the strange beauty of logical positivism and popular and academic misconceptions about it

There are two common misconceptions about logical positivism. 1. the positivists, like New Atheists, set out to prove that non scientific theories like religion and metaphysics, are false and only science can be true, and 2. Logical positivism fails at its own criterion of meaningfulness. 

(1) has got LP backwards. LP allows religions and metaphysical systems as not just true, but necessarily true, while it's the scientific theories that are possibly false, not necessarily true at all. 

That's the strange beauty of LP. The difference LP draws between theories is not between the true or the false, but "meaningful" and "not meaningful", using a peculiar definition of "meaning".  LP doesn't touch on any other aspect or virtue of non verifiable theories, their aesthetic value, their mystery or charm or inspirational insight, their moral or social value. Just their meaning, where "meaning" in LP is used as a theoretical jargon for "phenomenal informational impact -- how the world of phenomena and events are and are not." The challenge of unpacking their use of "meaning" such that it isn't circular is the reason for (2).

(2) is flatly false. Apply LP to LP and it verifies. (2) also assumes that LP is a theory and not either a definition or description or an axiomatic system or merely a kind of practical advice like Popper's demarcation. 

The popular misconception has it that if LP is a theory it should apply to itself, but LP can't itself be verified. I think people who say this must not have tried to apply LP to LP, maybe because "it doesn't apply to itself" is self-reflexive and so clever-sounding that they don't bother to experiment to verify whether the clever is also true. Whatever their reason for why they don't apply it to itself, we can apply it here and now:

LP says that theories are meaningful (in the sense of "tells us how the world is or is not", "what's in the phenomenal world and what isn't") if their statements and predictions about the world are verifiable. Is this assertion verifiable? Sure. God is not a directly verifiable object. It's not meaningful in the LP sense of telling us how the phenomenal world is. Religion is meaningless in that sense. Are the bones of dinosaurs verifiable? Yes. Archeology is meaningful in the LP sense of telling us about the phenomenal world, in this case where to find dinosaur bones. LP is verified by both these cases. LP is meaningful in the LP sense of meaningful, telling us how the world is and is not. 

This is all crude and simplistic, but it shows how to apply LP to LP. Let's try again with something more substantial.

Creationism cannot predict the fossil record. There's no book of trilobites and dinosaurs in scriptures, and scriptures don't need them. It doesn't tell us how the world is, phenomenally. That's an unverifiable theory, and notice, it's a necessarily true theory -- no empirical evidence can prove it false. (The New Atheist will complain about it's internal contradictions, but those are logical disproofs, not evidential disconfirmations, and LP is concerned only with evidence. That's a huge difference.) Archeology does predict the fossil record. Treatises on trilobites and dinosaurs belong to science. Biology and archeology tell us what we will and will not find when we dig into the earth and find bones. Are they true? Well, not necessarily. They are the most likely theories of the topic given the evidence currently available. True? Who knows what we'll discover tomorrow? And that's one difference between the religious or metaphysical theories and the scientific theories -- according to LP.  What we discover tomorrow could trash our current science. It will never trash the religious or metaphysical theories. Are these differences between creationism and archeology verifiable? Yes, the difference seems to be verified. That difference is the LP criterion, the LP "theory". 

You may have already noticed that creationism is strictly ambiguous over verificationism, since it doesn't predict, so you could say verificationism can't apply to creationism. IOW, the problem is not that verificationism doesn't apply to verificationism -- it does -- it's that verificationism doesn't apply to the necessarily true and meaningless (in the LP sense) theories. 

Right at the outset, it's important to know that Karl Popper identified an essential flaw in LP. Verifiability runs into the inductive fallacy. Verifying a theory supports the theory but can't prove an explanatory theory, that is, it can't prove a theory that predicts the possible (as compared with post hoc descriptions of a closed set of observations). Popper replaced verificationism with falsificationism -- that a "meaningful" explanation must identify the conditions under which it would be false. The consequence is that scientific theories are never provably true, they are just the ones that haven't yet been proven false. There are weaknesses in falsifiability too, but it was an important advance over LP's primitive verificationism. LP was using its confirmation bias to confirm its theory of confirmationism. It's a typically human failure to use clear Bayesian reasoning. 

The weakness of LP is not alone its confirmationism. It also defined "meaningfulness" in terms of verificationism and vice versa. Their criterion of science was circular. That's because LP's use of "meaning" is not a theory at all. It's a definition or axiom or maybe a kind of practical advice. Definitions generally don't apply to themselves. "Blue" is not blue, and it would be irrelevant even if it were blue. "Blue" can be used as a kind of practical advice: you can view these objects as having this common property of being blue. "Look at these -- they all have a kind of similar hue. For convenience let's call them 'blue' so they're 'blue-ish'." That's all there is to a definition. And if you can provide for all or many possible additional individuals ("that thing there should be included in the set") all the better. Most definitions don't apply to themselves, though some do: the set of definitions that define themselves, for example. Not very practical. 

Another weakness of LP, most obvious in Wittgenstein's Tractatus, was the belief that there could be atomic facts, indivisible facts independent of any other facts or ideas or theories. But facts are partly theoretical -- they are conjectural, dependent on the likelihood of the theory -- and like theory, their value is their falsifiability. There's a frequency theory of hues that classes navy blue with sky blue although Russians distinguish them as distinct hues. When I was a child I refused to wear anything navy blue so repulsive to me was this color. My favorite color was sky blue. Which fact is relevant -- that navy blue and sky blue are opposite ends of a single color or that they are two colors? Depends on the theory and its purpose. (See "true but wrong" on this blog.) "Whales are giant fish" belongs to a biological taxonomy that sufficed for the deity in the Book of Job, and that book makes effective, memorable use of it. I have no problem with the "whales are giant fish" theory. It's just not useful for science, a predictive theory of what's out there in the phenomenal world and how it got there. People can wear different hats, you know. 

The lack of conjectural theory in LP led to Wittgenstein's private language argument, a kind of reductio ad absurdum of his verificationism, applying verificationism to the mind. Can experiential states be verified, he asks. Well, on a verificationsit model of truth, no. Rather than seeing this as a disproof of the  verificationist premise, and rather than seek a better conjecture, he oddly, and perversely, embraced the absurd result that the experiential is meaningless, and advocated for a kind of behaviorism that prevailed in philosophy and the sciences until Chomsky in 1956 demonstrated that such a behavioral program couldn't account for the productivity and inventiveness -- the creativeness -- of speech, that the mind played a necessary role in behavior. Chomsky's program was a better conjecture that led to a better understanding of the mind. 

The common view holds that Wittgenstein's later views is a rejection of his earlier logical positivism. I think that's another misconception. His later views are, I think, best understood as pushing his earlier views to their extreme and often counterintuitive and even absurd logical consequences, an insistence on biting the philosophical bullets one after another. It's a wonder he had any teeth left. 

So much for the strange beauty and the misconceptions. 

In sum, "It doesn't apply to itself" sounds like a clever dismissal of logical positivism from those who don't know much about it or don't want to know about it. Unfortunately, they miss everything interesting in it. There were a lot of flaws in logical positivism in its early efforts, but failure of reflexive application is not one. Those flaws are evident in Wittgenstein's early work and in his later work as well, where they produced logical absurdities when applied to the mind, leaving the philosophy of science in an impoverished behavioral model until Chomsky's 1956 Syntactic Structures. That that restrictive impoverishment led to extraordinary behavioral insights -- Ryle's criticism of the Cartesian ghost in the machine, Austin's speech act theory, among many many others -- might be a topic for another post. As Jerry Fodor thought and said, behaviorism was provably wrong, but brilliant. 

NYU imperatives workshop

Originally published on Language and Philosophy, March 21, 2016

Do scientists from differing disciplines have the same goals in addressing the same facts? Linguists attempt to accommodate all the natural language intuitions in their theoretical frameworks. That may lead them to extralogical means. Logicians have often taken on one or another natural language intuition and attempt to augment the logic to accommodate that intuition. In both cases there’s a question of purview: why not accommodate all the intuitions through the logical system, or how much of logic should accommodate the intuitions?

This became the battle at a workshop on imperatives at NYU today. Craige Roberts incorporated pragmatics into her analysis of imperatives to include a wide variety of natural language intuitions, while Kit Fine and Peter Vranas developed new logics to deal with some, but not all, intuitions. They both seemed to ignore that traditional logics are not just inadequate for linguistic intuitions, but also inadequate to basic facts about reality. If we assume that development of logics is still in its infancy, the attempt to accommodate each outstanding challenge is a step towards a more inclusive, flexible and useful logic.

Think of Kratzer’s lumps of thought. She observed that a single event can be represented in multiple descriptions that in sentential logic would imply multiple events, or if not implied, then at least failed to imply that that the descriptions were of the same event: if Sally made a painting that was a portrait depicting her sister, sentential logic would imply three events (or at least not imply one event) “Sally made a painting and painted a portrait and painted a picture of her sister.” That these three descriptions are of one event is not a linguistic intuition, it’s a fact of what Sally did. To formulate a logic that can identify these conjuncts as three descriptions of one event would be progress for logic, not for linguistics.

So it seems to me, logic is justified in picking its challenges independent of the needs of linguistics. The real test would be between AI and neurolinguistics — how are imperatives represented in the brain, how can they best be represented in a robotic program? I didn’t see anything from the linguists giving a brain representation argument the way, say, Chomsky did with syntax. There doesn’t seem to be an experimental program to follow, as there was with generative syntax. The logicians, on the other hand, were always mindful of the algorithmic value of their logic, but that’s why they are logicians.

There was also an interesting exchange on whether the background conditions of an imperative are factual or relative to the speaker or addressee. So “if it’s raining, take an umbrella” can be evaluated on whether it’s actually raining or whether the speaker thinks it’s raining. Does it matter whether it’s actually raining for the force of the imperative to hold? Roberts, the linguist, wants it to be contextual information of the speaker; Vranas wants to take this as factual so that the entailments can be validated within his three-valued logic. At first they seem to be different views — why should it matter whether it’s actually raining, since the imperative is the speaker’s insistence. But if there are only beliefs, and no facts, both views are the same. The force of the imperative, shared by the speaker’s intention and the addressee’s understanding of it, will shift if she comes to believe that it’s not actually raining.

The two talked past each other for about an hour. The problem is a really tough one. The entailments of speakers’ assertions are trivial. Sally said “I’m lying” just in case she said it. So as an assertion, it’s true. But the content, indexed to a speaker, is a paradox. It’s worth remembering that three-valued logic began with an attempt to incorporate the epistemic into the logic. The result is a loss of a distinction between the factual and the epistemic. But there’s an underlying problem: no one knows what is factual; all we know are our beliefs. Deductions from our beliefs will always be trivial; deductions from facts will require extralogical overlays for the epistemic. I worked out the problem a few years back here. I’ve complained that trivalence flattens modality here.

liar paradoxes, a problem with reductio proof and speech acts

Originally published on Language and Philosophy, June 20, 2013

It’s easy to mistake paradoxical sentences for liar paradoxes. “If this sentence is true, then it is false,” is a liar paradox. If the sentence is true, then the antecedent is true. If the antecedent is true, then the consequent must be false, the implication as a whole is false, so the sentence must be false. So if the sentence is true, then it is a contradiction and a falsehood. So the antecedent must not be true. If the sentence is false, antecedent is false, and the implication as a whole is true.

“If this sentence is false, then it is true,” however, is not a liar paradox. If it is false, then the antecedent is true and the implication fails, and the whole is false. If the sentence is true, then the antecedent is false, the implication holds, and the sentence is true. That’s not a paradox, it’s just a sentence the truth of which cannot be determined. It’s like the sentence, “This sentence is true.” Is it true or false? How could you tell?

Similarly, “The sentence I am now writing is true,” is indeterminate. “The sentence I am now writing is false” is provably a liar paradox, athough one could ask of these two sentences “true or false of what?” The deductive proof that yields a liar paradox of the latter, is a reductio: assume the sentence is true, you deduce that it is false; assume it’s false, you deduce it’s true. So if it’s true, it’s false and vice versa. But if you ask “true of what?” then you’re asking for an empirical answer — does the sentence corresponds to something, in this case to its own truth. Is truth a thing that can be pointed to? If it’s a correspondence with something, we’re stuck in an infinite recursion. So these sentences, on the one hand, lead to a questioning of the correspondence theory. But they also lead to questioning of the validity of deductive reductio argumentation, not unlike that questioning of the reductios that led Cantor to multiple levels of infinities, and the intuitionist rejection of the reductio in favor of proof by demonstration. Several directions from here: you can say these sentences don’t correspond to anything; or correspondence is not complete; or correspondence, even with its incompleteness is a better option than reductios that lead to liar paradoxes. 

Some performative acts are puzzling in relation to paradoxes and lead to a question about what their propositional content is. Suppose “I’m stating in writing that 2+2=5” is true by definition, because it is in fact true that I stated it in writing, even though the secondary content of what I stated is plainly false and wrong. Generally, speech acts (“I promise to…” “I deny…” “I insist…” “I’m saying that…”) could be viewed as true by virtue of stating them regardless of their secondary content, the proposition that is promised, denied, insisted upon or said. But what about “I’m stating in writing that I’m not stating this”? If it’s true that I stated that in writing (I just did, in fact), then it’s true. But in writing that sentence, I stated that I didn’t. Is this a contradiction or is it a paradox or something else?

It might be something else. It’s simply true that I stated and wrote the sentence. The secondary content is false. So the wrting act is true, but what I asserted was wrong. I am wrong, though the primary sentence is true. Think about a promise. Suppose you’d promised something that you can’t fulfill. It’s a promise, even though it’s a foolish promise. In these speech acts, there’s a relation not only to the content of what is promised, said, denied or insisted upon, but also a relation to the actor. It’s best not to conflate them.

So “I deny that this is a denial,” might still be considered a denial, and therefore true (Is the sentence a denial that I wrote? Yes.), even though the secondary content is false. I made a statement in the form of a denial, but what I said about that statement was wrong or false.

“I deny this” and “I deny denying this” are tricky cases. If they are true, they seem to deny themselves, so false; a liar paradox. Can these be treated as above — simple true denials with false secondary content? On this view, a lot rides on the little word “that.”

That’s one view. But what if these performative acts have no content? What if “I promise I’ll stop” means nothing more than “I’ll stop” and “I’ll stop” is in effect a promise not to stop? What if “I say it’s raining out” means nothing more than “It’s raining out” and “I deny I saw her” just means “I didn’t see her”? Then “I deny denying this” means “I deny this” which in turn means simply “This sentence is false.” Does this reduction matter? Is any content lost? Is it like the truth predicate — semantically inert (“‘2+2=4’ is true” adds no content to the simple statement “2+2=4”)?

“I’m not writing now” was plainly false when I wrote it just a moment ago. But surely the following sentence is meaningfully distinct from it: “I’m now writing ‘I’m not writing now’.”

Suppose there is no difference between those two sentences. Is there then also no difference between “I assert that I’m not writing now” and “I’m not now writing”? If there’s no difference, then either “I assert” somehow not count as language, or all utterances and written statements are true. In a previous post I claimed that that’s exaclty right — nothing of logical significance can be proved about the world by speech. Speech is nothing but opinions, only accidentally truths.

This same quandary arises with denials, since they are semantically an assertion (with a negative truth predicate): “2+2=5 s false” means “I deny 2+2=5” or “(~(2+2=5)) is true.”

Is there a difference between “I did not have sexual relations with that woman, Ms. Lewinsky” and “I deny that I had sexual relations with Ms. Lewinsky.” The first is a lie — not just false, but Bill knew it was false when he said it — the latter is true even though the secondary content is false. So the denial is true, but also involves a deception, by implication. That’s not to say the statement is false, but that it shouldn’t have been made, if honesty is required.

If these have distinct truth values, then utterances don’t have an implicit assertion predicate, contrary to my previous post.

non epistemic possibility and the lay of the land

Originally published on Language and Philosophy, May 31, 2012

The peculiarity of classical notion of possibility is that it has a relation to the actual world as well as a relation to the irreal world of conditions counter to the actual and the epistemic world of certainty and uncertainty. Lukasiewicz’ notion of possibility seems to apply only to uncertainty — it seems essentially epistemic.

So here’s the lay of the land, as I see these two modal programs:

Classical

actual=>possible (simpliciter) [because the actual is one instance (though merely one)]

necessary=>possible (simpliciter)

contingent (possible & possible-not)=>not necessary

contingent (possible & possible-not)<=>not necessary & not necessarily not

actual & not necessary=>contingent [possible-not in some plausible world beyond the actual world]

The actual and its entailments are non epistemic; contingency and necessity are epistemic.

Non classical

actual=>necessary=>not contingent

necessarily not=>actually not (i.e., determinate falsehoods)=>not contingent

possible<=>possibly-not<=>contingent<=>not necessarily & not necessarily not

The actual again is nonepistemic; the contingent/possible are epistemic; by implication, necessity is also nonepistemic, but perhaps not always or only epistemic.

Alternatively, the many-valued logic could abandon actuality altogether and treat the entire realm of assertions as epistemic except the logical truths. This makes a lot of sense to me, since I don’t believe that there are any certainties beyond maybe analytical truths or truths by definition. On this model the tautologies have a probability of 1, contradictions 0 and everything else is somewhere on the scale in between. I would dispense with the notion of “actual” entirely, since, on the one hand, tautologies do not imply actual instances of anything in the real world besides their tautological truth — all unicorns are unicorms, but there’s nothing in the actual world that bears on this truth or that this truth bears on actuality; and on the other hand, the empirical world of phenomena are not entirely certain — why fool ourselves with a pretense of knowledge? So:

necessary=>tautological

necessarily not=>contradictory

non necessary & not necessarily not<=>degrees of probability <1 and >0

Here necessity is analytical; everything else is epistemic. This model seems completely consistent with the classical model, where the actual is not necessary.

It has a philosophical consequence: on this view of modality, actual empirical events are never completely certain. So there is no “actual world” among possible worlds. Instead, there are apparent experiential worlds — phenomenal worlds in a more or less Kantian sense; subjective worlds of relative conviction. Along with the experiential worlds there are possible worlds of conjecture. Necessary truths will be true in all of them, etc., etc.

This seems much closer to the realm in which my mind lives where nothing is absolutely certain but tautologies and contradictions.

possible or not

Originally published on Language and Philosophy, May 10,2012

Is it possible to swim the Atlantic?
An ex neighbor points out that if “possibly” doesn’t imply also “possibly not” then how is “possibly” different from necessity? Doesn’t “Life on Mars is possible” mean “It’s also possible that there’s no life on Mars”? And doesn’t it also mean “Life on Mars isn’t necessary”?
Grice gave an answer to this question, and I’ve written about it elsewhere in this blog, but I think there’s more to be said and I want to try to sort all of them out.
Suppose Goldbach’s postulate is possibly true. Suppose someone proves it. Now it’s necessarily true. Is it no longer possibly true?
My neighbor says no. I think Lukasciewicz agreed with him.
But suppose we have a set of conditions that fulfill the capacity to swim the Atlantic, and some person satisfies that set. Now it’s possible for someone to swim the Atlantic, and the question of whether it has been done is irrelevant.
One way to cash out the notion of truth is a saying of what is, and the notion of false, a saying of what is not. In this dichotomy, where does possibility lie?
One place is knowledge. There’s what we know exists, what we know doesn’t, and what we don’t know about is the possible.
But isn’t this Lukasiewicz trichotomy conflating knowledge with fact?
A lot depends on whether determinism holds — if every actuality is necessary, then everything that happens is necessary. But whether determinism is true is itself uncertain.
It seems to me that there are many meanings of “possible” besides “unsure.” There’s the abstract conditions of conceivability, “if someone were strong enough, someone could swim the Atlantic;” the actually satisfiable conditions of possibility, “there is someone strong enough, so it is possible,” and the actual itself, “she swam the Atlantic, proving that it is possible.”
If you choose a Lukasiewicz trichotomy, is there space for all these? It seems well suited to determinism, but not to modality. The classic Aristotelian four-way modality (possible+negation yielding possible, not possible, possibly not, not possibly not) along with the conjunction of “possible & possibly not” accommodates a distinction between actuality and necessity, where we can be agnostic about determinism and entertain possible worlds that will never be.

Aymara, trivalence, competing satisfaction and modality

Originally published on Language and Philosophy, January 6, 2012

In his monograph on the trivalent logic of Aymara, Ivan Guzman de Rojas sets out to show that a trivalent logic can reach conclusions unavailable to bivalent logic. I want to tease out the import and accuracy of this extraordinary claim, and try to understand its significance for modal logic.

Consider two circumstances: p) there’s smoke; q) there’s fire.

And consider these two premises:

X) if there’s smoke, there’s fire;

¬p) there’s no smoke.

From these two premises in a bivalent logic using the standard definition of implication, you can conclude that (X) is true, but no conclusion can be reached as to whether (q) holds or not. Using a Rojas matrix:

p  T   |  T  |  F  |  F

q  T  |  F  |  T  |  F

X  T  |  F  |  T  |  T

¬p F  |  F  |  T  |  T

The third and fourth columns reflect that the two cases in which the premises are both true, the circumstance described in (p), that there is smoke, clearly is false, but the circumstance described by (q), that there is fire, can hold or not.

Before looking at the trivalent deduction, I’d like to consider what the bivalent deduction means for bivalency. There’s a question just how to describe the consequences of the deduction. Can one deduce from the premises: (q v ¬q)? Of course, but that’s a tautology, and would be true even if (q) were provable from the premises. So, for example, from the premises it can still be concluded that (p v ¬p), even though we know from the premises that (p) is false and (¬p) holds.

Another way to express this would be (q & ¬q), a contradiction, which is always false. But that was so before the premises were stated. Similarly, we can conclude from the premises that(p & ¬p), even though the premises state (¬p).

So what is deduced from these premises with respect to (q)?

Using common sense English, the deduction sounds something like this: either of (q) or its negation could be true; (q) and (¬q) satisfy the combined premises; if (q) is true, then the premises are both true but also if (¬q) is true, both premises are still true. Let’s tentatively describe this with a modal notion of “possibly” notated as ◇. Can (◇q & ◇¬q) be concluded from the premises? Well, the premises don’t mention any modal notions, but in the meta-language of our reasoning over the premises, let’s allow modal notions. Is (◇q & ◇¬q) a tautology or a contradiction? Neither. And if we restrict the world of possibilities to the circumstances relative to the premises in our argument, then (◇p & ◇¬p) seems unequivocally false in our meta-language, since we deduced — prove — that (p) is true, and the world of possibilities are restricted only to those defined by the premises.

If this seems a leap, we could use a different modality, epistemic modality, to render the meta-language function explicit: K(◇q & ◇¬q) where “K” means “know” and to “know” means to have understood a set of sentences and made deductions. So we know that q is possible and not-q is also possible, but we also know that p is not possible. So, if we substitute p for q, K(◇p & ◇¬p) turns out obviously false, since we know (p) is true.

In plainer English, these premises and the circumstances together lay out a set of possible circumstances of what we can believe could hold.

So now how about trivalence?

Here’s a truth table in a trivalent system

p  T  |  T  |  T  |  F  |  F  |  F  |  0  |  0  |  0

q  T  |  F  |  0  |  T  |  F  |  0  |  T  |  F  |  0

X  T  |  F  |  0  |  T  |  T  |  T  |  T  |  0  |  0

¬p F  |  F  |  F  |  T  |  T  |  T  |  0  | 0  |  0

The only cases in which the circumstances and the premises are all satsified are the fourth, fifth and sixth columns. The result is that we still know that p is false, but we now have three evaluations of (q): T, F and 0.

On the face of it, the deductive consequences are the same as the bivalent system, except that it is expressed with more truth values. It’s instructive for understanding the bivalent system in which we were left with two inconclusive “conclusions.” The common sense response to that was, we are uncertain which holds, q or not-q, based on the premises. The trivalent system has made this explicit, though somewhat redundantly, and so at the expense of elegance: the trivalent system allows three uncertainties. We are uncertain that q is true and uncertain that q is false and we either are certain that q is uncertain, or maybe we are uncertain of that as well, depending on what we mean by 0.

Let’s clean up the last option. If we treat all these deductions in the meta-language, we’ve got to say that from the premises we know that q is possible, we know ¬q is possible and we know that q is uncertain. Let’s use “~” to designate “uncertain” just as “¬” designates negation. So based on our knowledge of the premises and our deductive ability, we deduce in our metalanguage K(◇q & ◇¬q & ◇~q), or, in other words, we know that all three circumstances are possible given the world of possibilities given by the premises.

If in a trivalent system the third truth value means “neither true nor false,” then in our example, trivalence has simply added another possibility in our epistemic world of possibilities. If the third truth value means “uncertain,” then in our example, trivalence has simply added a redundancy.

There’s a difference between “uncertain” and “neither true nor false.” The former is a genuine epistemic value — “we don’t know q.” The latter is not modal — “q is neither true nor false.” Looking at the meaning of the premises, we want to be careful about this difference, since it could be significant. Remember that “q” is short for “there’s fire.” Now, in any circumstance, there’s either fire or not. It could be that we don’t know which, or it could be that, given some set of premises, we can’t conclude which, but nevertheless there’s either fire or not, and, if we have any deductive ability, we know that there’s either fire or not.

I conclude from this fact of the world — not a fact of logic, but of reality — that this use of a third truth value is intrinsically epistemic. It means not “neither true nor false” but rather “uncertain whether true or false.” If that’s so, then this use of trivalence, and its “~” operator, are redundant.

Not all uses of trivalence are redundant. Sentences that are semantically anomalous can be usefully designated as neither true nor false. “Obama stopped beating Michele” under the assumption that he never beat her, would be such a candidate for 0. It is true that either Obama beats her or not, but to have stopped beating her is to add the presuppositional premise that he once did. So there are three possible facts of the world: he used to beat her and doesn’t now; did and still does; never did. The first makes the sentence true, the second false; and the third is neither. It’s the presupposition that is either so or not; the sentence itself contains it as an assumption.

Since there is no operator that takes a sentence and turns it into a semantic anomaly, the kind of trivalence that Rojas proposed won’t work. So I find a dilemma for this kind of trivalence: it is either epistemically modal and redundant (interpreting 0 as “uncertain”) or it can’t have an operator.

Just to be comprehensive, let’s try a couple more premise sets. Suppose we replace ¬p with ~p (uncertain p).

p  T  |  T  |  T  |  F  |  F  |  F  |  0  |  0  |  0

q  T  |  F  |  0  |  T  |  F  |  0  |  T  |  F  |  0

X  T  |  F  |  0  |  T  |  T  |  T  |  T  |  0  |  0

~p 0  |  0  |  0  |  0  |  0  |  0  |  T  | T  |  T

Remember the common sense meaning: if there’s smoke, there’s fire; but we don’t know whether there’s smoke. Now according to Rojas, the Aymara would conclude the seventh column — that if we aren’t sure that there’s smoke, we still conclude that there’s fire. This is patently wrong, just for the simplest common sense. If we aren’t sure whether there’s smoke, the most confident conclusion is that it’s uncertain whether there is also fire. Now if it turns out that there’s fire, then of course, that would satisfy the premises, but that is not a certainty of the premises. The only certain conclusion is that fire is uncertain.

So what’s the mistake in the matrix? In a trivalent system, not only must all the T values be considered with respect to the conditional, but also the 0 values too.  So all three last columns are possible worlds of deduction based on the premises. And those last three include each of q is true, q is false, and q is uncertain (or to insist stubbornly on a nonmodal view, neither true nor false). And that is the intuitive, obvious, common sense conclusion — the certain, unequivocal conclusion that the lack of smoke doesn’t say a damned thing about fire at all; could be fire, could be no fire but definitely fire is uncertain.

If the Aymara conclude that there’s fire from “if there’s smoke then there’s fire; and there’s no smoke” all the worse for them. They wouldn’t be the only people who have trouble understanding the conditional (it’s commonly mistaken for iff). But I’d seriously wonder how they built Tihuanaco, Puma Puncu and all those astonishing and inexplicable wall structures. So I’m confident that the Aymara don’t use such a deductive method, and are using epistemic modality much like English:

p is either T or F

uncertain of p is K(◇p & ◇¬p).

I don’t think all the above exhaust the problems with a trivalent system interpreted as “neither true nor false” with an operator “neither truly nor falsely x.” But I’d like to work them out in epistemic modality before judging.

fun with Gödel

Originally published on Language and Philosophy, November 3, 2011

There’s an easy answer to this question, but if you replace c) 60% with 0%, then you get a liar paradox, a Gödel-type statement — it has no numerical answer and can only be evaluated outside its terms.

http://flowingdata.com/2011/10/28/best-statistics-question-ever/

Gödel, btw, once proved Anselm’s ontological argument, the argument that proves god exists. His version, so far as I can tell, removes all modality in Anselm’s argument, and I think that’s why it works. It’s a complicated version, but I think that even extremely simple, non modal versions work too, e.g., if “god” by definition is that which nothing is greater, then whatever is greatest is that thing. This version works for any model in which there is at least one object. So if nothing exists, then it doesn’t prove anything, but since something patently exists, we can safely assume that there is a greatest thing.

It’s been pointed out to me that there may be many notions of greatness. I think this is a problem in the logical semantics, not in the syntax, unless there is no such notion as “the greatest.” The only important of the many notions is the notion of “greatest of greatest things.” If that can’t be evaluated, then the designation of “god” is indeterminate, but not necessarily nonexistent. I mean, the argument is valid, so far as it goes; it just doesn’t tell you which notion of greatness is in fact the greatest.

There are a number of ways to treat this notion of greatness. Maybe the notion is inherently relative, like “good” — a good chess player might not be a good bricklayer. If so, then there is no idea of “the greatest.” But assuming there are such notions as “the good” and “the greatest,” I don’t see any contradiction within them. So one can assume there is such, without determining which.

I am fond of Anselm’s proof in part because it seems to make everyone nervous, non believers and believers alike. It makes non believers uncomfortable because they don’t want to admit they are wrong and they don’t want to believe in god. I’m not sure which is worse to them. It makes the believers nervous because the god it proves might not be the comfortable personal one they’ve been living with. They want to believe in their god, so it’s convenient that there is no logical proof of god that might be out of their control. And that is as it should be: the notion of god should be a #$%ing scary one, if you actually take it seriously for a moment. The Old Testament got that one right. People don’t take it as seriously as it should be. They should be crossing their fingers that there is no god.

If you look at the argument more closely, the character of the god that it proves is such a cipher, that it steals away all the personal comfort of the deity they want or need. That’s one reason Aquinas didn’t include it as one of his proofs. His proofs are all friendly ones like the order of the cosmos and its beauty. Nothing so cold as logical truth.

I delight in Anselm’s proof. What it says about god is utterly trivial — so I’m happy to live with it — and the argument strikes terror in everyone else. That’s almost as good as having a personal god working for you.

The logical positivists would have approved: the argument is true, but meaningless. So god exists in all his greatestness. So what? It’s no different than pantheism, which is basically indistinguishable from atheism.

Everything and more

Originally published on Language and Philosophy, April 9, 2011

Btw, Wallace’s book on infinity, Everything and More, is an excellent, lucid treatment of the problems within mathematics (which implies scientific theory in general) and its application to the real world.

There are limits at the boundaries not only of mathematics, but also in logics — not only modal logics but even simple first order logic. Reminds me of a Kantian remark Russell made somewhere to the effect that our descriptions of phenomena only approach the phenomena from our descriptive perspective. The things themselves remain utterly mysterious. Worse, our descriptive apparatus is limited. Even ourselves as phenomena (pace Schopenhauer) cannot gain access to ourselves beyond our own descriptive apparatus — language, sensibility, logic and science. We can, at best, observe behavior and derive a few conclusions. Schopenhauer, prior to Darwin and armed only with Eastern religion, mistook that behavior for the thing in itself, when, really, it was just a character of evolutionary survival, not of the entire cosmos of phenomena or noumena. With sufficient scientific research, we approach explanation of both sentient and non sentient behavior … but substance itself? What can you say about the limits of knowledge? Is there a something beyond it or not? I’ve never been impressed with Wittgenstein’s cavalier gnome “It’s not a something, but it’s not a nothing either.” Well, so what? I think Rumsfeld said it better. We can have no access to it, and we don’t even know if there’s a there there.

Which brings me back to the notion of explanation in the sciences: the theory of evolution has explanatory value for psychology (despite Fodor’s just complaint that it is, at this point, merely post hoc and not predictive) because it is a theory independent of emotions or sensibility. It is not just a statistical account of emotions under conditions (the behaviorist model). It is a theory of species development in general. I think Fodor is right that it is post  hoc and unpredictive, but it still has explanatory power, just post hoc. Maybe that’s the best place to rest on Fodor’s complaint: natural selection is explanatory but not yet predictive.

Wallace’s solution

Originally published on Language and Philosophy, April 9, 2011

I’m a little uncomfortable reading Wallace’s book since it was a youthful work not intended for publication, was never published while he lived, and is being published now without his permission. And he left no later comments on it and can’t respond to critics.

Wallace’s solution depends on a scope difference in an alethic tensed modality. Using Taylor’s example: “if the battle occurred, then the admiral must have ordered it” is ambiguous between

1. if the battle occurred, then yesterday it was the case that the admiral must have ordered it

2. if the battle occurred, then it must have been the case that the admiral ordered it

Wallace admits that (1) entails fatalism, but points out that (2) doesn’t. According to (1), the admiral must have ordered the battle, and so had no choice. In (2) the admiral ordered it, but not under duress, as it were, of necessity (must). He had a choice — he might have contemplated several possible worlds in which he orders and several in which he declines to order. It’s just that none of those possible worlds turn out to have been real. That is, yesterday’s world in which the battle was ordered, turns out to have been the only possible world.

But if that’s the only possible world, why is it the only? Wallace seems to show successfully that the answer cannot be the logic alone.

Suppose you are at the moment of ordering. That moment excludes any moment in which you decline to order. That moment includes only moments in which you order the battle. The difference seems to be between whether you have free will or whether you have freedom. Wallace’s draws a nice distinction between fatalism and a kind of post hoc determinism.

Is this a difference without a distinction? If the admiral knows that the moment determines his order (he has no freedom), what does it serve him to have free will? Nothing in the real world. But that accords with our experience: no matter how we plan for the world, the consequences are beyond our ability to control.

The utilitarian/consequentialist effects of determinism and fatalism are equally discouraging. But the entailments for (Kantian)  moral sensibility are completely distinct. Determinism is consistent with holding moral sentiments. Not a fatalist, and that’s why even philosophers spurn it.

On the other hand, while Wallace has found a distinction, I’m not sure that it is telling against Taylor’s view. Relations of necessity among physical effects depend on circumstances, and these are explicit in Taylor’s assumptions. If there is no world in which the order for battle is not given, then the only possible worlds are those in which he chooses to order it. That entails a strange paradox: he is free to choose, but he can only choose one option, not the alternate choice.

How the logic of implication works entirely depends on how you set up the modal system — its axiomata or its inferential rules or both — and its consistency. What makes a system meaningful, assuming it’s consistent, is its usefulness or accuracy. Wallace uses our natural language notions of “it couldn’t have happened” and “it can’t have happened.” That’s good for his system, but not telling against Taylor, since Taylor is specifically using logic against natural language notions which, he is attempting to show, are wrong. And on the other hand,  Wallace’s distinction seems to violate our linguistic, and maybe real-world, understanding of “free.” It may be that the logical syntax should include an inference from

must yesterday order

to

yesterday must order

or it may be that the inference should be dealt with in the semantics, in the model — in any world in which there is only one option, there is no free choice.