Logica Docens – Page 10 – Maverick Philosopher

The Truth Operator and the Truth Predicate

This is an addendum to our earlier discussion which I hope will advance it a step or two. We heard Alan Rhoda claim that the following sentence is false: 'If nothing exists, then it is true that nothing exists.' Let's think further about this. We first note that 'If nothing exists, then it is true that nothing exists' can be parsed in two ways:

1. If nothing exists, then it is true that (nothing exists).

2. If nothing exists, then it is true (that nothing exists).

Call (1) the operator construal. 'It is true that ( )' is a sentential operator the operand of which is a sentence. The result of the operation is itself a sentence. If the operand is true, then the resulting sentence is true. If the operand is false, then the resulting sentence is false. Please note that prefixing 'It is true that' to a sentence cannot change the truth-value of the sentence. In this respect, the truth operator 'It is true that ( )' is unlike the negation operator 'It is not the case that ( ).' Assuming Bivalence — as I have been doing throughout — if you negate a true sentence you get a false one, and vice versa.

Call (2) the predicate construal. The consequent of (2) is of course a sentence, but it is not the result or product of a sentential operator operating upon a sentence. For what is within the parentheses is not a sentence. 'That nothing exists' is not a sentence. It does not have a truth-value. If I assertively utter it I do not convey a complete thought to my audience. 'That nothing exists' is the name of a proposition. It follows that 'it is true' in the consequent of (2) functions as a predicate as one can more clearly see from the equivalent

3. If nothing exists, then that nothing exists is true.

In (2) and (3) a predicate is attached to a name, whereas in (1) this is not the case: a sentential operator is attached to a sentence.

Not only are the parsings different, the ontological commitments are as well. (2) commits us to propositions while (1) doesn't. And (1) seems to commit us to operators while (2) doesn't.

Here is the place to comment on my asterisks convention. Putting asterisks around a declarative sentence forms a name of the proposition expressed by the sentence. 'The Moon is uninhabited' is a declarative sentence. '*The Moon is uninhabited*' is not a sentence but a name. It names an entity that has a truth-value, but it itself does not have a truth-value. (2) and (3) can also be rendered as

4. If nothing exists, then *Nothing exists* is true.

With the operator/predicate distinction under our belts we may be in a position to see how one philosopher (Alan) could reasonably reject 'If nothing exists, then it is true that nothing exists' while another accepts it. The one philosopher gives the original sentence the predicate construal which is committed to propositions. This philosopher then reasons that, if nothing exists, then no propositions exist either, and are therefore not available to instantiate the property of being true. The other philosopher gives the original sentence the operator construal and finds it impossible to understand how anyone could reject the original sentence so construed. This philosopher insists that if nothing exists, then it is true that nothing exists; that this truth is not nothing, and that therefore it is something, which implies that it cannot be the case that nothing exists.

A Counterexample to P –> It is True that P?

Alan Rhoda e-mails:

In a recent post you write:

The objector is inviting us to consider the possible situation in which beings like us do not exist and no truths either. The claim that this situation is possible, however, is equivalent to the claim that it is true that this situation is possible.

I think there's a mistake here. In general, p does not entail it is true that p. The envisioned scenario is a case in point. The sense in which the situation is admitted to be possible is purely negative in that absent truths, no contradiction results. To say, however, that it is true that the situation is possible, where truths are supposed to depend on cognizers, requires that the situation be possible in a positive sense, i.e., it requires that something be the case, not merely that contradictions not be the case.

Thanks, Alan. Let's rehearse the dialectic. I argued in a standard self-referential way that *There are truths* is not just true, but necessarily true. (For *There are no truths,* if true is false, and if false is false, hence is necessarily false, so its negation is necessarily true.) I then asked whether the necessity of its truth is unconditional or rests on a condition such as the existence of thinking beings. (In other words: is the necessity of truth merely a transcendental presupposition without which we cannot operate as thinking beings, or is the necessity of there being truths metaphysically grounded in rerum natura?) If the existence of truths is merely a transcendental presupposition, then it would seem that the following scenario is possible: there are no thinking beings and no truths either. If this scenario is possible, then the necessity of *There are truths* would be conditional. I then tried to show that the scenario is not possible by invoking the principle Necessarily, for any p, p –> it is true that p. My thought was that if it is possible that there be no thinking beings and no truths either, then it is true that this is possible. But if it is true that this is possible, then it is true independently of what anyone thinks. But then truth as something more than a transcendental presupposition is being presupposed.

I am afraid I don't understand your criticism of the reasoning. The principle p –> it is true that p strikes me as self-evident. Its 'intellectual luminosity,' if you will, will trump any putative counterexample. If snow is white, then it is true that snow is white; if grass is green, then it is true that grass is green; if it is possible that no thinkers and no truths exist, then it is true that it is possible that no truths and no thnkers exist. Now the point is that this last truth says how things are in a situation in which no thinkers exist; therefore it is a truth that cannot exist only if thinkers exist. It exists whether or not thinkers exist.

You write, "The sense in which the situation is admitted to be possible is purely negative in that absent truths, no contradiction results." I don't follow you. The situation is possible assuming that truth is a mere transcendental presupposition. Now suppose the possibility is actual. Then it will be true both that it is possible and that it is actual. So once again truth cannot be a mere transcendental presupposition.

You then say, " To say, however, that it is true that the situation is possible, where truths are supposed to depend on cognizers, requires that the situation be possible in a positive sense, i.e., it requires that something be the case, not merely that contradictions not be the case." But I am not claiming that truths are dependent on cognizers; I am refuting that view. If the existence of truths depends on cognizers, and cognizers exist contingently, then it is possible that there be no truths. But this is not possible since if, per impossibile, it were the case that there are no truths, then this would be the case, i.e., would be true.

The ComBox is open if you want to discuss this further.

More on the Law of Non-Contradiction and its Putative Empirical Refutability

A reader's e-mail with my comments in blue:

Nice post on the LNC. That topic is a real quagmire, isn't it?

I’ve lost the link to the Science Daily report of the Cleland experiment, so the details of how he confirmed the superposition are lost to me, but I’m really struck by the fact that you are defending LNC as a transcendental, not transcendent, principle. Kant doesn’t take this route in the First Critique, does he? LNC is not some form of sensibility, is it?

That's right, I am defending LNC as a transcendental, not a transcendent principle, and for two reasons. First, I believe that LNC is well-nigh unassailable if presented as a transcendental a priori condition of the possibility of (i) meaningful discourse and (ii) experience of the objects of Sellar's manifest image or of Kant's phenomenal world, with (i) being more unassailable than (ii). Second, the transcendental defense is all I need to turn aside what I take to be your conclusion from the Cleland experiment, namely, that there are macro-objects of direct perceptual acquaintance that serve as counterexamples to LNC. To show that LNC applies beyond our thought and beyond our experience to whatever lies beyond our thought and experience, if anything, is not so easy. One cannot just dogmatically assume that a law of thought is automatically a law of reality, especially since this has been denied by any number of philosophers. Aristotle in Metaphysics Gamma, 3, 4, attempts a proof by retortion of LNC, but as far as I can see, all he establishes is that LNC is a necessary condition of meaningful thinking and speaking, not that its validity extends beyond thought and speech and their objects to things in themselves.

I would also urge in passing against certain dogmatic Thomists that the Critical Problem — the problem of showing how a priori conditions of thinking apply to things external to us — is already present in nuce in Aristotle. But that's another long series of posts.

LNC is surely not a form of sensibility for Kant, but it is a form of understanding. Since there is for Kant no experience (Erfahrung) without a 'marriage' of sensibility (Sinnlichkeit) and understanding (Verstand), it seems reasonable to impute to Kant the view that no macro-object of experience can violate LNC.

One thing I think Cleland would say is that observing the paddle in the normal sense, i.e, bombarding it with lots of photons, disturbs the superposition and collapses the ambivalent quantum state into a moving or a not moving state. So he would seem to agree with you as far “seeing” in the ordinary sense goes. We don’t see something moving & not moving—and one could add: our eyes and brains are just not designed to experience such objects even if we could do so without disturbing them. But, seeing is not the same as sensing, and presumably the paddle in its quantum state has effects (on us) that are unambiguously different from its effects in states where the superposition has collapsed. So, as you say, no naked eye observations of superposition, but perhaps that’s too narrow a focus and we should admit that we might experience a superposition is some other unique way.

You seem to be assuming the Copenhagen interpretation of QM. But as you know, it is not the only game in town. Bill Hill, a U.K. immunologist, e-mailed me the following, which is very helpful:

There are two main interpretations of quantum mechanics which are popular in the physics community (there are a few others, but they are mostly propounded by eccentrics). The first is the "Copenhagen" interpretation, in which quantum events really do exist in multiple incompatible states at the same time, but only when there is no outside observer looking at them. I am not making this up, though I should add that by "observer", they do not just mean conscious beings but any information-carrying system (such as a sensor) which can report data about the quantum event. Though it is implausible at first glance, this interpretation does in fact solve the boundary problem that so vexes many scientists. Because sub-atomic particles are too small for us to see, they are free to exhibit this behaviour. But people, planets and so on are so large that they are always under observation in some sense, they cannot behave in this way. Hence, when the little bit of metal in the article is observed, it will either appear moving or not moving to the person looking at it, but when nobody is looking it is in fact doing both. This raises enormous questions about perception and causality, and many people are very unhappy with it as a result. The important point is that your suggestion that there cannot be an empirical counterexample to the Law of Non-Contradiction remains intact under the Copenhagen interpretation.

The most popular alternative to Copenhagen is the "Many-Worlds" interpretation, in which the universe splits into two duplicates every time a quantum event occurs. So when the little bit of metal in the article is put into its quantum state, in one universe it is moving, and in the other it is not. Of course, it is impossible for us to tell which one we are in. Many people (rightly, in my opinion) think that this is just silly, and embrace Copenhagen on grounds of parsimony. However, it is consistent with the data, and also with the Law of Non-Contradiction, since two incompatible states cannot exist in the same universe under Many-Worlds.

So as far as I can tell from my limited experience, you are correct and neither the Copenhagen nor the Many-Worlds interpretations of quantum mechanics offer an empirical counterexample to the Law of Non-Contradiction, whatever other fascinating philosophical questions they may raise in their own right.

The salient point is that, on the 'many worlds' interpretation of QM there is no violation of LNC not even on the micro-level let alone on the macro-level. Given that there is no one settled interpretation of QM accepted by all physicists, the case against LNC at either level is bound to be weak.

This is very tricky stuff, but I think it is the paddle, a macro object that we can directly observe under other conditions, that is now in the superposition state of moving and not moving. We in fact have put it into this state. The paddle is not some ding an sich, but an ordinary object that can transition from existing “normally” in one state or its opposite to existing at once in both contradictory states. In principle any macro-object could be reduced to such a quantum ground state but we just can’t physically do so.

I am afraid that you are not making sense. You have already granted that the paddle that we see with the naked eye cannot be seen by the naked eye to be both moving and not moving, But now you are saying that that very visible paddle — and not some invisible micro-constituents of it – has been put by the experimental apparatus into a state in which it is both moving and not moving. This implies that one and the same visible paddle is both (moving & not moving) and not (moving & not moving). Which is is higher -order contradiction.

Are you saying that there are two paddles? Then they can't both be visible.

Furthermore, if you say, as you do above, following the Copenhagen interpretation, that observation of the paddle forces it into one state or the other, then cannot also say that that very same visible paddle is in both states.

I am afraid that the science journalist's report on the Cleland experiment has delivered us into a realm of rank gibberish.

Your second point that LNC is also a “form of intelligibilty” is surely right, and it just invites incomprehension to say that the paddle is both moving and not moving. I guess we need to learn the jargon of the physicists here. I’m not sure exactly what they say but something like the paddle in its quantum ground state is in a superposition of motion and no motion. That I get, and it says something remarkable about the really weird universe we apparently live in. I’m saving up my money and moving to a good old Newtonian universe at the first opportunity!

But now you are sounding like certain Trinitarian theologians who say that we should just repeat the creedal formulae without worrying whether or how they make any bloody sense. It is curious that defenders of the coherence of the Trinity often bring up QM. You of course grant no authority to the Bible or the Church. Why then do you genuflect before the authority of scientists when they spout gibberish? I am being intentionally provocative. ComBox is open if you care to counterrespond.

An Empirical Refutation of the Law of Non-Contradiction?

Nice work if you can get it! Here we read:

A team of scientists has succeeded in putting an object large enough to be visible to the naked eye into a mixed quantum state of moving and not moving.

Andrew Cleland at the University of California, Santa Barbara, and his team cooled a tiny metal paddle until it reached its quantum mechanical "ground state"– the lowest-energy state permitted by quantum mechanics. They then used the weird rules of quantum mechanics to simultaneously set the paddle moving while leaving it standing still. The experiment shows that the principles of quantum mechanics can apply to everyday objects as well as as atomic-scale particles.

So we have a little object, visible to the naked eye, that is simultaneously moving and not moving. Is that possible? Yes, if one part is moving and a distinct part is not moving. Presumably that is not what is meant above. What is meant is that the whole object is simultaneously both moving and not moving. That too is possible if 'simultaneously' or 'at the same time' is being applied to an interval of time. Consider a temporal interval five seconds in duration. Let 't' refer to that interval. It is surely possible that object O, the whole of it, be at rest at t and in motion at t. But this triviality is also not what is meant above.

Continue reading “An Empirical Refutation of the Law of Non-Contradiction?”

Seldom Seen Slim on ‘Tautologies’ That Ain’t

Seldom Seen Slim in a characteristic back-to-the-camera pose evaluates the shooting skills of the man we call 'Doc' (in allusion to Doc Holliday). Slim writes:

Whilst I'm mulling over your thoughts on souls and salvation, here's a trifle you might agree with.

You write "There are many examples of the use of tautological sentences to express non-tautological propositions."

Indeed, my favorite ordinary language example is the use of the double identity "a=a and b=b" to assert that a and b are quite different (in some salient respect under discussion), and to imply that the listener is rashly ignoring this obvious fact!

"Why did she do that?" Men are men and women are women.

"The hell they are!" does not reject the identities, but the salient difference.

An exercise I used to give to my (brighter) logic students was to formalize what "men are men and women are women" is trying to assert in the Predicate Calculus.

……………

'Men are men and women are women,' which appears to be a conjunction of two tautologies and thus a tautology, is, however, typically used to express the non-tautological proposition that men and women are different as Slim suggests. The idea is not that each man is numerically different from each woman, but that there are properties had by men, but not by women, which render men and women qualitatively different. So perhaps we can symbolize the intended non-tautological proposition as follows using second-order predicate logic:

There is a property P such that for every x and every y, if x is a man, then x has P, and if y is a woman, then y does not have P. Symbolically:

(EP)(x)(y)[(Mx –>Px) & (Wy –>~Px)]

where 'E' is the existential quantifier, 'x' and 'y' are individual variables, 'M' and 'W' are predicate constants, 'P' is a predicate variable, '&' is the sign for truth-functional conjunction, and '–>' denotes the material or Philonian conditional.

But on second thought, this doesn't seem right. For when we say that men are men and women women, we do not mean that there is one particular property that all men have and all women lack that renders them qualitatively different; what we we mean is that there are some properties which render them different, allowing that these could be different properties for different men and women. To illustrate: consider a universe consisting of just two men and two women: Al, Bill, Carla, and Diana. The property of having lousy social skills might be had by both Al and Bill and lacked by both Carla and Diana. But it might also be that there are two properties, the property of being ornery and the property of being highly unemotional such that Al is highly unemotional but not ornery and Bill is ornery but not highly unemotional while neither of the ladies has either. In that case there would be no one propery that distinguishes the men from the women.

So let's try:

(x)(y)[(Mx & Wy –> (EP) (Px & ~Py)]

What say you, Slim?

A Quiz on Alienans Adjectives

First read study the post Alienans Adjectives. Then take the quiz. Answers below the fold. Classify the adjectives in the following examples as either specifying (S), alienans (A), or neither (N). Much of course depends on the context in which the phrase is used. So imagine a plausible and common context.

1. Deciduous tree. 2. Alleged assailant. 3. Imaginary friend. 4. Material implication. 5. Contemptible leftist. 6. Infrared radiation. 7. Hypothetical medium of the transmission of electromagnetic signals. 8. Postal service. 9. Imaginary number. 10. Male chauvinist.

Continue reading “A Quiz on Alienans Adjectives”

Alienans Adjectives

A reader inquires:

I find your blog interesting and educational. A while ago you mentioned that there is a term for an adjective which is used not to specify a particular sort of the noun which it modifies, but rather a thing which does not meet the definition of that noun. (I've likely somewhat mangled the description of this term in trying to recall it.) For example 'polished leather' and 'red leather' are kinds of leather, but 'artificial leather' refers to things which aren't leather at all. I have tried to find the post that talked about this but I forgot what the topic was when you mentioned it. Can you please tell me the name for this?

'Artificial' in 'artificial leather' functions as an alienans adjective. It 'alienates' the sense of the noun it modifies. In the case of specifying adjectives, an FG is a G, where F is an adjective and G a noun. Thus a nagging wife is a wife, a female duck is a duck, cow's leather is leather, and a contingent truth is a truth. But if 'F' is alienans, then either an FG is not a G, or it does not follow from x's being an FG that x is a G. For example, your former wife is not your wife, a decoy duck is not a duck, artificial leather is not leather, and a relative truth is not a truth. Is an apparent heart attack a heart attack? It may or may not be. One cannot validly move from 'Jones had an apparent heart attack' to 'Jones had a heart attack.' So 'apparent' in 'apparent heart attack' is alienans.

Note that I was careful to say 'artificial' in 'artificial leather' is an alienans adjective. For it does not function as such in every context. 'Artificial' in 'artificial insemination' is not alienans: you are just as inseminated if it has come about artificially or naturally.

Two more examples of alienans adjectives that I borrow from Peter Geach: 'forged' in 'forged banknote' and 'putative in 'putative father.' If x is a forged banknote it does not follow that x is a banknote. And if x is the putative father of y, it does not follow that x is the father of y. Here is an example I got from the late Australian philosopher Barry Miller: 'negative' in 'negative growth.' If my stock portfolio is experiencing negative growth, then it is precisely not experiencing growth.

Of course, I am not suggesting that every adjective (as employed in some definite context) can be classified as either specifying or alienans. Consider the way 'mean-spirited' functions in 'mean-spirited Republican.' In most contexts, the implication is not that some Republicans are mean-spirited and some are not; the implication is that all are. To be a Republican is just to be mean-spirited. Is there a name for that sort of adjective? I don't know. But there ought to be, and if I ever work out a general theory of adjectives, I'll give it one.

Now consider 'Muslim terrorist.' A politically correct idiot might take offense at this phrase as implying that all Muslims are terrorists or even that all and only Muslims are terrorists. But no intelligent person would take it this way. If I say that Hasan is a Muslim terrorist , then the plain meaning to anyone with his head screwed on properly is that Hasan is a Muslim and a terrorist, which obviously does not imply that all Muslims are terrorists.

When Is a Tautology Not a Tautology?

My Aunt T. was married to a gruff and taciturn Irishman who rejoiced under the name of 'Morris.' Thinking to engage Uncle Mo in conversation during one of my infrequent visits to the Big Apple, and knowing that Morris drove a beer truck, I once made some comment about the superiority of German over American beer. Uncle Mo, not to be seduced into the bracing waters of dialectic, replied, "Beer is beer." End of conversation.

But the beginning of an interesting line of thought. A tautology is a logical truth. To be precise, a tautology is a logical truth within the propositional calculus. (Every tautology is a logical truth, but not every logical truth is a tautology. The logical truths of the predicate calculus are not tautologies, strictly speaking.)

Continue reading “When Is a Tautology Not a Tautology?”

A Modal Fallacy to Avoid: Confusing the Necessity of the Consequence with the Necessity of the Consequent

No one anywhere can utter 'I am talking now' without saying something true. Indeed, that is necessarily the case: it doesn't just happen to be the case. Letting T = 'I am talking now,' we can write

1. Necessarily, for any speaker S, if S utters T, then T is true.

But it would be a mistake to infer

2. For any speaker S, if S utters T, then T is necessarily true.

The same goes for 'I exist now.' It cannot be tokened, in language or thought, without it being the case that a truth is expressed; but it does not follow that the one who tokens it necessarily exists. Its negation, 'I do not exist now,' cannot be tokened in language or thought without it being the case that a falsehood is expressed; but it does not follow that the nonexistence of the one who tokens it is impossible.

Validity, Invalidity, and Logical Form

When we say that an argument is valid we are saying something about its logical form. To put it epigrammatically, validity is a matter of form. We are saying that its form is such that no (actual or possible) argument of that form has true premises and a false conclusion. Validity is necessarily truth preserving. I just used the expression, 'its form.' But since an argument can have two or more forms, a better formulation is this:

1. An argument is valid iff it instantiates a valid argument-form.

Given (1), some will be tempted by

2. An argument is invalid iff it instantiates an invalid argument-form.

But (2) is false. After all, every (noncircular) argument instantiates an invalid form. 'Some cameras are digital devices; therefore, some digital devices are cameras,' which is obviously valid, instantiates the invalid form p therefore q. Similarly, every valid syllogism has the invalid form p, q, therefore r. Consider this argument:

The Metaphysics 101 Argument for Propositions

In his SEP entry on propositions, Matthew McGrath presents what he calls the 'Metaphysics 101' argument for propositions. Rather than quote him, I will put the argument in my own more detailed way.

1. With respect to any occurrent (as opposed to dispositional) belief, there is a distinction between the mental act of believing and the content believed. Since believing is 'intentional' as philosophers use this term, i.e., necessarily object-directed, there cannot be an act of believing that is not directed upon some object or content. To believe is to believe something, that the door has been left ajar, for example. Nevertheless, the believing and the believed are distinct.

Logic and Meditation: Complementary Disciplines

Logic is an attempt at disciplining the discursive mind from within the discursive mind. Meditation is an attempt at transcending, by silencing, the discursive mind by using a resource that lies beyond it. Logic is disciplined thinking; meditation is disciplined nonthinking.

Ataraxia and Non-Contradiction

What is the highest good? To be a bit more precise, what is the highest good attainable by us though our own (individual or collective) efforts? One perennially attractive, if unambitious, answer is that of the Pyrrhonian skeptics: our highest good lies in ataraxia. The term connotes tranquillity, peace of mind, freedom from disturbance, unperturbedness. Other Hellenistic schools also identified the summum bonum with ataraxia, but let us confine ourselves to skepticism as represented by Sextus Empiricus.

The Pyrrhonian skeptic, then, seeks ataraxia as the summum bonum. This freedom from disturbance is supposed to be achieved by an epoché (ἐποχή) or suspension of doxastic commitments of a certain sort. One is supposed to achieve the happiness of tranquillity by suspending one's belief on a certain range of issues, those issues that typically cause contention, enmity, and bloodshed. Among these will be found philosophical, theological, and political issues. My elite readers can easily supply their own examples.

Continue reading “Ataraxia and Non-Contradiction”

Why Be Consistent? Three Types of Consistency

A reader inquires:

This idea of the necessity to be consistent seems to be the logician's "absolute," as though being inconsistent was the most painful accusation one could endure. [. . .] What rule of life says that one must be absolutely consistent in how one evaluates truth? It is good to argue from first principles but it can also lead one down a rat hole.

Before we can discuss whether one ought to be consistent, we need to know which type of consistency is at issue. There are at least three types of consistency that people often confuse and that need to be kept distinct. I'll call them 'logical,' 'pragmatic,' and 'diachronic.' But it doesn't matter how we label them as long as we keep them separate.

Continue reading “Why Be Consistent? Three Types of Consistency”