Disarming Stove's Paradox: In Defence of Formal Logic
R. Rodrigo Soberano
The thesis of formal logic can be stated very simply thus: The validity or invalidity of an argument rests on the form or structure of the argument rather than on its content. It is the form, more precisely the specific form of an argument which makes it valid or makes it invalid.
It has come however that in the philosophical scene this what used to be a very respectable thesis is now an object of vicious attack. Author D. C. Stove for example has in his book (1) one chapter entitled 'The Myth of Formal Logic'. In the said chapter Stove presents specimen arguments to debunk the formalist creed: The specimen arguments are such that expectedly formal logic true of course to the formalist creed would judge them valid but are, as per Stove at least, actually invalid or nonetheless problematic.
One of these specimen arguments comprises what I call Stove's paradox. (2) This paradox constitutes, I think, Stove's heaviest specific strike against formal logic. Accordingly, in defense of formal logic, the task I set to do in this paper is to neutralize, dissolve, disarm Stove's paradox.
REALIZING STOVE'S PARADOX
Allow me to start by of course presenting Stove's paradox. Let us read the pertinent passage from Stove:
If (d) is invalid then our syllogistic rule is false straight off. If (d) is valid, then its conclusion is false, and so one of its premisses must be false. Then the problem is to find the false premiss. The first premiss is true. So is the second part (we are supposing) of the second premiss. The falsity must therefore be in the first part of the second premiss: but where? Indeed, since the conclusion, if false, is necessarily false, and since the first premiss is necessarily true,and the second part of the second premiss is necessarily true: please find the necessary falsity which is asserted by the first part of the second premiss, (the part which says both premisses are true). (3)
Stove is explicit that he intends argument (d) to be not simply a counterexample but even a counterexample-or-paradox. That a paradox is somehow generated is undeniable and in fact we will heighten the sense of this paradox as we proceed with our analysis. In the meantime we must realize just in what precise sense Stove's example is a counterexample to the claim of formal logic.
The intended counterexample is, of course, at first glance at least seemingly an argument of the form formal logic claims always yields a valid argument. Accordingly, formal logic must claim the argument to be valid, so to say, by its form alone.
But consider the content of the argument. Consider just what the different sentences in the argument are saying given the values of x, F and G. Most remarkable among these is what the conclusion is asserting: the argument is invalid flatly in contradiction to the claim of formal logic!
How about the premises? The first premise asserts something anyone who knows the basic notion of invalidity in logic would not question. Among the different sentences in the argument the first premise is the least subject to question.
How about the second premise? Next to the conclusion, what the second premise says is also remarkable: the argument is one with true premises and a false conclusion. This together with the first premise enables the drawing of the conclusion seemingly validly that the argument is invalid. But the second premise also says such conclusion is false. The paradox must by now have started to smell.
The point of Stove in the present case seemingly is this: If one does not content oneself with just the form of the argument to say the argument is valid as formal logic is wont to do, if one does consider, so to say, the content of the argument one surely realizes its being a problematic and even a paradoxical case. The content of the argument is surely such that the claim, as Stove would have it, of formal logic that the argument is valid is put in bad light.
Let us now fully realize the paradox in the argument. Present the argument again:
Just where is or in what consist the paradox? The first premise is analytically true. What it says is not sensically subject to question whether in or out of the argument. It is one formal logicians and even opponents of formal logic would agree with.
So we go to the second premise. The second premise is a sentence whose subject term makes reference to the very argument wherein the sentence is a premise, to wit, argument (d). But how does one know this? One knows this from the ominous '(d)' one very first comes across as one scans the presented argument.
By way of making reference to argument (d), the second premise tells of itself that it (the second premise) is true.
Even as the second premise tells of itself that it is true (and the first premise also true) (premise 1 is true anyway regardless of what premise 2 says of premise 1) premise 2, also by way of making reference to argument (d), tells of the conclusion that the conclusion is false.
Now, (d) is, apparently, formally valid. Premise 2 in it tells that the premises are true. Hence, the conclusion must also be true.
So the situation is this: premise 2 asserts that the conclusion is false, but premise 2 together with the first premise also entails (implies) that the conclusion is true. Letting 'A' stand for the first premise, 'B' stand for the second premise, and 'C' for the conclusion, the situation can be depicted in symbolic terms thus:
Any good enough student of symbolic logic knows the system, the conjunction, of the four well-formed formulae makes for a contradiction. Hence, the paradox.
DISARMING STOVE'S PARADOX
So what now? Shall we melt before the paradox? Stove himself says that the present paradox is a relative of the Liar. (Perhaps the paradox is a liar?) The Liar paradox is relative to more technical paradoxes like the Russell's paradox and the paradox of the greatest cardinal number. Way before the present paradox, Russell has worked on these other paradoxes, believes that they have a common feature, and proposes as solution his famous theory of types to these paradoxes paradoxes including the present one insofar as such solution perhaps applies. This is not to say that the Russellian solution is inadequate. Rather I just would like to do my own part in exposing the nature specifically of the present paradox and how it may be neutralized or resolved.
The argument form being attacked in Stove's paradox is the following:. It is not my concern here however to make an exposition of the Russellian solution to these
A straightforward example of an argument of this form is none other than the logician's favorite:
In this paradigm example, x = Socrates, F = human, and G = mortal. Everything is clean in this example. Not a whiff of a paradox.
Now let x = (d), F = argument with true premises and false conclusion, and G = invalid. And we have:
Does the above have semblance with Stove's counterexample? Answer: Yes. Does the above completely depict Stove's counterexample? Answer: No. For compare the above with:
There is an essential difference between the above two arguments where the former is not paradoxical whereas the latter is paradoxical. The difference is signified by the ominous '(d)' one very first comes across in scanning the latter. This ominous '(d)' is of course absent in the former.
Pointing out the significance of the ominous '(d)' should not be actually very surprising at this juncture. Afterall this ominous '(d)' has already been referred to in the analysis of the paradox in the counterexample. But it serves to emphasize the presence of this ominous '(d)' for much as the paradox is not realized without this ominous '(d)' the neutralization of the paradox will turn on the question of the legitimacy of the presence of this ominous '(d)'.
I proceed to attack Stove's counterexample by considering the genesis, the coming to be, the coming to existence, of this alleged counterexample. As I may put it, there is both literally and figuratively a sleight of hand involved in the generation of this alleged counterexample. In other words, this counterexample owes its being to a trick. The trick is not obvious at all. It is what I proceed to expose.
To expose the trick, let us start by drawing certain relevant point-to-point comparisons between the paradigm case about Socrates and Stove's counterexample about (d). Afterall, as Stove would have it, the counterexample is supposed to be of the same form as the paradigm case.
About the first premises of both arguments, there are no problems. But compare the second premises. In the sentence 'Socrates is human' the word 'Socrates' is unquestionably a proper name. And, as may be reiterated, a proper name is genuinely a name, is meaningful as a name, only if it is the name of something, that is, only if it successfully denotes or refers to something. It is otherwise meaningless (For a Russellian at least). And we take the word 'Socrates' as meaningful and not as meaningless mark or sound because we take ourselves in principle at least to be capable of pointing to the historical Socrates. To elaborate, we take the word 'Socrates' as meaningful and not as a meaningless mark or sound because we are quite sure it is referentially meaningful. And we are quite sure it is referentially meaningful because we are quite sure that, so to say, Socrates really existed, i.e. it is an undisputed historical fact that there really was someone who fit such definite descriptions as 'the master of Plato', 'the philosopher who drank the hemlock', 'the philosopher whom logicians assert to be mortal', etc. (4)
Consider now the sentence '(d) is an argument with true premises and false conclusion.' Just like 'Socrates' is a proper name, '(d)' here is supposed to be a proper name, to wit, the name of the something claimed to be an argument with true premises and false conclusion. Without this something '(d)' refers to it is just a meaningless mark.
I go now for the jugular vein. The sentence 'Socrates is human' is meaningful within or without the argument in which it occurs as a second premise. How about the sentence '(d) is an argument with true premises and false conclusion' as it occurs as a second premise in Stove's attempted counterexample? Is it meaningful within or without the argument in which it is portrayed as the second premise?
Since '(d)' as we have said is intended to refer to the argument in which the sentence '(d) is an argument ...' is a premise, '(d)' fails to refer to anything without this argument, and so is meaningless without this argument. The special case of '(d)' here is that its meaningfulness rests on its being able to refer to the whole (the argument) in which it is a mere part. (A Fregean may claim to find sense in '(d)' but I do not and I doubt very much whether the reader does.)
There is somehow a difficulty in thinking of a mere part of a linguistic expression being able to refer to the whole of the linguistic expression. How can that be? If someone utters the sentence 'This is false', the word 'This' here is a pure demonstrative. And a successful or meaningful use of a demonstrative is achieved only if it actually denotes or refers to something. A pure demonstrative must be meaningful whether in or out of the larger linguistic expression in which it occurs.
In the sentence 'Socrates is human' the proper name 'Socrates' is meaningful independently of the whole sentence 'Socrates is human'. Consider in turn the sentence 'This is false'. The word 'This' here is a demonstrative intended to point or refer to something just like a genuine proper name is supposed to do. Normal construal of the sentence 'This is false' is that the word 'This' refers to something even without the sentence, and so is meaningful as pure demonstrative even outside of the sentence. The part does not refer to the whole in which it is mere part. Similarly if '(d)' as a proper name be capable of normal construal '(d)' as a proper name must be meaningful independently of the argument in which it occurs as subject term of one of the argument's premises. This surely is not the case in Stove's counterexample.
Consider how one would normally and meaningfully use the sentence 'This is false'. In writing or uttering the sentence one starts with the word 'This'. But even before one can write or utter another word, anyone may interpose, and legitimately, the question 'What this? What this are you talking about?', so that it may be pointed out to such person the referent of the word 'This'. So to say, the word 'This' must be meaningful even as the whole sentence in which it is a mere part is not yet completed.
But grant that no such interposing occurs, so that one is able to complete the whole sentence 'This is false'. Suppose that it is only now that the question 'What this?' is asked, and one answers, 'The very sentence I have previously uttered.' If the reader follows all of what have been said so far, the reader must of course understand what the case is at this point: The term 'This' in the sentence 'This is false' refers to the whole sentence 'This is false' in which it is a mere subject term. The part has come to refer to the whole. The part can no longer be meaningful outside of the whole. (And it does not take much to realize further that a full blown paradox, much like the liar paradox, has thereby attained being.)
The case of the sentence 'This is false' in which the 'This' is used to refer to the whole sentence is obviously a lower level analog of the argument (d) where '(d)' is used to refer to the entire argument. In both cases, the paradoxical situation results from the fact that the part is made to refer to the whole.
But if the part is meaningful only by its referring to the whole, how could the part have been meaningful when there was as yet no whole to refer to as the whole is still in the process of construction? How could the word 'This' have been meaningful when the sentence 'This is false' was not yet complete? How could the symbol '(d)' have been meaningful when the argument (d) was not yet there? The existence of a whole linguistic expression cannot precede the existence of the parts of that linguistic expression. An argument's existence cannot precede the existence of its conclusion and premises. The existence of the part must have preceded the existence of the whole. But if the part's existence preceded the whole's existence and it is insisted also that the part's meaningfulness is in its referring to the whole how could the part have had meaningful existence when there was as yet no whole to speak of?
It is now time to give an account of how Stove's counterexample must have been generated into existence. And thereby be more definite as to what sleight of hand or trick must have been involved in the process.
The construction of the counterexample must of course have started with setting down the premises. Afterall in a genuine argument backed by an actual inference the premises must precede the conclusion insofar as the conclusion is really drawn from the premises. There is no such thing as conclusion without premises. And there is also no such thing as an argument without premises and conclusion.
More detailedly, the counterexample must have started its genesis with the filling in by the constructor the places of 'F', 'G', and 'x' in the argument form
Afterall it is this argument form and formal logic's claim of it that it always yields a valid argument for any x, any F, and any G that is the focus of attack in the present case. Note that in this argument form there is no place for the ominous '(d)'. The argument form in no way demands the presence of the ominous '(d)'. The places to be filled in are simply those of 'F', 'G', and 'x'. The construction of the counterexample to this argument form must have started with, for it must start with, filling in the places of 'F', 'G', and 'x', specifically the place of 'F', 'G', and 'x' in the premise sentence forms.
Trouble starts of course when the place of 'x' in the sentence form 'x is F' is filled in by '(d)' to generate the second premise for '(d)' is then supposed to be a proper name, and not only a proper name in the ordinary sense of 'proper name' but a pure demonstrative: For it to be meaningful it must be the name of something. But what is it name to? Surely not an argument which does not exist. We may drive home the point by interposing at this crucial juncture the question: What (d) is being talked about? What (d) is being claimed to be an argument with true premises and false conclusion? The constructor then either is or is not able to pick out the something he calls '(d)'. If he picks out the something he calls '(d)' we can nevertheless be sure it is not what the ominous '(d)' in Stove's counterexample refers to for the obvious reason that this does not yet exist. The '(d)' then in the second premise points to some other and once so assigned cannot on later occasion be assigned to point to something else on pain of equivocation. But if the constructor is not able to pick out what he calls '(d)' as would surely be the case if '(d)' is to be the ominous '(d)' in Stove's counterexample, then there is a hiatus of meaningfulness and the constructor may proceed only on pain of using '(d)' as just a meaningless mark. Equivocation if not spotted at the proper time may function as a trick to serve a purpose. On the other hand the use of what is actually a meaningless mark in discourse, if not discerned on time, may comprise a trick to serve a purpose. To sum up, the generation of Stove's counterexample must have taken either the route of equivocation undetected or the route of undetected hiatus of meaningfulness. In either case the constructor's proceedings must smell of cheating. Stove's counterexample-or-paradox is, as should be realized by now, something that could not have been legitimately constructed.
(1) D. C. Stove, The Rationality Of Induction (Clarendon Press, Oxford, 1986).
(2) Stove, op. cit., 125. Stove here actually acknowledges debt to a certain Mr.Peter Kintominas for the specimen argument comprising what I call Stove's paradox. Thus, perhaps a better name for the paradox is 'Kintominas-Stove paradox'. Stove's role is of course that of valiantly commissioning the paradox in the attack against formal logic.
(4) Here the problem solved by Russell in his celebrated theory of definite descriptions does not quite occur. For the problem of Russell there was that of the meaningfulness of sentences containing definite descriptions whose referential meaningfulness is strongly disputed or strongly disputable. In the end Russell was yet able to uphold the referential theory of meaning. But how he accomplished the feat need not so much concern us here. For in a clear and quite direct way the name 'Socrates' and allied descriptions are referentially meaningful and their being referentially meaningful is not something disputed.