular]
*There isn’t every coin in my purse at all.  [aboutness: number, charge: negative, quantification: general]

There is one exception to this: when uttered in a counterfactual context, “any x at all” appends well formed formulae to yield new wff’s whenever x acts as a noun or noun phrase.  The gramaticallity of this construction depends only  on the counterfactual status and not on positivity, quantity, or aboutness.  

You might have been any animal at all.
Unicorns could live anywhere at all. (where “where” acts as a noun)
There couldn’t be any man who breathes air in space at all.

The quantity of “unicorns could live anywhere at all” is actually generalization of a particular.  In fact, it would seem that the quantity of all counterfactual sentences with adverbial “at all” is nested.  At all is a sticky problem for semanticians, because in order to understand what’s really going on we’re going to have to work out words like “could”.

What exactly is the problem with “could”?  The problem is the truth value of sentences containing it.  “Unicorns could live anywhere at all,” seems to be asserting the truth of the intended proposition just like any other statement, but if the characteristic function for the truth value of counterfactuals in the actual world exists, it is inaccessible.  It’s not that we don’t know the graph of the relevant function because cataloging all the things of such and such a class along with their truth conditions is impractical or would take an infinite amount of time--it’s that such a process can’t even begin for the class of non-actualized possibilities.  Therefore, semanticians trying to hang out in some kind of formalistic limbo will have to take a metaphysical stance on the nature of semantic practice to work out the meaning of counterfactuals and by extension the meaning of “at all”.

I see a few options for rendering “at all” meaningful.

• Platonism (aka “the lazy way out”): the sentence has a truth value of true or false and the characteristic function exists mind-independently.  “Unicorns could live anywhere at all” means that the proposition to which it refers exists in Plato’s heaven.  It doesn’t matter that we can’t know the function’s graph; we can employ the function intentionally anyway.
• Modal realism: the sentence has a truth value of true or false and the characteristic funciton exists mind-independently.  “Unicorns could live anywhere at all” means that for every place in the actual world there exists a possible world containing a counterpart of that place and at least one unicorn lives in it.  As above, we can use the function without being able to know its graph.
• Conceptual semantics: much like modal realism, but fully mind-dependent.  “Unicorns could live anywhere at all” means that the pluriverse existing representationally in the mind of the speaker is such that the modal realistic interpretation is represented as true (and no relationship to a mind-independent pluriverse is relevant).
• Intuitionism: the truth value needn’t be either “true” or “false”, in which case we get to keep sober realism and counterfactuals are unproblematic.  Obviously, this would mean a complete overhaul of semantics to account for a non-finite t domain.  I say we go for it.  Any takers?