The Discursive Framework, Logic, and Whether the Via Negativa is the Path to Nowhere

The Historian of Logic comments:

It seems to me that what you call the ‘Discursive Framework’ is what I and others call ‘logic’, and that it reflects a Kantian view of logic that prevailed before Russell and Frege, namely that logic reflects the ‘laws of thought’ only. Are you mooting the possibility of beings which defy conception under these laws, or realms where the laws do not apply?

I was re-reading Kant’s Logic last week and it is full of this stuff.

Logic. I would define logic as the normative science of inference. Science: scientia, study of. Inference: the mental process of deriving a proposition (the conclusion) from one or more propositions (the premises). Normative: logic is not concerned with how people think as a matter of fact, which is a concern of psychology, but with how they ought to think if they are to arrive at truth and move from known truths to further truths. The task of logic is to set forth the criteria whereby correct inferences may be distinguished from incorrect inferences.

The above definition is neutral with respect to any number of ontological questions. Thus I used 'proposition' above innocuously without presupposing any theory as to what propositions are. I spoke of inference as a mental process, but this too is innocuous inasmuch as one could be a mind-brain identity theorist and agree with me about logic. (But if you are an eliminativist about the mental, then you 'get the boot': it is a Moorean fact that there are inferences.)

Discursive Framework. This is not the same as logic, pace the Historian, even though it does contain such logical principles as LNC and LEM, and indeed the whole of standard logic. (We can argue about 'standard' in the ComBox.) The DF also contains principles that are not strictly logical — they are not logical truths — but are better classifiable as metaphysical, as propositions of metaphysica generalis. Examples:

a. Everything exists: There are no nonexistent items. Pace van Inwagen, the negation of this is not a logical contradiction.

b. Everything has properties. (Partisans of bare or thin particulars do not deny this.)

c. Nothing has a property P by being identical to P. (The 'is' of predication is not assimilable to the 'is' if formal identity.)

d. Principles of logic, such as For any x, x = x, are not just true of objects of thought qua objects of thought, but are also true of mind-independentally real items. Thus the principles of logic are not merely principles of thought but principles of reality as well. Not merely logical, they are also ontological. There is a jump here, from the logical to the ontological, that Aristotle was aware of. With that jump comes the problem of justifying it.

e. The thinking of ectypal intellects such as ourselves is necessarily such as to involve a distinction between subject and predicate. There are no simple thoughts/propositions if by that we mean thoughts/proposition lacking sub-propositional structure. Every proposition is internally structured, e.g. Fa, Rab, (x)Fx, etc.

Laws of Thought but No Psychologism. Kant, Husserl, and Frege all rejected psychologism in logic. Are the laws of logic laws of thought? Yes, of course. What else would they be? But this is not to say that they are laws of human psychology. They are laws that govern the thinking of any actual or possible ectypal intellect. They might also be laws of reality, all reality, with no exceptions. But surely it would be uncritical simply to assume this. It wants proof, or at least argument. As I said, Aristotle had already seen the problem.

Are you mooting the possibility of beings which defy conception under these laws, or realms where the laws do not apply?

Yes, that is what I am doing, although I wouldn't speak of beings. That's plural, and the singular-plural distinction is part and parcel of the Discursive Framework. My aim is to make philosophy safe for mysticism. My aim is to show that while remaining in philosophy, in the DF, one can come to descry the 'possibility' of a, or rather, THE transdiscursive realm. I deny that the via negativa is the road to nowheresville or u-topia.

I am not attempting anything new; the novelty is merely in the way I go about it. And there is nothing illogical about it. Or can you find non sequiturs or other strictly logical mistakes in the above or in recent cognate posts? If there is a suprarational realm would it not be sloppy thinking, and thus 'illogical,' to assume that it must be infrarational?

Nothing new: we have seen this sort of thing in the Far East in Buddhist schools like that of Nagarjuna and in Taoism; in the ancient and medieval Western world, e.g., Pseudo-Dionysus the Areopagite, and in the modern period with Kant and then again in the early Wittgenstein.

The Doctrine of Divine Simplicity (DDS) Helps Focus The Issue. Duality is unavoidable on the discursive plane. To think is to judge, and to judge at the most basic level is to judge of a that it is or is not F. At a bare minimum, then, there is the duality of subject and property. (Brentano transformations of predications into existential sentences avail nothing: the duality of existence-nonexistence remains.) As I said above, no thought/proposition, no content of an act of thinking, is simple. But God is simple according to DDS. He is identical to his attributes, which implies that each attribute is identical to every other one. If he weren't then he would be dependent on his attributes for his nature, and he would not be the absolute reality. He could not possess aseity. He could not be uniquely unique. If God were unique only in the sense that he is necessarily one of a kind, then he would one of a class of such beings, and a greater could be conceived, namely, a being uniquely unique, i.e., unique in the sense of transcending the very distinction between instance and kind.

Now if God is simple, then how can our talk and thought, which is necessarily discursive, be literally true of him? One response is that God talk is literal but analogical. This needs exploring in a separate post. But if we cannot accept the doctrine of analogy, then the simple God lies entirely beyond the DF.

Circular Definitions, Arguments, and Explanations

In the course of our discursive operations we often encounter circularity. Clarity will be served if we distinguish different types of circularity. I count three types. We could label them definitional, argumentative, and explanatory.

A. The life of the mind often includes the framing of definitions. Now one constraint on a good definition is that it not be circular. A circular definition is one in which the term to be defined (the definiendum) or a cognate thereof occurs in the defining term (the definiens). 'A triangle is a plane figure having a triangular shape,' though plainly true, is circular. 'The extension of a term is the set of items to which the term applies' is an example of a non-circular definition.

B. Sometimes we argue. We attempt to support a proposition p by adducing other propositions as reasons for accepting p. Now one constraint on a good argument is that it not be circular. A circular argument in is one in which the conclusion appears among the premises, sometimes nakedly, other times clothed for decency's sake in different verbal dress. Supply your own examples.

C. Sometimes we explain. What is it for an individual x to exist? Suppose you say that for x to exist is for some property to be instantiated. One variation on this theme is to say that for Socrates to exist is for the haecceity property Socrateity to be instantiated. This counts as a metaphysical explanation, and a circular one to boot. For if Socrateity is instantiated, then it is is instantiated by Socrates who must exist to stand in the instantiation relation. The account moves in a circle, an explanatory circle of embarrassingly short diameter.

Suppose someone says that for x to exist is for x to be identical to something or other. They could mean this merely as an equivalence, in which case I have no objection. But if they are shooting for a explanation of existence in terms of identity-with-something-or-other, then they move in an explanatory circle. For if x exists in virtue of its identity with some y, then y must exist, and you have moved in an explanatory circle.

Some philosophers argue that philosophers ought not be in the business of explanation. I beg to differ. But that is a large metaphilosophical topic unto itself.

Neither the Existence Nor the Nonexistence of God is Provable

A post of mine ends like this:

To theists, I say: go on being theists. You are better off being a theist than not being one. Your position is rationally defensible and the alternatives are rationally rejectable. But don't fancy that you can prove the existence of God or the opposite. In the end you must decide how you will live and what you will believe.

About "Don't fancy that you can prove the existence of God or the opposite," Owen Anderson asks:

How would we know if that claim is itself true? Isn't it is possible that one or the other can indeed be proven?

To formulate my point in the declarative rather than the exhortative mood:

P. Neither the existence nor the nonexistence of God is provable.

How do I know (P) to be true? By reflection on the nature of proof. An argument is a proof if and only if it satisfies all of the following six requirements: it is deductive; valid in point of logical form; free of such informal fallacies as petitio principii; possesses a conclusion that is relevant to the premises; has premises each of which is true; has premises each of which is known to be true.

I say that an argument is a proof if and only it is rationally compelling, or rationally coercive. But an argument needn't be rationally compelling to be a more or less 'good argument,' one that renders its conclusion more or less rationally acceptable.

Now if my definition above gives what we ought to mean by 'proof,' then it is clear that neither the existence nor the nonexistence of God can be proven. Suppose you present a theistic or anti-theistic argument that satisfies the first five requirements. I will then ask how you know that the premises are true. Suppose one of your premises is that change is the conversion of potency into act. That is a plausible thing to maintain, but how do you know that it is true? How do you know that the general-ontological framework within which the proposition acquires its very sense, namely, Aristotelian metaphysics, is tenable? After all, there are alternative ways of understanding change. That there is change is a datum, a Moorean fact, but it would be an obvious mistake to confuse this datum with some theory about it, even if the theory is true. Suppose the theory is true. This still leaves us with the question of how we know it is. Besides, the notions of potency and act, substance and accident, form and matter, and all the rest of the Aristotelian conceptuality are murky and open to question. (For example, the notion of prime matter is a necessary ingredient in an Aristotelian understanding of substantial change, but the notion of materia prima is either incoherent or else not provably coherent.)

To take a second example, suppose I give a cosmological argument the starting point of which is the seemingly innocuous proposition that there are are contingent beings, and go on to argument that this starting point together with some auxiliary premises, entails the existence of God. How do I know that existnece can be predicated of concrete individuals? Great philosophers have denied it. Frege and Russell fanmously held that existence vannot be meaningfully predicated of individuals but only of cncepts and propositional functions. I have rather less famoulsy argued that the 'GFressellina' view' is mstaken, but this is a point of controversy. Furtrhertmore, if existence cannot be meaningfully predicated of individuals, how can individuals be said to exist contingently?

The Appeal to Further Arguments

If you tell me that the premises of your favorite argument can be known to be true on the basis of further arguments that take those premises as their conclusions, then I simply iterate my critical procedure: I run the first five tests above and if your arguments pass those, then I ask how you know that their premises are true. If you appeal to still further arguments, then you embark upon a vicious infinite regress.

The Appeal to Self-Evidence

If you tell me that the premises of your argument are self-evident, then I will point out that your and my subjective self-evidence is unavailing. It is self-evident to me that capital punishment is precisely what justice demands in certain cases. I'll die in the ditch for that one, and pronounce you morally obtuse to boot for not seeing it. But there are some who are intelligent, well-meaning, and sophisticated to whom this is not self-evident. They will charge with with moral obtuseness. Examples are easily multiplied. What is needed is objective, discussion-stopping, self-evidence. But then, how, in a given case, do you know that your evidence is indeed objective? All you can go on is how things seem to you. If it seems to you that it is is objectively the case that p, that boils down to: it seems to you that, etc., in which case your self-evidence is again merely subjective.

The Appeal to Authority

You may attempt to support the premises of your argument by an appeal to authority. Now many such appeals are justified. We rightly appeal to the authority of gunsmiths, orthopaedic surgeons, actuaries and other experts all the time, and quite sensibly. But such appeals are useless when it comes to PROOF. How do you know that your putative authority really is one, and even if he is, how do you know that he is eight in the present case? How do you know he is not lying to you well he tells you you need a new sere in your semi-auto pistol?

The Appeal to Revelation

This is the ultimate appeal to authority. Necessarily, if God reveals that p, then p! Again, useless for purposes of proof. See Josiah Royce and the Paradox of Revelation.

Move in a Circle?

If your argument falls afoul of petitio principii, that condemns it, and the diameter of the circle doesn't matter. A circle is a circle no matter its diameter.

Am I Setting the Bar Too High?

It seems to me I am setting it exactly where it belongs. After all we are talking about PROOF here and surely only arguments that generate knowledge count as proofs. But if an argument is to generate a known proposition, then its premises must be known, and not merely believed, or believed on good evidence, or assumed, etc.

"But aren't you assuming that knowledge entails certainty, or (if this is different) impossibility of mistake?" Yes I am assuming that. Argument here.

Can I Consistently Claim to Know that (P) is true?

Owen Anderson asked me how I know that (P) is true. I said I know it by reflection on the concept of proof. But that was too quick. Obviously I cannot consistently claim to know that (P) if knowledge entails certainty. For how do I know that my definition captures the essence of proof? How do I know that there is an essence of proof, or any essence of anything? What I want to say, of course, is that it is very reasonable to define 'proof' as I define it — absent some better definition — and that if one does so define it then it is clear that there are very few proofs, and, in particular, that there are no proofs of God or of the opposite.

"But then isn't it is possible that one or the other can indeed be proven?"

Yes, if one operates with a different, less rigorous, definition of 'proof.' But in philosophy we have and maintain high standards. So I say proof is PROOF (a tautological form of words that expresses a non-tautological proposition) and that we shouldn't use the word to refer to arguments that merely render their conclusions rationally acceptable.

Note also that if we retreat from the rationally compelling to the rationally acceptable, then both theism and atheism are rationally acceptable. I suspect that what Owen wants is a knock-down argument for the existence of God. But if that is what he wants, then he wants a proof in my sense of the world. If I am right, that is something very unreasonable to expect.

There is no getting around the need for a decision. In the end, after all the considerations pro et contra, you must decide what you will believe and how you will live.

Life is a venture and an adventure. You cannot live without risk. This is true not only in the material sphere, but also in the realm of ideas.

Kripke, Belief, Irrationality, and Contradiction

London Ed comments:

I also note a confusion that has been running through this discussion, about the meaning of ‘contradiction’. I do not mean to appeal to etymology or authority, but it’s important we agree on what we mean by it. On my understanding, a contradiction is not ‘the tallest girl in the class is 18’ and ‘the cleverest girl in the class is not 18’, even when the tallest girl is also the cleverest. Someone could easily believe both, without being irrational. The point of the Kripke puzzle is that Pierre seems to end up with an irrational belief. So it’s essential, as Kripke specifies, that he must correctly understand all the terms in both utterances, and that both utterances are logically contradictory, as in ‘Susan is 18’ and ‘Susan is not 18’.

Do we agree?

Well, let's see. The Maverick method enjoins the exposure of any inconsistent polyads that may be lurking in the vicinity. Sure enough, there is one:

An Inconsistent Triad

a. The tallest girl in the class is the cleverest girl in the class.
b. The tallest girl in the class is 18.
c. The cleverest girl in the class is not 18.

This trio is logically inconsistent in the sense that it is not logically possible that all three propositions be true. But if we consider only the second two limbs, there is no logical inconsistency: it is possible that (b) and (c) both be true. And so someone, Tom for example, who believes that (b) and also believes that (c) cannot be convicted of irrationality, at least not on this score. For all Tom knows — assuming that he does not know that (a) — they could both be true: it is epistemically possible that both be true. This is the case even if in fact (a) is true. But we can say more: it is metaphysically possible that both be true. For (a), if true, is contingently true, which implies that it is is possible that it be false.

By contrast, if Tom entertains together, in the synthetic unity of one consciousness, the propositions expressed by 'Susan is 18 years old' and 'Susan is not 18 years old,' and if Tom is rational, then he will see that the two propositions are logical contradictories of each other, and it will not be epistemically possible for him that both be true. If he nonetheless accepts both, then we have a good reason to convict him of being irrational, in this instance at least.

Given the truth of (a), (b) and (c) cannot both be true and cannot both be false. This suggests that the pair consisting of (b) and (c) is a pair of logical contradictories. But then we would have to say that the contradictoriness of the pair rests on a contingent presupposition, namely, the truth of (a). London Ed will presumably reject this. I expect he would say that the logical contradictoriness of a pair of propositions cannot rest on any contingent presupposition, or on any presupposition at all. Thus

d. Susan is 18

and

e. Susan is not 18

form a contradictory pair the contradictoriness of which rests on their internal logical form — Fa, ~Fa — and not on anything external to the propositions in question.

So what should we say? If Tom believes both (b) and (c), does he have contradictory beliefs? Or not?

The London answer is No! The belief-contents are not formally contradictory even though, given the truth of (a), the contents are such that they cannot both be true and cannot both be false. And because the belief-contents are not formally contradictory, the beliefs themselves — where a belief involves both an occurrent or dispositional state of a person and a belief-content towards which the person takes up a propositional attitude — are in no theoretically useful sense logically contradictory.

The Phoenix answer suggestion is that, because we are dealing with the beliefs of a concrete believer embedded in the actual world, there is sense to the notion that Tom's beliefs are contradictory in the sense that their contents are logically contradictory given the actual-world truth of (a). After all, if Susan is the tallest and cleverest girl, and the beliefs in question are irreducibly de re, then Tom believes, of Susan, that she is both 18 and not 18, even if Tom can gain epistemic access to her only via definition descriptions. That belief is de re, irreducibly, is entailed by (SUB), to which Kripke apparently subscribes:

SUB: Proper names are everywhere intersubstitutable salva veritate.

A Second Question

If, at the same time, Peter believes that Paderewski is musical and Peter believes that Paderewski is not musical, does it follow that Peter believes that (Paderewski is musical and Paderewski is not musical)? Could this conceivably be a non sequitur? Compare the following modal principle:

MP: If possibly p and possibly ~p, it does not follow that possibly (p & ~p).

For example, I am now seated, so it is possible that I now be seated; but it is also possible that I now not be seated, where all three occurrences/tokens of 'now' rigidly designate the same time. But surely it doesn't follow that it is possible that (I am now seated and I am now not seated). Is it perhaps conceivable that

BP: If it is believed by S that p and it is believed by S that ~p, it does not follow that it is believed by S that (p & ~p)?

Has anybody ever discussed this suggestion, even if only to dismiss it?

In What Sense Does an Indefinite Noun Phrase Refer?

London Ed propounds a difficulty for our delectation and possible solution:

Clearly the difficulty with the intralinguistic theory is its apparent absurdity, but I am trying to turn this around. What can we say about extralinguistic reference? What actually is the extralinguistic theory? You argue that the pronoun ‘he’ inherits a reference from its antecedent, so that the pronoun does refer extralinguistically, but only per alium, not per se.

Mark 14:51 And there followed him [Jesus] a certain young man (νεανίσκος τις) , having a linen cloth (σινδόνα) cast about his naked body; and the young men laid hold on him. 14:52 And he left the linen cloth, and fled from them naked.

So the pronoun ‘he’ inherits its reference through its antecedent. But the antecedent is the noun phrase ‘a certain young man’. On your theory, does this refer extralinguistically? That’s a problem, because indefinite noun phrases traditionally do not refer, indeed that’s the whole point of them. ‘a certain young man’ translates the Latin ‘adulescens quidam’ which in turn translates the Greek ‘νεανίσκος τις’. Here ‘certain’ (Latin quidam, Greek τις) signifies that the speaker knows who he is talking about, but declines to tell the audience who this is. Many commentators have speculated that the man was Mark himself, the author of the gospel, which if true means that ‘a certain young man’ and the pronouns, could be replaced with ‘I’, salva veritate. But Mark deliberately does not tell us.

So, question 1, in what sense does the indefinite noun phrase refer, given that, on the extralinguistic theory, it has to be the primary referring phrase, from which all subsequent back-reference inherits its reference?

A. First of all, it is not clear why Ed says, ". . . indefinite noun phrases traditionally do not refer, indeed that’s the whole point of them." Following Fred Sommers, in traditional formal logic (TFL) as opposed to modern predicate logic (MPL), indefinite noun phrases do refer. (See Chapter 3, "Indefinite Reference" of The Logic of Natural Language.) Thus the subject terms in 'Some senator is a physician' and 'A physician is running for president' refer, traditionally, to some senator and to a physician. This may be logically objectionable by Fregean lights but it is surely traditional. That's one quibble. A second is that it is not clear why Ed says "that's the whole point of them."

So the whole point of a tokening of 'a certain young man' is to avoid making an extralinguistic reference? I don't understand.

B. Ed says there is a problem on my view. A lover of aporetic polyads, I shall try to massage it into one. I submit for your solution the following inconsistent pentad:

a. There are only two kinds of extralinguistic reference: via logically proper names, including demonstratives and indexicals, and via definite descriptions.
b. The extralinguistic reference of a grammatical pronoun used pronominally (as opposed to quantificationally or indexically) piggy-backs on the extralinguistic reference of its antecedent. It is per alium not per se.
c. 'His,' 'him,' and 'he' in the verse from Mark are pronouns used pronominally the antecedent of which is 'a certain young man.'
d. 'A certain young man' in the verse from Mark is neither a logically proper name nor a definite description.
e. 'A certain young man' in the verse from Mark refers extralinguistically on pain of the sentence of which it is a part being not true.

The pentad is inconsistent.

The middle three limbs strike me as datanic. So there are two possible solutions.

One is (a)-rejection. Maintain as Sommers does that indefinite descriptions can refer. This 'solution' bangs up against the critique of Peter Geach and other Fregeans.

The other is (e)-rejection. Deny that there is any extralinguistic reference at all. This, I think, is Ed's line. Makes no sense to me, though.

I wonder: could Ed be toying with the idea of using the first four limbs as premises in an argument to the conclusion that all reference is intralinguistic? I hope not.

Like a Moth to the Flame

Jean van Heijenoort was drawn to Anne-Marie Zamora like a moth to the flame. He firmly believed she wanted to kill him and yet he travelled thousands of miles to Mexico City to visit her where kill him she did by pumping three rounds from her Colt .38 Special into his head while he slept. She then turned the gun on herself. There is no little irony in the fact that van Heijenoort met his end in the same city as Lev Davidovich Bronstein, better known as Leon Trotsky. For van Heijenoort was Trotsky's secretary, body guard, and translator from 1932 to 1939.

The former 'Comrade Van' was a super-sharp logician but a romantic fool nonetheless. He is known mainly for his contribution to the history of mathematical logic. He edited From Frege to Godel: A Source Book in Mathematical Logic, 1879-1931 (Harvard University Press 1967) and translated some of the papers. The source book is a work of meticulous scholarship that has earned almost universally high praise from experts in the field.

One lesson is the folly of seeking happiness in another human being. The happiness we seek, whether we know it or not, no man or woman can provide. And then there is the mystery of self-destruction. Here is a brilliant, productive, and well-respected man. He knows that 'the flame' will destroy him, but he enters it anyway. And if you believe that this material life is the only life you will ever have, why throw it away for an unstable, pistol-packing female?

One might conclude to the uselessness of logic for life. If the heart has its reasons (Pascal) they apparently are not subject to the discipline of mathematical logic. All that logic and you still behave irrationally about the most important matters of self-interest? So what good is it? Apparently, van Heijenoort never learned to control his sexual and emotional nature. Does it make sense to be ever so scrupulous about what you allow yourself to believe, but not about what you allow yourself to love?

SOURCES (The following are extremely enjoyable books. I've read both twice.)

Anita Burdman Feferman, Politics, Logic, and Love: The Life of Jean van Heijenoort, Boston: Jones and Bartlett Publishers, 1993.

Jean van Hejenoort, With Trotsky in Exile: From Prinkipo to Coyoacan, Harvard UP, 1978.

Related: Trotsky's Faith

The Last Words of Leon Trotsky

Trotsky, Frida Kahlo, with van Heijenoort standing behind Frida.

God, Proof, and Desire

From a reader:

. . . I’m confused by some of your epistemic terms. You reject [in the first article referenced below] the view that we can “rigorously prove” the existence of God, and several times say that theistic arguments are not rationally compelling, by which you mean that there are no arguments “that will force every competent philosophical practitioner to accept their conclusions on pain of being irrational if he does not.“

Okay, so far I’m tracking with you. But then you go on to say that “[t]here are all kinds of evidence” for theism (not just non-naturalism), while the atheist “fails to account for obvious facts (consciousness, self-consciousness, conscience, intentionality, purposiveness, etc.) if he assumes that all that exists is in the space-time world. I will expose and question all his assumptions. I will vigorously and rigorously drive him to dogmatism. Having had all his arguments neutralized, if not refuted, he will be left with nothing better than the dogmatic assertion of his position."

So how is the atheist not irrational on your view, assuming he is apprised of your arguments? Perhaps the positive case for theism and the negative case against naturalism don’t count as demonstrations in a mathematical sense, but I’m not sure why they’re not supposed to be compelling according to your gloss on the term.

The term 'mathematical' muddies the waters since it could lead to a side-wrangle over what mathematicians are doing when they construct proofs. Let's not muddy the waters. My claim is that we have no demonstrative knowledge of the truth of theism or of the falsity of naturalism. Demonstrative knowledge is knowledge produced by a demonstration. A demonstration in this context is an argument that satisfies all of the following conditions:

1. It is deductive
2. It is valid in point of logical form
3. It is free of such informal fallacies as petitio principii
4. It is such that all its premises are true
5. It is such that all its premises are known to be true
6. It is such that its conclusion is relevant to its premises.

To illustrate (6). The following argument satisfies all of the conditions except the last and is therefore probatively worthless:

Snow is white
ergo
Either Obama is president or he is not.

On my use of terms, a demonstrative argument = a probative argument = a proof = a rationally compelling argument. Now clearly there are good arguments (of different sorts) that are not demonstrative, probative, rationally compelling. One type is the strong inductive argument. By definition, no such argument satisfies (1) or (2). A second type is the argument that satisfies all the conditions except (5).

Can one prove the existence of God? That is, can one produce a proof (as above defined) of the existence of God? I don't think so. For how will you satisfy condition (5)? Suppose you give argument A for the existence of God. How do you know that the premises of A are true? By argument? Suppose A has premises P1, P2, P3. Will you give arguments for these premises? Then you need three more arguments, one for each of P1, P2, P3, each of which has its own premises. A vicious infinite regress is in the offing. Needless to say, moving in an argumentative circle is no better.

At some point you will have to invoke self-evidence. You will have to say that, e.g., it is just self-evident that every event has a cause. And you will have to mean objectively self-evident, not just subjectively self-evident. But how can you prove, to yourself or anyone else, that what is subjectively self-evident is objectively self-evident? You can't, at least not with respect to states of affairs transcending your consciousness.

I conclude that no one can prove the existence of God. But one can reasonably believe that God exists. The same holds for the nonexistence of God. No one can prove the nonexistence of God. But one can reasonably believe that there is no God.

The same goes for naturalism. I cannot prove that there is more to reality than the space-time system and its contents. But I can reasonably believe it. For I have a battery of arguments each of which satisfies conditions (1), (2), (3) and (6) and may even, as far as far as I know, satisfy (4).

"So how is the atheist not irrational on your view, assuming he is apprised of your arguments?"

He is not irrational because none of my arguments are rationally compelling in the sense I supplied, namely, they are not such as to force every competent philosophical practitioner to accept their conclusions on pain of being irrational if he does not. To illustrate, consider the following argument from Peter Kreeft (based on C. S. Lewis), an argument I consider good, but not rationally compelling. I will argue (though I will not prove!) that one who rejects this argument is not irrational.

The Argument From Desire

Ecstasy of St. Teresa by Gianlorenzo Bernini (Permission by Mark Harden; http://www.artchive.com)

Premise 1: Every natural, innate desire in us corresponds to some real object that can satisfy that desire.
Premise 2: But there exists in us a desire which nothing in time, nothing on earth, no creature can satisfy.
Conclusion: Therefore there must exist something more than time, earth and creatures, which can satisfy this desire.

This something is what people call "God" and "life with God forever."

This is surely not a compelling argument. In fact, as it stands, it is not even valid. But it is easily repaired. There is need of an additional premise, one to the effect that the desire that nothing in time can satisfy is a natural desire. This supplementary premise is needed for validity, but it is not obviously true. For it might be — it is epistemically possible that — this desire that nothing in time can satisfy is artificially induced by one's religious upbringing or some other factor or factors.

Furthermore, is premise (1) true? Not as it stands. Suppose I am dying of thirst in the desert. Does that desire in me correspond to some real object that can satisfy it? Does the existence of my token desire entail the existence of a token satisfier? No! For it may be that there is no potable water in the vicinity, when only potable water in the immediate vicinity can satisfy my particular thirst. At most, what the natural desire for water shows is that water had to have existed at some time. It doesn't even show that water exists now. Suppose all the water on earth is suddenly rendered undrinkable. That is consistent with the continuing existence of the natural desire/need for water.

But this is not a decisive objection since repairs can be made. One could reformulate:

1* Every type of natural, innate desire in us corresponds to some real object that can satisfy some tokens of that type of desire.

But is (1*) obviously true? It could be that our spiritual desires are not artificial, like the desire to play chess, but lacking in real objects nonetheless. It could be that their objects are merely intentional. Suppose our mental life (sentience, intentionality, self-awareness, the spiritual desires for meaning, for love, for lasting happiness, for an end to ignorance and delusion and enslavement to base desires) is just an evolutionary fluke. Our spiritual desires would then be natural as opposed to artificial, but lacking in real objects.

Why do we naturally desire, water, air, sunlight? Because without them we wouldn't have come into material existence in the first place. Speaking loosely, Nature implanted these desires in us. This is what allows us to infer the reality of the object of the desire from the desire. Now if God created us and implanted in us a desire for fellowship with him, then we could reliably infer the reality of God from the desire. But we don't know whether God exists; so it may be that the natural desire for God lacks a real object.

Obviously, one cannot define 'natural desire' as a desire that has a real and not merely intentional object, and then take the non-artificiality of a desire as proof that it is natural. That would be question-begging.

My point is that (1) or (1*) is not known to be true and is therefore rationally rejectable. The argument from desire, then, is not rationally compelling.

As for premise (2), how do we know that it is true? Granting that it is true hitherto, how do we know that it will be true in the future? The utopian dream of the progressives is precisely that we can achieve here on earth final satisfaction of our deepest desires. Now I don't believe this for a second. But I don't think one can reasonably claim to know that (2) is false. What supports it is a very reasonable induction. But inductive arguments don't prove anything.

In sum, the argument from desire, suitably deployed and rigorously articulated, helps render theistic belief rationally acceptable. But it is not a rationally compelling argument.

Denying the Antecedent?

While traipsing through the Superstition foothills Sunday morning in search of further footnotes to Plato, I happened to think of James Madison and Federalist #51 wherein we read, "If men were angels, no government would be necessary." My next thought was: "Men are not angels." But I realized it could be the formal fallacy of Denying the Antecedent were I to conclude to the truth, "Some government is necessary." (I hope you agree with me that that is a truth.)

The first premise is a counterfactual conditional, indeed, what I call a per impossibile counterfactual. To keep things simple, however, we trade the subjunctive in for the indicative. Let this be the argument under consideration:

1. If men are angels, then no government is necessary.
2. Men are not angels.
ergo
3. Some government is necessary.

A prima vista, we have here an instance of the invalid argument-form, Denying the Antecedent:

If p, then q
~p
ergo
~q.

But I am loath to say that the argument (as opposed to the just-depicted argument-form) is invalid. It strikes me as valid. But how could it be valid?

Approach One

One could take the (1)-(3) argument to be an enthymeme where the following is the tacit premise:

1.5 If no government is necessary, then men are angels.

Add (1.5) to the premises of the original argument and the conclusion follows by modus tollendo tollens.

Approach Two

Might it be that 'if ___ then ___' sentences in English sometimes express biconditional propositions? Clearly, if we replace (1) with

1* Men are angels if and only if no government is necessary

the resulting argument is valid.

Approach Three

One might take the (1)-(3) argument as inductive. Now every inductive argument is invalid in the technical sense of 'invalid' in play here. So if there are good inductive arguments, then there are good invalid arguments. Right? If the (1)-(3) argument is inductive, then I think we should say it is a very strong inductive argument. It would then be right churlish and cyberpunkish to snort, "You're denying the antecedent!"

The question arises: are there any good examples from real argumentative life (as opposed to logic text books) of Denying the Antecedent? I mean, nobody or hardly anybody argues like this:

If Jack ran a red light, then Jack deserves a traffic citation.
Jack did not run a red light.
ergo
Jack does not deserve a traffic citation.

Is it a Contradiction?

London Ed writes,

I am interested in your logical or linguistic intuitions here. Consider

(*) There is someone called ‘Peter’, and Peter is a musician. There is another person called ‘Peter’, and Peter is not a musician.

Is this a contradiction? Bear in mind that the whole conjunction contains the sentences “Peter is a musician” and “Peter is not a musician”. I am corresponding with a fairly eminent philosopher who insists it is contradictory.

Whether or not (*) is a contradiction depends on its logical form. I say the logical form is as follows, where 'Fx' abbreviates 'x is called 'Peter'' and 'Mx' abbreviates 'x is a musician':

LF1. (∃x)(∃y)[Fx & Mx & Fy & ~My & ~(x =y)]

In 'canonical English':

CE. There is something x and something y such that x is called 'Peter' and x is a musician and y is called 'Peter' and y is not a musician and it is not the case that x is identical to y.

There is no contradiction. It is obviously logically possible — and not just logically possible — that there be two men, both named 'Peter,' one of whom is a musician and the other of whom is not.

I would guess that your correspondent takes the logical form to be

LF2. (∃x)(∃y)(Fx & Fy & ~(x = y)) & Mp & ~Mp

where 'p' is an individual constant abbreviating 'Peter.'

(LF2) is plainly a contradiction.

My analysis assumes that in the original sentence(s) the first USE (not mention) of 'Peter' is replaceable salva significatione by 'he,' and that the antecedent of 'he' is the immediately preceding expression 'Peter.' And the same for the second USE (not mention) of 'Peter.'

If I thought burden-of-proof considerations were relevant in philosophy, I'd say the burden of proving otherwise rests on your eminent interlocutor.

But I concede one could go outlandish and construe the original sentences — which I am also assuming can be conjoined into one sentence — as having (LF2).

So it all depends on what you take to be the logical form of the original sentence(s). And that depends on what proposition you take the original sentence(s) to be expressing. The original sentences(s) are patient of both readings.

Now Ed, why are you vexing yourself over this bagatelle when the barbarians are at the gates of London? And not just at them?

Is Anything Real Self-Identical?

I am sometimes tempted by the following line of thought. But I am also deeply suspicious of it.

Are the 'laws of thought' 'laws of reality' as well? Since such laws are necessities of thought, the question can also be put by asking whether or not the necessities of thought are also necessities of being. It is surely not self-evident that principles that govern how we must think if we are to make sense to ourselves and to others must also apply to mind-independent reality. One cannot invoke self-evidence since such philosophers as Nagarjuna and Hegel and Nietzsche have denied (in different ways) that the laws of thought apply to the real.

Consider, for example, the Law of Identity:

Id. Necessarily, for any x, x = x.

(Id) seems harmless enough and indisputable. Everything, absolutely everything, is identical to itself, and this doesn't just happen to be the case. But what does 'x' range over? Thought-accusatives? Or reals? Or both? What I single out in an act of mind, as so singled out, cannot be thought of as self-diverse. No object of thought, qua object of thought, is self-diverse. And no object of thought, as such, is both F and not F at the same time, in the same respect, and in the same sense. So there is no question but that Identity and Non-Contradiction apply to objects of thought, and are aptly described as laws of thought. (Excluded Middle is trickier and so I leave it to one side.) What's more, these laws of thought hold for all possible finite, discursive, ectypal intellects. Thus what we have here is a transcendental principle, at least, not one grounded in the contingent empirical psychology or physiology of the type of animals we happen to be. Transcendentalism maybe, but no psychologism or physiologism!

But do Identity and Non-Contradiction apply to 'reals,' i.e., to entities whose existence is independent of their being objects of thought? Are these transcendental principles also ontological principles? Is the necessity of such principles as (Id) grounded in the transcendental structure of the finite intellect, or in being itself? Are the principles merely transcendental or are they also transcendent? (It goes without saying that I am using these 't' words in the Kantian way.)

The answer is not obvious.

Consider a pile of leaves. If I refer to something using the phrase, 'that pile of leaves,' I thereby refer to one self-identical pile; as so referred to, the pile cannot be self-diverse. But is the pile self-identical in itself (apart from my referring to it, whether in thought or in overt speech)?

In itself, in its full concrete extramental reality, the pile is not self-identical in that it is composed of many numerically different leaves, and has many different properties. In itself, the pile is both one and many. As both one and many, it is both self-identical and self-diverse. It is self-identical in that it is one pile; it is self-diverse in that this one pile is composed of many numerically different parts and has many different properties. Since the parts and properties are diverse from each other, and these parts and properties make up the pile, the pile is just as much self-diverse as it is self-identical. The pile is of course not a pure diversity; it is a diversity that constitutes one thing. So, in concrete reality, the pile of leaves is both self-identical and self-diverse.

If you insist that the pile's being self-identical excludes its being self-diverse, then you are abstracting from its having many parts and properties. So abstracting, you are no longer viewing the pile as it is in concrete mind-independent reality, but considering it as an object of thought merely. You are simply leaving out of consideration its plurality of parts and of properties. For the pile to be self-identical in a manner to exclude self-diversity, the pile would have to be simple as opposed to complex. But it is not simple in that it has many parts and many properties.

The upshot is that the pile of leaves, in concrete reality, is both one and many and therefore both self-identical and self-diverse. But this is a contradiction. Or is the contradiction merely apparent? Now the time-honored way to defuse a contradiction is by making a distinction.

One will be tempted to say that the respect in which the pile is self-identical is distinct from the respect in which it is self-diverse. The pile is self-identical in that it is one pile; the pile is self-diverse in that it has many parts and properties. No doubt.

But 'it has many parts and properties' already contains a contradiction. For what does 'it' refer to? 'It' refers to the pile which does not have parts and properties, but is its parts and properties. The pile is not something distinct from its parts and properties. The pile is a unity in and through a diversity of parts and properties. As such, the pile is both self-identical and self-diverse.

What the above reasoning suggests is that such 'laws of thought' as Identity and Non-Contradiction do not apply to extramental reality. No partite thing, such as a pile of leaves, is self-identical in a manner to exclude self-diversity. Such things are as self-diverse as they are self-identical. So partite things are self-contradictory.

From here we can proceed in two ways.

The contradictoriness of partite entities can be taken to argue their relative unreality. For nothing that truly exists can be self-contradictory. This is the way of F. H. Bradley. One takes the laws of thought as criterial for what is ultimately real, shows that partite entities are not up to this exacting standard, and concludes that partite entities belong to Appearance.

The other way takes the lack of fit between logic and reality as reflecting poorly on logic: partite entities are taken to be fully real, and logic as a falsification. One can find this theme in Nietzsche and in Hegel.

Are Burden-of-Proof Considerations Relevant in Philosophy?

1. The question this post raises is whether it is at all useful to speak of burden of proof (BOP) in dialectical situations in which there are no agreed-upon rules of procedure that are constitutive of the 'game' played within the dialectical situation. By a dialectical situation I mean a context in which orderly discussion occurs among two or more competent and sincere interlocutors who share the goal of arriving as best they can at the truth about some matter, or the goal of resolving some question in dispute. My main concern is with dialectical situations that are broadly philosophical. I suspect that in philosophical debates the notion of burden of proof is out of place and not usefully deployed. That is what I will now try to argue.

2. I will begin with the observation that the presumption of innocence (POI) in an Anglo-American court of law is never up for grabs in that arena. Thus the POI is not itself presumptively maintained and subject to defeat. If Jones is accused of a crime, the presumption of his innocence can of course be defeated, but that the accused must be presumed innocent until proven guilty is itself never questioned and of course never defeated. The POI is not itself a defeasible presumption. And if Rescher is right that there are no indefeasible presumptions, then the POI is not even a presumption. The POI is a rule of the 'game,' and constitutive of the 'game.' The POI in a court room situation is like a law of chess. The laws of chess, as constitutive of chess, cannot themselves be contested within a game of chess. In a particular game a dispute may arise as to whether or not a three-fold repetition of position has occurred. But that a three-fold repetition of position results in a draw is not subject to dispute. The reason there is always a definite outcome in chess (win, lose, or draw) is precisely because of the non-negotiable chess-constitutive laws. These laws, of course, are not inscribed in the nature of things, but are conventional in nature.

As I pointed out earlier, defeasible presumption (DP) and burden of proof are correlative notions. The defeasible presumption that the accused is innocent until proven guilty places the onus probandi on the prosecution. Therefore, from the fact that the POI is not itself defeasible in a court of law, it follows that neither is the BOP. Where the initating BOP lies — the BOP that remains in force and never shifts during the proceedings — is never subject to debate. It lies on the state in a criminal case and on the plaintiff in a civil case. If you agree to play the game, then you agree to its constitutive rules. Since these rules are constitutive of the game, they cannot be rejected on pain of ceasing to play the particular game in question.

3. But in philosophy matters are otherwise. For in philosophy everything is up for grabs, including the nature of philosophical inquiry and the rules of procedure. (This is why metaphilosophy is not 'outside of' philosophy but a branch of same.) And so where the BOP lies in a debate between, say, atheists and theists is itself a matter of debate and bitter contention. Each party seeks to put the BOP on the other, to 'bop' him if you will. The theist is inclined to say that there is a defeasible presumption in favor of the truth of theism; but of course few atheists will meekly submit to that pronunciamento. If the theist is right in his presumption, then he doesn't have to do anything except turn aside the atheist's objections: he is under no obligation to argue positively for theism any more than the accused is under an obligation to prove his innocence.

Accused to accuser: "I don't have to prove my innocence; you have to prove my guilt. I enjoy the presumption of innocence; you bear the burden of proof."

Theist to atheist: "I don't have to prove that God exists; you have to prove that God does not exist. Theism enjoys the presumption of being true; atheism bears the burden of proving that theism is not true." (This assumes that BOP and DP are legitimately deployed within broadly philosophical precincts — which I am denying.)

Note that if the theist invokes the above presumption he needn't be committing the ad ignorantiam fallacy. He needn't be saying that theism is true because it hasn't been proved to be false. Surely the following deductive argument is invalid:

No one has ever proved that God does not exist
ergo
God does exist.

Just as the presumption of innocence does not entail that the accused is innocent, the presumption of truth does not entail that the proposition presumed true is true. So the mere fact that I have the presumption on my side does not amount to an argument that what I am presuming is true. If I have the presumption on my side, then my dialectical opponent bears the BOP. That's all.

4. Now we come to my tentative suggestion. There is no fact of the matter as to where the BOP lies in any dialectical context, legal, philosophical or any other: it is a matter of decision and agreement upon what has been conventionally decided. In chess, for example, the rules had to be decided and the players have to agree to accept them. No one thinks that these rules are inscribed in rerum natura. The same goes for BOP and DP. It had to be decided that in court room discourse and dialectic the accused enjoys the DP and the accuser(s) the BOP.

In philosophical discourse, however, there are no procedural rules regarding DP and BOP that we will all agree on.

For example, according to Douglas N. Walton, ". . . the basic rule of burden of proof in reasonable dialogue is: He who asserts must prove." (Informal Logic, p. 59) That is clearly false. If I assert that that you left the door open, there is no need for me to prove my assertion. A proof is an argument having premises and conclusion. Surely there is no need to argue for matters evident to sense perception. In fact, it would be unreasonable to do so. Or suppose I assert the Law of Noncontradiction. There is no way I can (non-circularly) prove it. So I cannot be under any epistemic obligation to prove it. 'Ought' implies 'can.'

And how would this work in a dispute between theist and atheist? I assert that God exists and you assert that God does not exist. We both assert. So we both bear the BOP, and we both enjoy DP? But then BOP and DP have no application in this area.

I have heard it said that the BOP lies on the one who makes a positive (affirmative) assertion. But surely both theist and atheist make positive assertions about reality. 'Reality is such that God exists.' 'Reality is such that God does not exist.' Both propositions are logically affirmative.

Suppose our atheist denies God by saying 'God is an unconscious anthropomorphic projection.' Logically, that is an affirmative proposition. Will you conclude that the BOP is on the atheist?

Some say that presumptions are essentially conservative: there is a presumption in favor of the existing and the established and against the novel, the far-out, and what runs contrary to prevailing opinion. "If it ain't broke, don't fix it." Suppose I give the following speech:

There is a presumption in favor of every existing institution, long-standing way of doing things, and well-entrenched and widespread way of belief. Now the consensus gentium is that God exists. And so I lay it down that there is a defeasible presumption in favor of theism and that the burden of proof lies squarely on the shoulders of the atheist. Theism is doxastically innocent until proven guilty. The theist need only rebut the atheist's objections; he needn't make a positive case for his side.

Not only would the atheist not accept this declaration, he would be justified in not accepting it, for reasons that are perhaps obvious. For my declaration is as much up for grabs as anything else in philosophy. And of course if I make an ad baculum move then I remove myself from philosophy's precincts altogether. In philosophy the appeal is to reason, never to the stick.

The situation in philosophy could be likened to the situation in a court of law in which the contending parties are the ones who decide on the rules of procedure, including BOP and DP rules. Such a trial could not be brought to a conclusion. That's the way it is in philosophy. Every procedural rule and methodological maxim is further fodder for philosophical Forschung. (Sorry, couldn't resist the alliteration.)

My tentative conclusion is as follows. In philosophy no good purpose is served by claims that the BOP lies on one side or the other of a dispute, or that there is a DP in favor of this thesis but not in favor of that one. For there is no fact of the matter as to where the BOP lies. BOP considerations are usefully deployed only in dialectical situations in which there is an antecedent conventional agreement on the rules of procedure, rules that constitute the dialectical 'game' in question, and that are agreed upon by the players of the game and never contested by them while playing it.

Quod Gratis Asseritur, Gratis Negatur and Petitio Principii

It occurred to me this morning that there is a connection between the two.

Suppose a person asserts that abortion is morally wrong. Insofar forth, a bare assertion which is likely to elicit the bare counter-assertion, 'Abortion is not morally wrong.' What can be gratuitously asserted may be gratuitously denied without breach of logical propriety, a maxim long enshrined in the Latin tag Quod gratis asseritur, gratis negatur. So one reasonably demands arguments from those who make assertions. Arguments are supposed to move us beyond mere assertions and counter-assertions. Here is one:

Infanticide is morally wrong
There is no morally relevant difference between abortion and infanticide
Ergo
Abortion is morally wrong.

Someone who forwards this argument in a concrete dialectical situation in which he is attempting to persuade himself or another asserts the premises and in so doing provides reasons for accepting the conclusion. This goes some distance toward removing the gratuitousness of the conclusion. THe conclusion is supported by reasons that are independent of the conclusion. But suppose he gave this argument:

Abortion is the deliberate and immoral termination of an innocent pre-natal human life
Ergo
Abortion is morally wrong.

The second argument is a clear example of petitio principii, begging the question. While the premise entails the conclusion, it does not support it with a reason independent of the conclusion. The argument 'moves in a circle' presupposing the very thing it needs to prove.

So the second 'argument' merely appears to be an argument: it us really just an assertion in the guise of an argument, and a gratuitous assertion at that. But what is gratuitously asserted can be gratuitously denied.

So there we have the connection between Quod gratis asseritur, gratis negatur and Petitio principii.

Morris Raphael Cohen: Logical Thought as the Basis of Civilization

This just over the transom from David Marans:

Recognizing your praise for Critical Rationalism and Morris Raphael Cohen, I believe his page (and also the Karl Popper page) in my PDF Logic Gallery will interest you.

Of course, I hope the book's entire theme/content will also interest you.

Your comments will surely interest ME.

In these dark days of the Age of Feeling, when thinking appears obsolete and civilization is under massive threat from Islamism and its 'liberal' and leftist enablers, it seems fitting that I should repost with additions my old tribute to Morris Raphael Cohen. So here it is:

Tribute to Morris R. Cohen: Rational Thought as the Great Liberator

Morris Raphael Cohen (1880-1947) was an American philosopher of naturalist bent who taught at the City College of New York from 1912 to 1938. He was reputed to have been an outstanding teacher. I admire him more for his rationalism than for his naturalism. In the early 1990s, I met an ancient lady at a party who had been a student of Cohen's at CCNY in the 1930s. She enthusiastically related how Cohen had converted her to logical positivism, and how she had announced to her mother, "I am a logical positivist!" much to her mother's incomprehension.

We best honor a thinker by critically re-enacting his thoughts. Herewith, a passage from Cohen's A Preface to Logic, Dover, 1944, pp. 186-187:

…the exercise of thought along logical lines is the great liberation, or, at any rate, the basis of all civilization. We are all creatures of circumstance; we are all born in certain social groups and we acquire the beliefs as well as the customs of that group. Those ideas to which we are accustomed seem to us self-evident when [while?] our first reaction against those who do not share our beliefs is to regard them as inferiors or perverts. The only way to overcome this initial dogmatism which is the basis of all fanaticism is by formulating our position in logical form so that we can see that we have taken certain things for granted, and that someone may from a purely logical point of view start with the denial of what we have asserted. Of course, this does not apply to the principles of logic themselves, but it does apply to all material propositions. Every material proposition has an intelligible alternative if our proposition can be accurately expressed.

These are timely words. Dogmatism is the basis of all fanaticism. Dogmatism can be combatted by the setting forth of one's beliefs as conclusions of (valid) arguments so that the premises needed to support the beliefs become evident. By this method one comes to see what one is assuming. One can also show by this method that arguments 'run forward' can just as logically be 'run in reverse,' or, as we say in the trade, 'One man's modus ponens is another man's modus tollens.' These logical exercises are not merely academic. They bear practical fruit when they chasten the dogmatism to which humans are naturally prone.

In Cohen's day, the threats to civilization were Fascism, National Socialism, and Communism. Today the main threat is Islamo-totalitarianism, with a secondary threat emanating from the totalitarian Left. Then as now, logic has a small but important role to play in the defeat of these threats. The fanaticism of the Islamic world is due in no small measure to the paucity there of rational heads like Cohen.

But I do have one quibble with Cohen. He tells us that "Every material proposition has an intelligible alternative…" (Ibid.) This is not quite right. A material proposition is one that is non-logical, i.e., one that is not logically true if true. But surely there are material propositions that have no intelligible alternative. No color is a sound is not a logical truth since its truth is not grounded in its logical form. No F is a G has both true and false substitution-instances. No color is a sound is therefore a material truth. But its negation Some color is a sound is not intelligible if 'intelligible' means possibly true. If, on the other hand, 'intelligible' characterizes any form of words that is understandable, i.e., is not gibberish, then logical truths such as Every cat is a cat have intelligible alternatives: Some cat is not a cat, though self-contradictory, is understandable. If it were not, it could not be understood to be self-contradictory. By contrast, Atla kozomil eshduk is not understandable at all, and so cannot be classified as true, false, logically true, etc.

So if 'intelligible' means (broadly logically or metaphysically) possibly true, then it is false that "Every material proposition has an intelligible alternative . . . ."

A Truthmaker Account of Validity

If you accept truthmakers, and two further principles, then you can maintain that a deductive argument is valid just in case the truthmakers of its premises suffice to make true its conclusion. Or as David Armstrong puts it in Sketch of a Systematic Metaphysics (Oxford UP, 2010), p. 66,

In a valid argument the truthmaker for the conclusion is contained in the truthmaker for the premises. The conclusion needs no extra truthmakers.

For this account of validity to work, two further principles are needed, Truthmaker Maximalism and the Entailment Principle. Truthmaker Maximalism is the thesis that every truth has a truthmaker. Although I find the basic truthmaker intuition well-nigh irresistible, I have difficulty with the notion that every truth has a truthmaker. Thus I question Truthmaker Maximalism. (The hyperlinked entry sports a fine photo of Peter L.)

Armstrong, on the other hand, thinks that "Maximalism flows from the idea of correspondence and I am not willing to give up on the idea that correspondence with reality is necessary for any truth." (63) Well, every cygnet is a swan. Must there be something extramental and extralinguistic to make this analytic truth true? And let's not forget that Armstrong has no truck with so-called abstract objects. His brand of naturalism excludes them. So he can't say that there are the quasi-Platonic properties being a cygnet and being a swan with the first entailing the second, and that this entailment relation is the truthmaker of 'Every cygnet is a swan.'

The Entailment Principle runs as follows:

Suppose that a true proposition p entails a proposition q. By truthmaker Maximalism p has a truthmaker. According to the Entailment Principle, it follows that this truthmaker for p is also a truthmaker for q. [. . .] Note that this must be an entailment. If all that is true is that p –> q, the so-called material conditional, then this result does not follow.

I would accept a restricted Entailment Prinicple that does not presuppose Maximalism. To wit, if a proposition p has a truthmaker T, and p entails a proposition q, then T is also a truthmaker for q. For example, if Achilles' running is the truthmaker of 'Achilles is running,' then, given that the proposition expressed by this sentence entails the proposition expressed by 'Achilles is on his feet,' Achilles' running is also the truthmaker of the proposition expressed by 'Achilles is on his feet.'

More on Values and Variables and Logical Form: An Aporetic Hexad

David Brightly comments:

. . . my old copy of Alan Hamilton, Logic for Mathematicians, CUP 1978, uses 'statement variables' in his account of the 'statement calculus', as here. The justification for 'variable' is surely that statements have values, namely truth and falsehood. The truth value of a compound statement is calculated from the truth values of its component simple statements by composition of the truth functions corresponding to the logical connectives. This is analogous to the evaluation of an arithmetic expression by composition of arithmetic functions applied to the values of arithmetic variables.

I detect a possible conflation of two senses of 'value.' There is 'value' in the sense of truth value, and there is 'value' in the sense of the value of a variable.

If I am not mistaken, talk of truth values in the strict sense of this phrase enters the history of logic first with Gottlob Frege (1848-1925). Truth and Falsity for him are not properties of propositions, but values of propositional functions. Thus the propositional function denoted by 'x is wise' has True for its value with Socrates as argument, and False for its value with Nero as argument. Please note the ambiguity of 'argument.' We are now engaging in MathSpeak. The analogy with mathematics is obvious. The squaring function has 4 for its value with 2 or -2 as arguments. Propositional functions map their arguments onto the two truth values.

But we also speak in a different sense of the value of a variable. The bound variables in

(x)(x is a man –> x is mortal)

range over real items. These items are the values of the bound variables but they are not truth values. Therefore, one should not confuse 'value' in the sense of truth value with 'value' in the sense of value of a variable. When Quine famously stated that "To be is to be the value of a [bound] variable" he was not referring to truth values.

Brightly says that "The justification for 'variable' is surely that statements have values, namely truth and falsehood." I think that is a mistake that trades on the confusion just exposed. Agreed, statements have truth values. But it doesn't follow that that placeholders for statements are variables.

I was pleased to see that Hamilton observes the distinction I drew several times between an abbreviation and a placeholder. He uses 'label' for 'abbreviation,' but no matter. But I distinguish a placeholder from a variable while Hamilton doesn't.

To appreciate the distinction, first note that with respect to variables we ought to make a three-way distinction among the variable, say 'x,' the value, say Socrates, and the substituend, say 'Socrates.' Now consider the argument:

Tom is tall or Tom is fat
Tom is not tall
——-
Tom is fat

This argument has the form of the Disjunctive Syllogism:

P v Q
~P
——-
Q.

Obviously, 'P' and 'Q' are not abbreviations (labels); if they were then the second display would not display an argument form. It would be an abbreviated argument. But it doesn't follow that 'P' and 'Q' are variables. For if they were variables, then they would have both substituends andf values. But while they have substituends, e.g., the sentences 'Tom is tall' and 'Tom is fat,' they don't have values. Why not? Because we are not quantifying over propositions (or statements if you prefer). There are no quantifiers in the form diagram. (This is not to say that one cannot quantify over propositions.)

'Tom' is tall' is one of many possible substituends for 'P.' But 'Tom is tall' is not the value of 'P.' For we are not quantifying over sentences. We are not quantifying over propositions either. So *Tom is tall* is also not a value of 'P.'

My thesis is that placeholders in the propositional calculus are arbitrary propositional constants. Since they are constants, they are not variables. It is a subtle distinction, I'll grant you that, but it seems necessary if we are to think precisely about these matters. But then one man's necessary distinction is another man's hair-splitting.

You also argue that London must wrongly decide that 'if roses are red then roses are red' (RR) is a contingency, because we say it can be seen as having the form 'P–>Q' and in general statements of this form are contingencies. Indeed they are. But we don't so decide. We say this is a special case in which P and Q stand for the same simple sentence, 'roses are red', not different ones. P and Q are therefore either both true or both false and either way the truth function for –> returns true. Hence this special case is tautologous. We disagree that the move from RR to 'P–>Q' must be seen as an abstraction. We retain the information that P and Q stand for specific substatements within RR, which may themselves have internal structure. 'Form' is a device for making such structure explicit.

So you are saying that 'P –> Q' has a special case that is tautologous. But that makes no sense to me if RR has both forms. A sentence (understood to have one definite meaning) is tautologous if its logical form is tautologous, and if RR has the form 'P–> Q' then it it is not tautologous as an instance of that form. So you seem committed to saying that RR is both tautologous and not tautologous.

Isn't that obvious? If one and same sentence (understood to have one definite meaning) has two logical forms, one tautologous and the other non-tautologous, then one and the same sentence is both tautologous and non-tautologous — which is a contradiction.

One solution, as I have suggested several times already, is to say that, while 'P –> P' is a special case of 'P –>Q,' namely the case in which P = Q, the two forms are not both forms of 'If roses are red, then roses are red.' Only one of them is, the first one. The second is a form of the first form, not a form of the English sentence.

Putting the problem as an aporetic hexad:

1. 'P –>P' is a special case of 'P –> Q'
2. If a proposition s instantiates form F, and F is a special case of form G, then s instantiates G.
3. 'P –> P' is a tautologous form.
4. 'P –> Q' is a non-tautologous form.
5. No one proposition instantiates both a tautologous and a non-tautologous form.
6. 'If roses are red, then roses are red' instantiates the form 'P –> P.'

The hexad is inconsistent. Phoenix and London agree on (1), (3), (4), and (6). The Phoenician solution is to reject (2). The Londonian solution is reject (5).

But the Phoenicians have an argument for (5):

7. The logical form of a proposition is not an accidental feature of it but determines the very identity of the proposition.
Ergo
8. If s instantiates form F, then necessarily, s instantiates F.
ergo
5. No one proposition instantiates both a tautologous and a non-tautologous form.