Maverick Philosopher

Now consider a richer example: (P1) We are creatures; (P2) There is no creature without a creator; therefore, (C) A creator exists. This argument begs the question in that it is impossible to know that (P1) is true without knowing that (C) is true. For only if I know that a creator exists can I know that I am a creature. The argument is not probative because it presupposes in (P1) what it needs to prove. (Of course, I am assuming that one is not equivocating on 'creature' and that one is using it in the sense in which it must be used for (P2) to be true; if one is equivocating, then naturally the argument is worthless for this reason.)

Now it might occur to someone to wonder whether logical arguments from evil for the nonexistence of God also beg the question. Is there anything to this notion?

Logical arguments from evil start with a fact, the fact of evil. No doubt evils exist. But for evils to prove the nonexistence of the omni-qualified God of traditional Judaism, Christianity, and Islam, these evils must be gratuitous. A gratuitous evil is one that cannot be reconciled with the existence of the omni-qualified God, for example, an evil that God has no morally sufficient reason to permit, or an evil that God cannot prevent because it is the deed of a free agent. A gratuitous evil, then, is one the existence of which is logically inconsistent with the existence of God conceived in the traditional way.

Granted, evils exist. But only gratuitous evils are inconsistent with the existence of God. Only they pose a problem for the existence of an all-good God. Consider the pain of a just punishment, the pain of imprisonment, say, that convinces a miscreant, an armed robber for example, to reform his life. (Few penal punishments are truly rehabilitating, but certainly some are.) The pain of punishment is evil. But in my example, it is necessary for the achievement of a greater good. Thus the evil in question is not gratuitous.

How do we know that there are gratuitous evils? To know this, we would have to know that God does not exist. Thus we would have to know the conclusion of the LAFE to be true in order to know that its key premise -- Gratuitous evils exist -- is true. But this is the very definition of begging the question.

Therefore, it seems that LAFE begs the question. To appreciate my point, compare:

1. If there are gratuitous evils, then God does not exist.

2. If creation exists, then a creator exists.

Both of these propositions are analytically true. So if one knows that there are gratuitous evils, then one knows that God does not exist. Likewise, if one knows that creation exists, then one one knows that a creator exists. But how does one know either of these things? Suppose I witness a brutal and unprovoked assault on a human being by another human being. Witnessing this, I witness an evil. But by what mark do I recognize the gratuitousness of the evil? By no empirical mark since what I observe is consistent both with the assault's being gratuitously evil and with its being nongratuitously evil.

Something similar holds for (2). How do I know that what I see, mountains and valleys, cacti and sky is creation or created nature as opposed to uncreated nature? By what mark do I know that nature has the status of having been created? By no mark.

Return to the brutal assault. Imagine it to be perpetrated against a loved one. Atheists will naturally take such crimes as 'proof' that there is no God. But it is no more 'proof' than the existence and order of nature is 'proof' of an intelligent creator. Both athetist and theist interpret the data in the light of a worldview that precedes the data and is imposed upon them. Each can fit the data into his scheme. But fitting data into a scheme is not the same as data proving a scheme. Thus the brutal assault fits the atheist's scheme but does not prove it, and the existence and order of nature fits the theist's scheme but does not prove it.

So here is my challenge to Peter and others: explain how LAFE does not beg the question against the theist.

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1. This sentence is false.

If (1) is true, then it is false, and if false, then true. This is an example of an antinomy. In pursuit of a taxonomy, we might call this Grade I of self-referential inconsistency. Grade I, then, is the class of self-referentially inconsistent sentences that issue in antinomies.

There are other self-referential sentences that are not antinomies, but imply their own necessary falsehood. These are such that, if true, then false, and if false, then false, and are therefore necessarily false. For example,

2. All generalizations are false.

If (2) is true, then, since (2) is itself a generalization, (2) is false. But its falsity does not imply its truth. So, if false, then false. Assuming Bivalence, it follows that (2) is necessarily false, whence it follows that its negation — Some generalizations are true — is necessarily true, and moreover an instance of itself. A second example might be

3. There are no truths.

If (3) is true, then it is false. And if false, then false. So, (3) is necessarily false, whence it follows that its negation — There are truths — is necessarily true.

Examples (2) and (3) belong to Grade II in my tentative taxonomy. These are self-referential sentences that entail their own necessary falsehood. Grade III comprises those self-referential sentences that are such that if true, then neither true nor false, and if false, then false. For example,

4. There are no truth-bearers.

If (4) is true, then, since (4) is a truth-bearer, (4) is neither true nor false. But if false, then false. If we define the cognitively meaningful as that which is either true or false, then (4) is either cognitively meaningless or false. A more interesting example that seems to belong in Grade III is the Verifiability Principle of the Logical Positivists:

5. Every cognitively meaningful sentence is either analytic or empirically verifiable in principle.

If (5) is supposed to be cognitively (as opposed to emotionally) meaningful, and thus not a mere linguistic recommendation or pure stipulation, then it applies to itself. So if (5) is true, then (5) — which is clearly neither analytic nor verifiable — is meaningless. So, if true, then meaningless, and if false, then false. Therefore, either meaningless or false. Not good!

Grade IV comprises those self-referential sentences that can be described as self-vitiating (self-weakening) though they are not strictly self-refuting. For example,

6. All truths are relative.

If (6) is true, then, assuming that (6) is intended to be true, it follows that (6) is relative, i.e., relatively true. It is not the case that if (6) is true, then (6) is false. So (6) is not self-refuting. Nevertheless, (6) is self-vitiating in that it relativizes and thus weakens itself: if true, it cannot be absolutely true; it can only be relatively true. Other examples which seem to fall into this category:

7. Every statement is subject to revision. (Quine)
8. Every theory reflects class interests. (Marxism)
9. All theory is ideology. (Marxism)
10. Nothing can be known.
11. Nothing is known.
12. Nothing is certain.
13. All truth is historical.
14. All is opinion.

What is wrong with self-vitiating propositions? What does their weakness consist in? Consider (8). If (8) is true, then the theory that every theory reflects class interests itself reflects class interests. Suppose (8) reflects the class interests of the proletariat. Then what is that to me, who am not a proletarian? What is it to anyone who is not a proletarian? If (8) is true only for you and those with your interests, and your interests are not my interests, then I have been given no reason to modify my views. The trouble with (7)-(14) and their ilk is that they make a claim on our rational attention, on our common rational interest, while undercutting that very claim.

It seems we need a fifth category. The sentences of Grade V are such that, if they are true then they are, not false, but non-assertible. Consider

15. No statement is negative.

(15) applies to itself and so at first appears to refute itself: if (15) is true, then it is false. And if false, then false; hence necessarily false. But consider a possible world W in which God destroys all negative statements and makes it impossible for anyone to make a negative statement. In W, (15) is true, but non-assertible. (15) does not prove itself to be false; it proves itself to be non-assertible.

Can the same said of

16. All is empty (Buddhism)?

I think not, for reasons supplied here.

Finally, we consider

17. All memory reports are deceptive.

This is subject to the retort that one who asserts (17) must rely on memory, and so must presuppose the reliability of the faculty whose reliability he questions by asserting (17). For if anyone is to be in a position to affirm (17), he must remember that on some occasions he has misremembered. He must remember and remember correctly that some of his memories were merely apparent. It seems obvious, then, that the truth of (17) is inconsistent with its being affirmed. If true, it is unaffirmable as true. But this is different from saying that (17), if true, is false. Although (17) is unaffirmable or non-assertible if true, it seems that (17) could be true nonetheless.

Here is a first rough formulation of the problem. A proposition is a whole of parts. Thus the proposition expressed by the declarative sentences 'Sokrates albus est' and 'Socrates is white' is a whole the parts of which are the semantic values of 'Socrates,' 'is,' and 'white.' Now the whole in question has the property of being either true or false, a property that the parts taken distributively do not have. Thus the semantic value of 'Socrates' is neither true nor false. And the same holds for the other constituents of the proposition. It is also the case that the constituents taken collectively, as a collection, is neither true nor false.

Let v('x') denote the semantic value of the expression 'x.' Then my point is that the set {v('Socrates'), v('is'), v('white')} is neither true nor false. No set is either true or false. The same holds for the corresponding mereological sum. No sum is either true or false. And if you balk at the notion of semantic values and think more nominalistically, then the point is that the declarative sentence 'Socrates is white' is toto caelo different from the list: 'Socrates,' 'is,' 'white.' The difference is that, whereas the sentence is either true or false, the list is neither true nor false. No list is either true or false. A proposition/declarative sentence, then, is not a set, sum or list of its constituents.

The problem, then, is to specify what it is about a declarative sentence that distinguishes it from a list, or what it is about a proposition that distinguishes it from a set or sum or its constituents. This amounts to specifying what it is about a declarative sentence or a proposition that gives it the property of being either true or false. Since a proposition is a unity of constituents, the problem is to explain the peculiar unity of a proposition whereby its constituents become fit to be either true or false.

Since a proposition is a whole of parts, but is more than its parts, as their unity, a unity that is not that of a set, sum, or list, one might think to solve the problem of unity by introducing into the proposition a special unifying constituent, one that ties the primary constituents into a unity capable of being either true or false. But this leads to Bradley's regress. Suppose the primary constituents are S and P. The truth-valued whole SP is not identical to {S, P} or to the corresponding sum or list. This for the simple reason that the set can exist without the truth-valued whole existing. So we introduce c to connect S and P. But what makes these three entities into a truth-valued whole? The truth-valued whole ScP is not identical to {S, c, P} or to the corresponding sum or list. This is obvious from the fact that the set can exist withgout the truth-valued whole existing. Something more is needed to 'cement' the constituents S, c, and P together into a truth-valued whole. If a fourth constituent, c*, is introduced then the problem arises all over again inasmuch as there is a difference between the truth-valued whole Scc*P and the set {S, c, c*, P}.

We see that the attempt to solve the unity problem by introducing a unifying constituent leads to a regress that is both infinite and vicious. So how are we to solve the unity problem?

2. The contents of believings have properties that belief-states lack. For one thing, belief-contents are sharable. That the door has been left ajar, that Frege died in 1925, that both 2 and -2 are roots of 4, are contents to which more than one mind has access. But the psychological state that I am in when I believe that the door is ajar is not sharable in the same sense. Second, the belief-contents can be accepted, rejected, entertained, etc., which is not the case for the corresponding believings, unless of course these believings become the objects of higher order beliefs. Third, belief contents are either true or false, which cannot be said in the same sense of believings.

3. There are occurrent beliefs.

Therefore

4. There are propositions.

To understand this argument, one must understand that no particular theory of propositions is being argued for. (It even allows for such wacky theories of propositions as that propositions are sets of possible worlds. That penetrating minds have championed such theories shows that such minds can descend into wackiness under various materialist and extensionalist pressures.) The argument is to the conclusion that something or other must play the roles of truth-bearer, object of such attitudes as knowing and believing, and ground of the possibility of two or more minds' coming to believe or know the same thing. It is an argument for the existence of propositions that leaves open their exact nature. (Analogy to be explored: in the way Aquinas' God-arguments leave open the exact (and in deed unknowable) nature of God.)

Perhaps the argument could be strengthened by restricting it to de dicto as opposed to de re beliefs. Compare 'S believes of some black that he is articulate' with 'S believes that some black is articulate.' The first is de re, the second de dicto. The first does not entail the second. Suppose S is a redneck who believes that no black is articulate. He hears a man on the radio speaking in an articulate manner, a man who, unbeknownst to S is black. It follows that S believes of some black man that he is articulate without believing that some black man is articulate.

Be this as it may. Like any argument, the Metaphysics 101 argument for propositions can be countered in several ways. Alan Rhoda discusses one way in his post, Propositions and Make-Believe. My own view is that the argument is more credible than any of its attempted counterings. More later.

This is correct except for one crucial detail. The 'modern view,' I take it, is the post-Fregean view. On this view propositions are indeed structured entities expressible by tokenings of indicative sentences. But a singular proposition such as 'Socrates is a man' need not have Socrates himself as one of its constituents. It could have some abstract representative of Socrates as its subject-constituent. In a Fregean proposition, it is not Socrates, but the sense of 'Socrates,' that enters as constituent into the proposition, while in a Russellian proposition it is Socrates himself who enters as constituent. Don't get hung up on these labels. If you don't like them, choose others. It is the distinction that counts, and it is not pegged to the actual views of the luminaries in question. (And as you know, these views changed, especially in the case of Russell.)

But Ockham is surely correct that on the traditional approach all propositions (as opposed to sentences) are general, i.e., either particular or universal. I made that point myself more than once. Ockham continues:

Singular propositions do not therefore exist as such in the traditional logic, although logicians after the scholastic period (such as Arnauld) treated them as universal propositions. Aquinas did not regard them, in our sense, as propositions at all. There are singular sentences, such as 'Socrates is wise'. But there is no proposition, in our sense, expressed by this sentence. Aquinas explains that (following Aristotle) the intellect only understands universal concepts such as 'man', 'mortal' and so on, by abstracting from the 'intelligible species' in perception. The intellect can only understand singulars in an indirect way, by what he calls 'turning round to sense-appearances'.

Now I come to my critical point. By failing to note that a singular proposition need not be Russellian, commenter Ockham falls into the error of thinking that there are no singular propositions at all. But this is belied by the plain fact that we reason with them.

Suppose I reason to the general conclusion that Some Athenian is wise from the singular premises, Socrates is wise and Socrates is an Athenian. One cannot reason from sensible marks, although one can reason about them. One reasons from senses. There has to be an intelligible structured entity, call it what you want, before the mind. The intellect must grasp (understand) a sense and this involves more than an eye detecting certain marks on a piece of paper or computer screen. So it is propositions that figure in the argument qua argument, not sentences. And two of these propositions are singular. Again, we reason from truth-bearers, not from physical strings that may or may not express these truth-bearers. These truth-bearers are propositions, and in the specimen of reasoning just given, two of these propositions are singular.

So we need singular propositions to explain the obvious datum I adduced. No doubt one can ask difficult questions about the subject-constituents of Fregean propositions, but these do not overturn the arguments for the postulation of Fregean propositions.

Despite the critical remark, I am inclined to agree with Ockham that propositions cannot be Russellian. When I understand that Marilyn is at the door, it is not Marilyn the concretum, all 130 lbs of her, warts and all, that is in my mind. And the same goes for my massive door. In or before my mind is a proposition all of the constituents of which are abstracta.

And now let me speculate a bit. If we can't speculate in philosophy, where the hell can we speculate? Our minds are finite and ectypal, which is why I can't get the whole of Marilyn into mine. But it might be a different story for an intellectus archtypus. For an infinite archetypal intellect, it may be that the Fregean/Russellian distinction collapses. For God there are perhaps no intermediate sense-structures. Sinn ist Sein und Sein ist Sinn.

Given the goal of truth, how ought I reason so as to attain truth and avoid falsehood? The question is not how I or anyone does reason, except insofar as this is relevant to the question of how we ought to reason. The question is how we ought to reason. This implies that logic is not an empirical discipline. It is not concerned with the empirical facts about how people (or any species of rational being) reasons. Suppose that 75% of a population reasons in accordance with this pattern:

Some Fs are Gs
-----
Some Fs are not Gs.

This fact, if a fact, is an interesting fact of concern to the psychology of reasoning. But it is not a fact that interests the logician qua logician. His concern is with the validity of the argument-pattern. He pronounces the pattern invalid, which does not mean that the pattern is not instantiated in the reasonings of actual people. He would judge the pattern invalid whether everyone or no one reasoned in accordance with it. Suppose there were no people at all. The pattern would still be invalid in the sense that if there were a person, and this person reasoned in accordance with the pattern, then the person would reason incorrectly.

Here is an analogy. Whether everyone kills someone for the fun of it, or no one kills anyone for the fun of it, or there are no people at all, it remains the case that killing people for the fun of it is morally wrong.

And the same goes for valid patterns. Suppose there is a necessarily truth-preserving pattern of reasoning in accordance with which it is possible for beings like us to reason, but in accordance with which no one of us in fact reasons, it would still be the case that the pattern is valid. In this sense, the valid patterns are norms. As such, they cannot be gleaned from the ways our minds in fact operate. For as a matter of fact, our minds operate in the damndest ways.

The normativity of logic is consistent with the fact that logical laws are not prescriptive as to their content. The Law of Non-Contradiction, applied to propositions, states that it is not the case that a proposition and its negation are both true: ~(p & ~p). LNC states a very general fact; it does not prescribe any action. But although LNC is not prescriptive in its content in the way that No felon shall own a firearm is prescriptive, or rather proscriptive, in its content, LNC does imply prescriptions. Thus it implies that one ought not affirm both a proposition and its negation.

This argument-form is technically valid, paradoxical as that might strike you. For a valid argument-form is one no actual or possible instance of which has all true premises and a false conclusion. Validity is the property of being necessarily truth-preserving: inferences and arguments are valid if (and only if) their forms are such that they (the inferences and arguments)never take one from truth to falsehood.

Ockham would have us conclude, however, that "there genuinely does exist a condition under which a [valid] standard form has an instance with TPFC." The condition in the present case is that 'Alexander' does not denote the same individual in both of its occurrences or tokenings. If this condition is satisfied, then it can easily be that both premises are true. 'TPFC' abbreviates 'has true premises and a false conclusion.' So what Ockham is urging is that the argument-form depicted above is only conditionally valid; it is valid, but only on condition that 'a' be taken to denote the same individual in both of its occurrences. The argument-form is not unconditionally valid because it has an instance, the Alexander argument above, that has true premises and a false conclusion.

My thesis is and has been all along that this distinction is bogus: there is no justification for any distinction between conditional and unconditional validity. Argument-forms are not arguments; argument-forms are the primary bearers of the properties of validity and invalidity; and the validity of an argument-form is in no way conditional upon whether terms in natural language are used univocally or not.

I turn aside Ockham's argument by denying what he takes to be "manifestly and undeniably" the case, namely, that the Alexander argument, given his proviso about the reference of 'Alexander,' is of the form depicted above. We need to be clear that 'a' in the form diagram is not an abbreviation for 'Alexander.' It is a place-holder. The distinction between abbreviations and place-holders was explained earlier. Although 'a' is an individual constant, and not an individual variable like 'x' or 'y,' it is an arbitrary individual constant. This means that any name for an individual (as opposed to a name for a property) can be its substituend. But since 'a' is a constant, whichever proper name is substituted for 'a' in one place in the argument must be substituted in every other place.

What's more (and this is crucial), the proper name substituted for 'a' must be used in the same way in each of its occurrences if it is to count as a genuine substituend of 'a.' It is an obvious fact that proper names are in one sense quite common: 'Alexander' can be used and has been used to refer to many different people and animals. No doubt some cats and dogs rejoice under this name. But 'Alexander' is not functioning as a proper name unless it is being used by a person in a context with the intention of referring to exactly one individual, and perhaps to exactly one individual in every possible world in which that individual exists.

Now 'a' is a place-holder for any proper name used as a proper name. It follows that 'Alexander' cannot be substituted for 'a' if it refers to a man in one occurrence and a dog in another, or to two different men.

My point contra Ockham is simply that the Alexander 'argument' does not instantiate the valid form given above. Since it does not, it cannot function as a counterexample to the validity of the form depicted. It cannot be used to show that the form is invalid or that it is only conditionally valid. For the condition Ockham wants to impose ab extra is already bulit into the form diagram. To repeat, 'a' is an arbitrary individual constant. Qua constant, it is a place-holder only for names used univocally.

It should also be obvious that existential generalization on a singular term such as 'a' presupposes that 'a' in all occurrences denotes the same individual. That is, to infer '(Ex)(Fx & ~Fx)' from 'Fa' and '~Fa,' by the inference rule of Existential Generalization, it must be assumed that 'a' denotes the same individual in both of its occurrences. Otherwise, what would follow from the premises would be '(Ex)(Fx) & (Ex)(~Fx)' which, of course, is not equivalent to the previous existentially quantified formula.

All occurrences of the variable 'x' in '(Ex)(Fx & ~Fx)' are bound by, and come under the scope of, one and the same existential quantifier. The binding captures the pronominal back-reference which in English is expressed by 'There is an F and it is not F.' But this would not be the case were 'a' interpreted as denoting distinct individuals.

What say you loyal readers? Have I refuted Ockham? Can we move on now?

This is an argument, not an argument-form. The following abbreviation of the argument is also an argument, not an argument-form:

1. P --> Q
2. Q
---
3. P

Both are arguments; it is just that the second is an abbreviation of the first in which sentences are replaced with upper-case letters and the logical words with symbols from the propositional calculus. But it is easy to confuse the second argument with the following argument-form:

1. p --> q
2. q
---
3. p

An argument-form is a one-over-many: many arguments can have the same form. And the same goes for its constituent propositional forms: each is a one-over-many. 'p --> q' is the form of indefinitely many conditional statements. But an argument, whether spelled out or abbreviated, is a particular, and as such uninstantiable. One cannot substitute different statements for the upper-case 'P' and 'Q' above.

Some of you will call this hair-splitting. But I prefer to think of it as a distinction essential to clear thinking. For suppose you confuse the second two schemata. Then you might think that the original argument, the one in 'canonical English,' is an instance of the formal fallacy of Affirming the Consequent. But the second schema, though it is an instance of the third, is also an instance of a valid argument-form:

(x)(Cgx)
---
(Ex)(Cgx).

In sum, the confusion of abbreviations with place-holders aids and abets the mistake of thinking that an argument that instantiates an invalid form is invalid. Validity and invalidity are asymmetrical: if an argument instantiates a valid form, then it is valid; but if it instantiates an invalid form, then it may or may not be invalid.

But talk of how people ought to reason, talk of what they ought to do and leave undone in matters of argument, presupposes that at least some belief formation comes under voluntary control. No one will contend that all our beliefs are subject to the control of the will, but if none of them are, then there is no possibility of reasoning, strictly speaking. And if there is no possibility of reasoning, strictly speaking, then there is no possibility of that normative study of reasoning called logic, which of course itself involves reasoning, namely, reasoning about correct and incorrect reasoning. Peirce:

Now, that which is uncontrollable is not subject to any normative laws at all; that is, it is neither good nor bad; it neither subserves an end nor fails to do so. But it is only the deliberate adoption of a belief in consequence of the admitted truth of some other proposition which is, properly speaking, reasoning. (p. 120)

If I see that p is true, and understand that p entails q, then I am logically required to accept the truth of q. This logical requirement, however, is not to be confused with any kind of natural necessity: though I am logically required to accept q, I am yet free to reject q. People do contradict themselves, and sometimes willfully: they will not accept the truth that they are logically required to accept given their other doxastic acceptances.

This suggests that our being rational beings is incompatible with our being deterministic systems. But to develop this properly would take a lot of scribbling. Instead, I'll just hand the ball to Victor Reppert and quit the field.

He seems to be saying that there are argument-forms that are valid but are not unconditionally valid. Well, consider modus ponens in the propositional calculus:

p --> q
p
---
q.

This argument-form is surely above reproach in point of validity.

Could this form be valid without being unconditionally valid? From what O. is saying above, it would be valid, but not unconditionally, if there were no proviso to the effect that the same indicative sentence be substituted for the place-holder 'p' in both of its occurrences and the same indicative sentence be substituted for 'q' in both of its occurrences. Without this proviso, the following could count as an instance of modus ponens:

If Wilson drowned, then Wilson is dead
Phil Spector lives in Alhambra, California
-----
Wilson is alive.

But surely two things are true: (i) this argument is not valid, and (ii) it is not an instance of modus ponens. There simply is no distinction, apart from a merely verbal distinction, between 'valid' and 'unconditionally valid.' Every valid argument-form is unconditionally valid, and conversely. For Ockham to justify his distinction, he must give us an example of an argument-form that is valid, but not unconditionally valid, or vice versa. I don't see how this could be done. But I am open to refutation.

I believe I can show how Ockham is going wrong. This debate began with the Pittacus argument:

Pittacus is a good man
Pittacus is a wise man
-----
Some wise man is a good man.

I said that this argument is valid, and O. objected that it is not "unconditionally valid." But surely the argument is valid and unconditionally so. The following is a crucial principle:

An argument is valid iff there is a valid argument-form that it instantiates.

To understand the bolded principle you must keep straight the distinction between an argument and an argument-form, and you must understand that a valid argument is valid in virtue of the validity of its logical form. It is argument-forms, not arguments, that are the primary bearers of the properties of validity and invalidity. An argument is said to be valid only insofar as it instantiates a valid argument-form. I hate to be pedantic, but just to make sure that we are on the same page, I will now make a list of some obvious corollaries:

No argument-form is an argument
Every argument has one or more forms
An argument can be valid even if it has an invalid form
Every argument-form is either valid or invalid
No argument-form is sound.

Now does the Pittacus argument instantiate a valid form? Of course. It instantiates this form:

Fa
Ga
---
(Ex)(Fx & Gx)

This form is provably valid within MPL. Therefore, the Pittacus argument is valid. It is valid because it instantiates a valid argument-form.

Now Ockham's reason for saying that the Pittacus argument is not unconditionally valid is presumably that 'Pittacus' might refer to different things in its two occurrences. Well, suppose it does. Then it is a different argument with a different form, the following invalid form:

Fa
Gb
---
(Ex)(Fx & Gx).

But none of this gives us any reason to distinguish between validity and unconditional validity. The original Pittacus argument is valid, as I showed, and unconditionally so. The second Pittacus argument, if you want to call it an argument at all, is invalid, and unconditionally so. That's all there is to it.

Larry: I am happy
Moe: I am not happy
-----
Curly Joe: Someone is both happy and not happy.

Now I deny that this is an argument for reasons already given. But even if it is an argument, call it the 'Stooge argument,' it certainly does not have the logical form that Ockham ascribes to it, namely:

Ha
~Ha
---
(Ex)(Ha & ~Ha)

This form is clearly valid: no argument of this form is such that the premises are all true and the conclusion false. (Obviously, the premises cannot both be true). So the argument-form is technically valid, though of course no argument of this form is sound. (A is sound =_df A has a valid form and true premises.)

Ockham reasons as follows: Since the Stooge argument has true premises and a false conclusion, the Stooge argument is invalid; but the Stooge argument instantiates the form just depicted; ergo, the form just depicted is invalid!

The Ockhamite argument is bad in my humble opinion, resting as it does on what I take to be a false presupposition, namely, that arguments are composed of sentences rather than propositions. The proposition that Larry expresses when he tokens 'I am happy' is or is equivalent to the proposition Larry is happy, while the proposition Moe expresses is or is equivalent to the proposition, Moe is not happy. You will have noticed that while 'I am happy' and 'I am not happy' appear to be contradictories, the corresponding Larry and Moe propositions are not contradictories, but are indeed logically independent.

Therefore, the form of the Stooge argument is:

Ha
~Hb
---
(Ex)(Ha & ~Ha)

which is invalid. So I consider myself to have won Round One of the debate. Round Two begins with an Ockhamite attack on the very idea of a proposition as an entity distinct from a sentence. What follows in italics is by Ockham:

(1) Propositions must be complex entities. Otherwise we could not use an existing, finite stock of words to express propositions that have never before been expressed.

Agreed. Barry Miller thinks that there are logically simple propositions, but I criticized him in print for this.

(2) There must be a part of a proposition corresponding to the proper name. For example, the sentence 'Etna is higher than Vesuvius', contains the name 'Etna', which occurs also in other sentences, e.g., in the sentence 'Etna is in Sicily'. There must therefore be something similar in the two corresponding propositions, that corresponds to the word 'Etna'. (But not in the propositions corresponding to 'Alexander did not seize Helen' and 'Alexander seized Helen', for reasons we agree on, of course).

Agreed.

(3) This cannot be something material and observable. Otherwise, the thought corresponding to 'Etna is not higher than Mont Blanc' would contain two large mountains, thus every piece of frozen, solidified lava in one, and every snowfield in the other.

This is true for Fregean propositions, though not true for Russellian propositions. But let's stick to Fregean propositions. I agree that the subject-constituents of such propositions cannot be physical objects. The propositions are not physical items and their parts cannot be physical either.

(4) But if not material or observable, how can we speak of propositions having the same 'form'? If two numerically different things possess the same form, it is because their shape is observable to the senses. In what sense can two unobservable things possess the same form?

You were doing so well up to now! But now you blunder by assuming, falsely in my humble opinion, that 'form' as when we speak of the 'logical form' of an argument or of a proposition must mean something like physical pattern or shape.

I of course reject that. And unless you have some extremely powerful arguments, you will not budge me from my view. The propositions Larry is taller than Moe and Moe is taller than Joe have the same logical form which might be depicted Rab or aRb. What's the problem? Each stooge is a physical particular, and the relation of being taller than is physically measurable. But the propositions are not physical, nor is their common form physical.

(5) Further, how does a propositional name represent the object that the corresponding spoken or written name is a name of? Is this representation imposed by convention? But if it is, we have an infinite regress. For I will use the 'Alexander argument' again. We could change the meaning of the propositional name 'Alexander' to signify Alexander of Troy instead of Alexander of Macedonia, and then the propositions 'Alexander did not seize Helen' and 'Alexander seized Helen' would contain the same propositional name, but both propositions true. If you say that the propositions signify other propositions, then I use the argument again, and so on.

You will have to spell this out for me to understand exactly what you mean. But I agree that one can legitimately ask how the subject-constituent of a Fregean proposition represents what it represents. Similarly, what makes my thought of you right now a thought of you? These questions about intentionality are interesting and important but I don't have to answer them to justify my introduction of propositions.

(6) But if the relation is not by convention, we have the absurdity that some abstract entity immutably and eternally represents Alexander, even before Alexander existed. Indeed, even if Alexander had never existed, the sentence 'Alexander seized Helen' expresses a proposition containing something eternally representing him. That's absurd.

Plantinga maintains a theory like this, so I would hesitate to say that it is absurd. Why can't proper names have senses? Why can't senses be Platonica? Why can't they exist uninstantiated? I am not endorsing all this, but where is the absurdity, i.e., the logical contradiction? Or are you using 'absurd' only in an emotive way?

(7) For similar reasons you previously rejected the idea of haecceities. Doesn't your 'solution' of my problem involve this very idea?

Interesting objection. I have indeed rejected nonqualitative haecceities. I find it hard to swallow the idea that before I was born there was a necessarily existent property, the property of being identical to BV, a property that can exist uninstantiated. So the proposition BV is happy cannot involve such an haecceity-property. Perhaps the proposition involves a qualitative haecceity. But this is a large topic.

The bottom line for me is that the arguments for the introduction of propositions (not necessarily Fregean propositions) are so strong, that the kinds of questions you raise do not amount to a good reason to dismiss them.

Alexander lived in Troy (where the expression 'Alexander' refers to Alexander of Troy)
Alexander did not live in Troy (where the expression 'Alexander' refers to Alexander of Macedonia)
-----
Someone lived in Troy and did not live in Troy.

'TPFC' abbreviates 'true premises false conclusion.' Ockham now argues as follows:

1. Any form of any TPFC argument is invalid
2. The Alexander argument is TPFC
3. Any form of which the Alexander argument is an instance, is invalid
4. If the Alexander argument is an instance of the form 'Fa, Ga, therefore (Ex)(Fx & Gx)', that form is invalid
5. The Alexander argument is an instance of the form 'Fa, Ga, therefore (Ex)(Fx & Gx)'
6. The form 'Fa, Ga, therefore (Ex)(Fx & Gx)' is invalid.

Ockham adds this remark:

Again, the argument is perfectly valid. Well, actually it's not, if it is sound, because its validity depends on the validity of forms, which the very same argument proves to be invalid. But leave that for now.

THE UNSOUNDNESS OF THE OCKHAMITE ARGUMENT

Premise (1) of the Ockhamite argument is of course true. If an argument has true premises and a false conclusion, then no form of such an argument is valid. But I see no reason to accept the truth of premise (2). And this for the simple reason that the 'Alexander argument' is not an argument!

What is an argument anyway? The word has of course several meanings. In logic it has two main meanings. Consider the propositional function 'x is wise.' This function has the value True for Socrates as argument. But this is not the presently relevant sense of 'argument.' An argument in the present context is a sequence of statements together with the claim that one of them, the conclusion, follows from the others, the premises. To argue is to adduce reasons in order to support, justify, prove or demonstrate a certain proposition. The reasons adduced are the premises of the argument and the proposition which they are supposed to support is the conclusion.

Now given this understanding of what an argument is, the 'Alexander argument' is not an argument. When a person gives an argument, he intends that all occurrences (tokens) of a proper name such as 'Zeno' should have the same reference. Thus if one argues, 'Philosophers respect reason; Zeno is a philosopher; ergo, Zeno respects reason,' the arguer is charitably understood to be referring or trying to refer to the same individual during the time in which the argument unfolds. But in the 'Alexander argument' the proper name 'Alexander' does not refer to the same individual in both of its occurrences. This 'argument' (sneer quotes!) is therefore a sequence of statements that does not amount to an argument. Obviously, not every sequence of statements is an argument.

Although (3) follows from (1) and (2), the falsity of (2) renders (3) false. (UPDATE 29 April: the falsity of (2) does not render (3) false. See retraction here

(4) may be granted, but (5) should not be granted. The 'Alexander argument' is not an instance of the argument-form depicted above, it is an instance of the following argument-forms:

Fa
Gb
---
(Ex)(Fx & Gx)

or better

Fa
~Fb
---
(Ex)(Fx & Gx)

both of which are obviously invalid.

Since (5) is false, the conclusion (6) is also false. I have just given two reasons why the Ockhamite argument is unsound: (2) is false; (5) is false. But as Ockham concedes, his argument is invalid! So this is a third reason to deem his argument unsound.

THE INVALIDITY OF THE OCKHAMITE ARGUMENT

The Ockhamite argument proves entirely too much: it proves among other things that modus ponens is invalid. Consider

If Zeno is breathing, then Zeno is alive
Zeno is breathing
-----
Zeno is alive.

Suppose 'Zeno' refers to my cat in its first three occurrences, and to the largest rock in my backyard in its fourth and final occurrence. Then the premises are true and the conclusion is false. By the Ockhamite reasoning, modus ponens is invalid since the above argument has the form of modus ponens and the above argument has true premises and a false conclusion.

Now if this isn't sophistry, what would be? And the sophistry speards since the 'argument' can be iterated for any argument-form including the argument-form instantiated by the Ockhamite argument. So the ockhamite argument is hoist by oys own petard: if sound, then unsound because invalid.

DIAGNOSIS

As explained in earlier posts in this chain, Ockham's troubles spring from a failure to distinguish between indicative sentences and the propositions they are used to express or the statements they are used to make. Some such distinction is simply unavoidable, and has the status of a datum denial of which leads to absurdity.

One who makes this unavoidable distinction sees immediately that the 'Alexander argument' does not have the logical form that Ockham thinks it has.

Ockham will demand to know what a proposition is. For present purposes, however, all we need to know is the self-evident point that propositions cannot be identified with indicative sentences. 'I am happy' said (assertively uttered) by me and said by Joe Blow express two different propositions. But it is one and the same sentence. Can anyone in his right mind deny that?

The definition just given allows for arguments that have two or more forms, one of which is invalid. Suppose argument A instantiates two forms, one valid the other invalid. The definition implies that A is valid.

Note the asymmetry between validity and invalidity. To be valid, it is sufficient that an argument instantiate a valid argument-form. To be invalid, however, it is not sufficient that an argument instantiate an invalid argument-form. For an argument that instantiates an invalid form may also instantiate a valid form. I gave examples of this in earlier posts in this chain.

Valid Form: None of its instances have true premises and a false conclusion.
Invalid Form: Some of its instances have true premises and a false conclusion, and some do not.

This suggests that there is room in logical theory for the concept of contravalidity:

Contravalid Form: All of its instances have true premises and a false conclusion.

Note that what we have here is a tripartition in the sense that the set of argument-forms is divided or partitioned into three disjoint and mutually exhaustive proper subsets, the valid, the invalid, and the contravalid.

According to David H. Sanford (If P, then Q: Conditionals and the Founbdations of Reasoning, Routledge 1989, p. 42), the above tripartition and the concept of contravalidity is traceable to Johannes de Celaya. But Professor Sanford says something that strikes me as incorrect. He writes, "All contravalid arguments are invalid, but not all invalid arguments are contravalid." (p. 42) But this can't be right given that the Celaya classification is a tripartition. If an argument is contravalid, then ALL instances of its form have true premises and a false conclusion. But if an argument is invalid, then SOME instances of its form have true premises and false conclusion and SOME do not. So if an argument is contravalid it is not invalid. Sanford makes it sound as if contravalid arguments are a proper subset of invalid arguments when in fact they are neither valid nor invalid. But it is easy to fall into confusion here since invalidity and contravalidity are usually not distinguished. What Sanford should have said is that all contravalid arguments are not-valid, the not-valid comprising both the invalid and the valid.

Here is an example of a contravalid argument-form:

If p, then either p or not p
-----
p if and only if not p.

Since the premise is a tautology and the conclusion a contradiction, every instance of this schema has a true premise and false conclusion. From a practical point of view, contravalidity is uninteresting -- which is why it is not discussed in logic textbooks. But it is theoretically interesting because its symmetry with validity throws into relief the asymmetry of validity and invalidity. Consider this argument:

If an argument has a valid form, then it is valid
-----
If an argument has an invalid form, then it is invalid.

This is a non sequitur, and the conclusion is false to boot. Someone who thinks that the conclusion is true can be said to be confusing invalidity with contravalidity. For if an argument has a contravalid form, then it is contravalid. So it does seem useful for purposes of logical theory to distinguish the valid, the invalid, and the contravalid.