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?

Phenomenology of Self-Interest

Re-posted from a Facebook note entitled "Is Self-Interest Rational?" by Robby Bensinger:

This is going to be exciting.

 
There are two ways to answer the question. First, is egoism valuable as a means? Assuming the self is an intrinsically worthy interest, is it rational to behave in a purely self-interested manner? I.e., is it actually in your interest to try to act exclusively in your own interest?

Put that way, the answer — to the extent that a general answer can be given at all — is clearly 'no.' Effectively all individuals in a society of self-interested individuals will be worse off than if they were in a society of more altruistic individuals. Society and economics are not zero-sum games; when each person is willing to sacrifice a little, everyone gains huge returns on their investment. We all sacrifice the immediate rewards we'd gain from exploiting and stealing from one another; in exchange, we gain the safety required for pleasures of greater duration and depth.

Yes, our brains are also hardwired to punish selfish, unfair, and corrupt behavior, so being an exploiter carries a high risk; but even if we were equipped to do the cost-benefit analysis each time an occasion for crime came up, we would rarely bother, because we are not crudely self-interested beings. We are not idealized economic profit-maximizers: We are messy complexes of conflicting drives and values, including the social emotions. And it is because of these social emotions that society is possible, that the benefits of exploitation do not tempt us into razing the world to reap the immediate rewards in our own lifetimes. Had we the power to make ourselves more 'rational' in this respect, it would be profoundly irrational to do so.

Second, is egoism even valuable as an end? Without assuming the value of self-interest, is it possible to rationally infer this value from a prior, more fundamental value that we all possess? So, sure, maybe it's counter-productive to act self-interested; but is the very premise of this project, the very assumption that we should be aiming to promote our own individual self-interest, grounded in anything?

Certainly it is a matter of fact that we do value ourselves. But we value many things. Sometimes we value things disproportionately; sometimes our priorities are screwed up. Hence the importance of recognizing which values are the most fundamental, which ones are most important and basic to our existence.


Regarding society's need for people to value their own future in order not to bring the world crashing down (as in the coke-addict hedonism of the subprime mortgage crisis), Dan Simonsön (2011) wrote:

"Should individuals sometimes sacrifice themselves now for themselves later?" of course, if you're a 65-year-old CEO, the answer to that question is "no" because there's not enough later left 

But this assumes that the only decision-making criterion for humans is personal self-interest. If you're a retiring executive who also values his children's or grandchildren's or friends' or country's or species' welfare, then other factors may complicate the proper calculus. And we desperately need these complications: The more every individual is willing to invest in every other individual, the more every individual benefits, overall.

So modernity's basic moral problem is twofold. First, people are factually deluded by the thought that a code of pure self-interest favors the individual (even in the abstract!), when in fact a healthy dollop of altruism favors the individual. Clearing up the facts is enough to settle the problem of the 65-year-old CEO who cares about others but is simply blinded by a Randian economic theory; but we'll have to enter deeper waters to confront the individual who simply does not sufficiently care, despite having the biological capacity to care (i.e., not being a psychopath). He turns a blind eye to suffering, even perhaps knowing that if he stared it in the face, he would be compelled to feel compassion. What is that blind eye missing? Why should this matter to an egoist?

The first aspect of our problem is that people aren't questioning the value of egoism as a means to satisfying other values (the human race's liberty, well-being, entrepeneurial progress, etc.). But the second, far more vexing aspect is that people aren't questioning the intrinsic value of egoism.

Why should I especially care about me?



The science and philosophy of mind have a little wisdom for us here. Our self, extended over time, is a construct. There is no purely logical reason for you to identify yourself now with yourself a year ago, or yourself a year from now; lacking an abiding essence (e.g., a set arrangement of cells, or of personality traits, or of memories), there is at best only a vague family resemblance between your 'self' at different stages of life. And perhaps this family resemblance is sufficient. But there is also a family resemblance between yourself and your family — including your extended family, the human species. So the relationship between your present self and your past and future selves differs only in degree from the relationship between your present self and the selves of 'other people.'

'But,' you might inquire, 'why should I sacrifice the pleasure I will actually experience for a transcendent pleasure I can never myself perceive, a pleasure existing only in another's mind?'

And this is to miss the point entirely. Life is a parade of little deaths. When you choose to go on a diet in order to benefit the future you, 'you' are not ever going to experience that future pleasure — 'you' will have radically changed by that point, will have been replaced by a cognitive sibling with only some of the same traits. To sacrifice present pleasure for greater future pleasure is the same type of act as sacrificing my pleasure for the Other's pleasure; in both case there is merely a leap of faith that I am connected to something larger, thus justifying an investment in a grander ecosystem of experiences than my own immediately accessible ones. Obviously force of habit and our basic instincts make it easier to help a future self than to help a perfect stranger — but the principle is still the same. Egoism is unstable because the ego is itself an unstable concept: Am I still the self whose actions I have completely forgotten? Should I sacrifice my current interests for the interests of a self who won't live for another 50 years, who will have completely forgotten even that I made this decision on his behalf? (The ungrateful cur!)

If we wish to be altogether consistent, our choice is not between egoism and altruism, but between participating in a larger reciprocal ecosystem or fixating exclusively upon the immediate. An individual who invests in his future selves ensures that every one of those selves lives a better life; he enters a sort of implicit contract with himself, saying, 'In exchange for the service my past selves have done for my present, I, this present self, will carry on doing good works for my future — and since all shall benefit by this cross-temporal agreement, why should any abstain from it and bring the edifice crashing down, even for the sake of a momentary increase in pleasure?'

But this arrangement, so clearly rational at the level of the community (the temporally extended self), doesn't make an iota of sense at the level of the momentary individuals who actually enact the arrangement. There's no rational reason any individual, momentary 'self' should continue to participate, since it will be a totally different self, not this self, that suffers the consequences if he defects right now. There's a reason that 'gratitude,' 'gratuitous' and 'grace' come from the same root: From the perspective of any individual self, it is unthinkable that you would help another rather than just cash in on others' investments in your well-being. Such gratitude can't be logically justified, if the only motivator of behavior is self-interest. Fortunately for everyone involved, this is not our only motivator: We are motivated by affection for and interest in our other selves (past, future, and in the larger human race). No self is built to exist in a vacuum; we are no more asocial than we are atemporal or anaerobic, which is why egoism is so ruinous, both materially and psychologically.

What is this experience that makes it possible for a self extended over time to flourish? What is the Now's feeling for its sister-moments? A certain affection. A certain familiarity. A trust; a continuity; a faith. A failure to altogether isolate my interests from another's. A genuine concern for this Other, both for its own sake and as an extension of myself. Inextricable intimacy. Impassable distance. Were we to try to designate this experience in a word, none could suffice but 'love.' Love, not 'rational' interest, is what holds the self together, what keeps every momentary self from defecting, and so what lifts up the well-being of the entire succession of lives of the human being.


This sort of self-sacrifice does not denigrate the individual for the sake of some mythic abstraction like Society or The World, or for that matter The Self. Just the opposite. It is precisely because every atom of mind is supremely precious that we must affirm Love as a value, as this self-sacrificing value is the one that makes it possible for life's individual droplets to thrive in a sustainable mutualism. The argument is abstract, but the best conduct for everyone is not ultimately justified by argument (for you could never argue someone into loving another; at best you could only clear up their delusion that they have a counter-argument against loving another); it is simply a byproduct of the fact that we are already in love with one another, that we have common values and desires — not totally identical values, but ones that overlap enough to justify a shared investment in the human project.

Self-love is a remarkable achievement; it is a good thing, for the love we bear at each moment for the totality of our obscurely linked experiences exalts the quality of each of those experiences immeasurably. But it is not an achievement to stop at self-love, to go no further. To extend our love to the family unit was an achievement, but to extend our love to a larger community is a still greater achievement, all the more remarkable and unlikely — and this achievement is a great boon both to the family and to the individual, the earlier stages of organization. To sacrifice our earlier gains in the interest of some abstract and aimless ideal of 'Progress' would be foolishness. To forget that the individual experience is the entire basis for the individual Self's value, or that the individual self is the entire basis for Society's value, would be to fixate upon the abstract and neglect our lived actuality. But altruism is not a sacrifice for nothing; it is a sacrifice for greater overall benefit. One may say that there is no rational way — rather, no selfish way, since 'rationality' is indifferent to self and other — to get the project of altruism started; but that if we can find some irrational way to somehow get the engine of reciprocity up and running, the 'rational' and selfish interests of every individual will be furthered immensely. This is the lesson of the prisoner's dilemma. We are blessed to live in a world where we do love, a world, therefore, where we have the capacity to wish for a better life for ourselves and for all beings that love and fear and strive.

To defend egoism as coherent is trivial. An infinity of logically consistent ethical programs may exist, but only a very few harmonize with our nature, which is love. In memory, love binds us to our past; in hypothesis, love binds us to our future; in the social, love binds us to our family; in art, love binds us to our very shadow. Love is the how, the why, and the what of a human being. Perhaps we can become egoists; and perhaps we can subsist off of nothing but apricots. What of it?

To defend egoism as more 'natural' than altruism — because it arises earlier in childhood development, say, or because it is easier to cultivate — is an absurdity. Eating marshmallows until one feels sick may be more 'natural' than exerting self-control, but that is no great reason to overeat; the health of the intersubjective ecosystem trumps immediacy.

Finally, to defend egoism — perhaps even the hyper-egoism of valuing only the self of the moment — as the pinnacle of some Buddhist ideal of Living in the Now — this risks a tragic misunderstanding. Setting aside speculations about the possibility of consumate perpetual bliss, we can say that the prosaic value of 'staying in the present moment' is to safeguard against our love for other selves mutating into psychic masochism or a total disregard for the immediate. The immediate is, after all, where every payoff will ultimately be found. If we fail to appreciate our hard-won moment-to-moment rewards, then long-term thinking is useless. But if we can attend to the present while remaining circumspect, if we can sail between the solipsistic Scylla of careless short-sightedness (hyperbolic, impractical 'Living in the Now') and the anxious Charybdis of fearful fixation upon distant selves, both our present and our future will be the better for it. Compared with the constructs of society and Self, the virtue of 'The Now' is a late-comer to the game, a phenomenological device appended to the edifice of what we as humans have become. It is a modern technology for improving perceptual information processing and alleviating corrosive mental states. It is not a return to some animalistic, asocial, or perfectly spontaneous state — if such a state has ever existed, it is not mindfulness as we understand it; and just because a lion is free of civilization and its discontents does not mean that a lion is happy or at peace.

But from a bird's-eye view, the greatest virtue of attending to the Now is not that it conduces to an unmediated hedonic upsurge, but that it allows us to better notice our indwelling affection and compassion, for others and for ourself. Coming full circle, the greatest evils of today are not caused by an abstract philosophical doctrine like egoism — egoism assists evil by serving as a rationalization, not as an original motive. Once we have pierced the façade of ideology, of rationalizations, we can confront the real root of evils like poverty, global warming, and the current economic crisis, which is our willful disregard for how much we ourselves can, and do, care for our larger world. The greater problem we face is not how to intelligently reason our way to short-term, narrowly focused pleasures, but how to intelligently cultivate the (in themselves unreasonable) faculties and habits of mind that provide fertile soil for the expansive joys of the lover and beloved.

(End Robby's note, begin my response).

I agree with you on nearly everything here and think this is a beautiful essay.  In fact, your crescendo paragraph  introducing love is stunningly gorgeous.  But I'm going to respond first to our point of contention.

Here is the false claim: "But this arrangement, so clearly rational at the level of the community (the temporally extended self), doesn't make an iota of sense at the level of the momentary individuals who actually enact the arrangement.  There's no rational reason any individual, momentary 'self' should continue to participate, since it will be a totally different self, not this self, that suffers the consequences if he defets right now."

Robby, you neglect that my immediate experience is more pleasureable when I believe in the temporally extended self.  The more vividly I imagine, for instance, a massage, the more pleasure I feel *right now*.  This is because I an create in my immediate experience an imaginary construction that is a body being massaged and can then subsume that construction into my current experience of "self".  This augmentation of identity brings *real* pleasure *now*.  There is no significant difference between imagining being massaged and now imagining later being thin due to my current decision to diet provided I incorporate sufficiently the imagining into my current experience of self.



In fact, there is evidence that the rewards we receive from *imagining* the pleasure of future people with whom we identify as "self" can be so immediately powerful that actual future selves are less likely to uphold the vividly imagined decisions, feeling as though the benefits have already been reaped.  If you tell someone, "I'm going to learn to play the guitar," you're less likely actually to do it than if you'd kept your mouth shut and simply done the deed in silence (or in plinky melody, as the case may be).

This accounts equally well for altruistic actions toward other organisms.  Empathy is exactly emotional identification with another.  We form a construct corresopnding to the experience of self-hood of another person and then either project our self-hood onto it or absorb it into our own, depending on the perspective, and conclude that what we feel as a result is what the other organism feels.  With our sense of self thus perferated, we believe we seek to relieve the suffering of another through our apparently altruistic actions when in fact it is our own suffering we seek to relieve.

It is important to remember there that all of this feeling and thinking is both by and about one immediate experience of self.  It is impossible actually to feel the emotions produced by another mind.  We feel the emotions of corresponding representations of others that in fact reside immediately in our own single (if convoluted) self construct.

Simmilarly, love for imaginings of "future/past selves" that present themselves *immediately* to conciousness is what produces pleasure when we believe we sacrifice for them.  We *do not love* actual future or past selves because not only are they subjectively inaccessable; they don't even exist. 

All that we may ever love, hate, reason about, or be are no more than idealized mental constructions presenting themselves to consciousness in an intricate, ephemeral entanglement whose central intersection we call the self.  Every self is perfectly selfish.


Nevertheless, your claim that love "makes it possible for life's individual droplets to thrive in a sustainable mutualism" is correct.  Love, or perhaps more appropriately "compassion" from an etymological perspective, is the over-arching phenomenon responsible for the illusion of temporally extended selfhood and for social behavior overall.  Regardless, all love is self love.  All compassion is feeling with none other than oneself.

For My Lovely Logicians-In-Training

(So I don't have to re-make this every time I get a new student)


Notes on Propositional Calculus

Syntax: the study of the principles for constructing grammatical entities in a language or logic.

Semantics: the study of the meanings of the symbols in a language or logic.  

Syntax in PC refers to the formal structure of arguments and their component sentences, while semantics refers to the truths or falsehoods of the same.

Valuation:

• is a semantic notion (it's about the *meanings* of the terms used in arguments).
• is a combination of truth value assignments to the atomic propositions in a sentence or argument.

Validity:

• is a syntactic notion (it's about the *structure* of an infinite set of arguments)
• does not depend on the actual truth values of the atomic sentences in the argument.
• An argument is valid if and only if any valuation rendering the premises true also renders the conclusion true.
• Operations in natural deduction (the proof system) depend on the presupposition that all the premises are simultaneously true.

Definition of propositional formula (plural formulae, pronounced form-you-lee)

• p,q,r, ect. are formulae (specifically atomic formulae)
• If X and Y are formulae,
• X*Y, XvY, X-->Y, and ~X are also formulae.

Definition of operators (aka connectives):

• *: T iff both conjuncts are T.
• v: T iff at least one disjunct is T.
• ~: T iff the negated formula is F.
• -->: F iff the antecedent is T and the conclusion F.

Related terms:

• Conjunct: a member of a conjunction
• Disjunct: a member of a disjunction
• Antecedent: the formula preceding the arrow in an implication
• Consequent: the formula following the arrow in an implication

Notes on Predicate Logic

Quantifiers:  

• (∀x) applies to every element of the defined set (general)
• (∃x) applies to at least one element of the defined set (particular)

When you hear... think...

• "unless": if not
• "Unless you like hate football you should watch the game," = "If not (hate football)--> (should watch the game)."
• "only if" or "only": following goes to consequent
• "Only cool people like jazz," = "(like jazz) --> (cool)"
• "You're a good dancer only if you love music." = "(good dancer)-->(love music)"
• if: following goes to antecedent
• "If you're happy and you know it clap your hands," = "If (you're happy and you know it)-->(clap your hands)."
• even if: (it or not it) then
• "Even if you're bad at math you might love logic," = "(bad at math or not bad at math) then (might love logic)
• not any: not some (negation of particular)
• "There are not any more M&M's," = "not (there are some M&M's)."
• no, no one, nothing: all not (general negative)
• "Nothing can stop me now," = "(for all x) not(x can stop me now)"
• "No snowman lives forever," = "(for all snowment) not (live forever)"
• there is/there exists/some: particular
• Some people like cake.
• "There are three agencies of government when I get there that are gone..." (EPA?)

Questions to ask yourself when translating:

• Is it an argument with premises or is it a single sentence?
• What's the main connective?
• What are the quantifiers?  To what parts of the sentence do they apply?
• What do I immediately know from key words in the sentence (like unless, for all, etc.)?
• When closing: again, to what parts of the sentence does the property apply?

Other notes: 

• Conditional arguments are your friends.  If the conclusion is an implication, assume its antecedent and derive its consequent.
• If the conclusion is a negation, assume the negated formula and derive an explicit contradiction.  Conclude the negation of your assumption.
• If you're having trouble with your derivation, assume the negation of the conclusion and derive an explicit contradiction.  Conclude the negation of your assumption and apply double negation elimination.
• If you're stuck on a particular derivation, stop thinking about it.  If possible, sleep before coming back to it.  Remember that there *is* an answer and it *will* come to you if you're patient.
• If you're confused or uncertain about a translation, find a similar sentence whose translation appears in the back of the book and study it, then come back to your translation.
• If you're feeling overwhelmed by the number of rules on your derivation cheat sheet, keep in mind that there are really only ten rules needed for any derivation in PC.  If needed, confine yourself to: assumption, conjunction introduction (conj), conjunction elimination (simplification), disjunction introduction (add), disjunction elimination (proof by cases), implication introduction (conditional proof), implication elimination (modus ponens), negation introduction (negation of assumption upon which a contradiction has been shown dependent), negation elimination (any formula, including the non-negated form of an assumed negation, can be concluded from an explicit contradiction), and double negation elimination.