This post is inspired by this discussion.
Is the following sentence true?
(1) If the borogoves were mimsy, then the borogoves were mimsy or green.
The following is, after all, true:
(2) If the kings of Antarctica were spherical, then the kings of Antarctica were spherical or green.
It seems like (1) unproblematically expresses an analytic truth. But of course it's not so simple. In order for (1) to express an analytic truth in the obvious way that it seems to, both occurrences of "borogoves", as well as of "mimsy", must have the same meaning. But "borogoves" and "mimsy" have no meaning, and hence in particular the multiple tokenings do not have the same meaning, and so (1) is not guaranteed. Unlike (2), which is unproblematically true, whether we read it as material or subjunctive.
So what?
Well, here is a puzzle. Take a bunch of ontological terms of art: "substance", "trope", "accident", "mode", "property", "universal", "relation", "essence", "form", "participation" and "bundle". These terms figure in different theories, some ancient and some modern. It is plausible that if one of these theories is false, then it is not only contingently false, but necessarily so. Moreover, it seems likely that if one of these theories is false, then the terms of art from it not only lack reference, but are actually nonsense. But if this is right, then how can we argue against one of these theories?
The typical way is by reductio: we assume the theory and derive a contradiction. Yes, but derive how? Obviously: logically. Yes, but how can we apply logic to nonsense? We get exactly the problem we saw in (1). It seems, thus, that if our argument against the theory succeeds, it cuts off the branch it was sitting on. And why should our opponent listen to an argument that, according to its own conclusion, makes no logical sense?
Maybe we can reason conditionally. If the words "borogoves" and "mimsy" had meaning, and if they were used univocally, then sentence (1) would be true. If so, then when we engage in a reductio of a theory that we think will ultimately be non-sense, we are really making a semantic statement. If theory T is true, then terms A, B and C have meaning. But if they have meaning, then theory T entails a contradiction. Hence, theory T is not true.
But getting the logic of this reductio right is a difficult affair, I think. Consider the first part, viz., the claim that if the theory were true, then certain words would have meaning. Where do we get that claim? From the theory itself? Typically not. Consider Platonism and its technical terms, "Form" and "participation". Platonism is a set of statements about Forms and participation. It is not a set of statements about the words "Form" and "participation". It is false to say that Platonism says that the words "Form" and "participation" (in the technical sense) make sense. Perhaps the most obvious way to see that it is false is to note that in Plato's time, the words "Form" and "participation" didn't make sense because there was no English language back then. Could we say that Platonism says that "eidos" (in the technical sense) makes sense? No, for Platonism would not have been a different theory had it been developed by people who spoke Hittite instead of Greek, but "eidos" (in the technical sense) would not have made sense.
A more complicated way of looking at this is that in the reductio, we look not at a theory considered as a set of propositions, but at a set of texts, or maybe of mental acts, and we are constructing an argument that if these texts follow the standard grammar of our language, then they contradict themselves and hence are false. But, the argument continues, these texts cannot merely be false--they can only be true or nonsense; so they must be nonsense. I think this kind of works if one is careful.
Our old friend the reductio is a complex beast.
16 comments:
It seems like (1) unproblematically expresses an analytic truth.
That's not analytic, Alex. If anything, it is a logical truth. It's truth is guaranteed by the structure of the proposition, not by the meaning of the constituent terms. You'll recall that Quine suggests at one point in Two Dogmas that we might reduce analyticity to logical truth, but finds the suggestion (ultimately) circular. I so no trouble in agreeing that it's a logical truth, since the nonsense terms are just placeholders in that case; compare (Fa v ~Fa) where neither 'F' nor 'a' are meaningful.
1. I doubt that a sentence expresses a proposition if its constituent terms are meaningless. Does "All borogoves are borogoves" express a proposition? If so, let p1 be that proposition. Since English isn't special, likewise "Tous les borogoves sont borogoves" expresses a proposition, call it p2. Now we have p1 and p2. But if so then there ought to be a fact of the matter whether p1=p2. However, there isn't, since that fact would require a meaning to be assigned to "borogoves" in English and "borogoves" in French to ensure that they're not false friends.
2. I think logical truths are a subset of analytic truths. But if that bothers one, one can instead just modify (1) to be (1*) "Green objects are extended and if the borogoves were mimsy, then the borogoves were mimsy or red objects are extended." (1*), if true, is true in virtue of the meanings of "green" and "red". If you don't think it's analytic that colored objects are extended, it's easy to come up with other examples.
3. I don't think "Fa or ~Fa" expresses a truth if we haven't assigned meaning to "F" and "a". At most we can say that it is a template for truths (and only if we posit that in the template things must be substituted non-equivocally).
I think logical truths are a subset of analytic truths. But if that bothers one, one can instead just modify (1) to be (1*) "Green objects are extended and if the borogoves were mimsy, then the borogoves were mimsy or red objects are extended." (1*), if true, is true in virtue of the meanings of "green" and "red".
The sentence then has an analytic claims as a first conjunct and a logical truth as a second. Still no problem that I can see.
Does "All borogoves are borogoves" express a proposition?
Yes. I take propositions to be sets of worlds. This just expresses the set of all worlds. We can argue over whether that individuates propositions in a sufficiently fine-grained way, but that's a different question entirely from whether it expresses one. It is not the question of whether that sentece expresses a proposition, but how to represent the proposition that it expresses.
there ought to be a fact of the matter whether p1=p2. However, there isn't, since that fact would require a meaning to be assigned to "borogoves" in English and "borogoves" in French to ensure that they're not false friends.
Not so! The logical structure of the two sentences is the same and they are both tautologies. That's all we need to show that they are logically equivalent, and so express the same proposition. Of course I realize that all necessary truths express the same proposition on this account, but this doesn't worry me.
I am worried about the idea that all necessary propositions are the same. One of the reasons for the worry is that one of the things we need propositions for is synonymy: Two sentences are synonymous iff they express the same proposition.
The other reason is that I do not think our semantics ought to change very much should we come to believe that Leibniz, Spinoza, Leslie and Rescher are right and there is only one possible world.
The other reason is that I do not think our semantics ought to change very much should we come to believe that Leibniz, Spinoza, Leslie and Rescher are right and there is only one possible world.
Good Lord, no! I'm sure they're wrong about that; that would be awful. But were they right, the semantics I endorse would be wrong. But even saying this raises the kind of worry you're posting about. If they are right, they are so necessarily and if wrong, they are wrong necessarily. This might be one good reason to go fictionalist about modal talk!
Here's another way not to tie semantics to contingency too much. Many if not most of the questions that interest me concern those propositions that are either necessarily true or necessarily false. Questions of ontology, mathematics and ethics are all like that. Many, though not all, of the most interesting questions of theology are like that, too. One can spend much of one's epistemic life living in the realm of necessary truths/falsehoods, and we philosophers do. And one can almost imagine, I think, spending all of one's epistemic life there.
Semantic theories for which it is essential that the everyday stuff we talk about is contingent are, I conjecture, unlikely to work in that realm. They are, thus, going to posit a deep divide between the semantics dealing with contingent facts and the semantics dealing with the facts that really interest me. But I don't buy that deep divide...
In the first sentence, I meant: "Here is another REASON not to..."
They are, thus, going to posit a deep divide between the semantics dealing with contingent facts and the semantics dealing with the facts that really interest me. But I don't buy that deep divide...
Alex, I confess I'm at a loss as to why you don't see a "deep divide". I can't see how interesting mathematical claims can be contingent and I cannot see how interesting botanical claims can be necessary (I take it you are referring to the reduction of contingency to necessity and vice versa on the assumption of a single world: viz. where |- p <-> Np). I see nothing but confusion from insisting on forcing everything into the contingent or everything into the necessary. Am I missing your point?
Alex,
I have encountered a very similar problem with reductio arguments while writing a paper about the Logical Argument from evil (LAFE). The problem is general and can be stated as follows:
a reductio argument is a deductive argument. Question: What are the soundness-conditions of a reductio argument? Well, such an argument is sound iff it is valid and all its premises are true. But, if a reductio argument is valid, then this fact is tantamount to proving that not all its premises are true. So, does it follow that a valid reductio argument cannot be sound?
But, how can that be?
My solution: reductio arguments typically have primary premises which are the targets of the reductio argument and auxiliary premises, which are imported in order to prove a contradiction. Well, we can now define the soundness conditions of a reductio argument as follows: A reductio argument is sound iff it is valid and all its auxiliary premises are true.
This solution conforms to the presuppositions behind the practice of reductio arguments where we do not assert that the target premises are true; nor do we present them as hypotheses that could turn out to be true. Rather we present the target premises for the sake of the argument and for the purpose of proving that they are not co-possible.
Note: I have read a couple of papers you presented on Problogian and wanted to comment but did not know how. oh! well, maybe it is just for the better.
Thanks
Peter Lupu
A reductio argument can always be transformed into a direct argument just by making use of conditionals. For instance, here is the Zenonian Achilles argument thus transformed.
1. If there is motion, then Achilles can catch the tortoise. (Premise)
2. If Achilles can catch the tortoise, then he can get to where the tortoise is now, by which time the tortoise would have moved away, and then he can get to the tortoise's new place, and so on. (Premise)
3. Thus, if Achilles can catch the tortoise, Achilles can make an infinite number of movements. (By 2)
4. Thus, if there is motion, Achilles can make an infinite number of movements. (By 1 and 3)
5. But Achilles cannot make an infinite number of movements. (Premise)
6. Thus, there is no motion. (By 4 and 5 and modus tollens)
The difficult questions are: (1) What it does to the validity of an argument that the sentences in the argument are nonsense by the lights of the person offering the argument, and (2) What if the auxiliary premises in the reductio argument make use of terms that are nonsense by the lights of the person offering the argument.
Here is something I just wrote on related issues.
Alex,
Two points:
a) about the question of arguments involving terms that lack sense (whether the argument is a reductio argument or direct one): call them "degenerate arguments".
From the point of view of formal logic, degenerate arguments cannot be represented in a formal system due to the stringent requirements of such systems on their vocabulary. So unless some provisional sense can be given to the terms involved, such arguments cannot be represented in a formal system.
The problem you pose reminds me of Quine's discussion about the question how can one deny that Pegasus exists, when one believes that Pegasus does not exists. Your problem is similar except it elevates the issue to the level meaning rather than reference.
So you might still ask: how does one deal with arguments which include meaningless terms, from one's point of view?
I think there is a way out that mimicks to some extent the requirement on the meaningfulness of terms in formal systems. Namely, one insists that the terms involved should be given at least provisionally a meaning or sense. The claims then can be assessed with respect to the provisional sense so given.
Now suppose one attempts to show that a certain philosophical theory (say Plato's theory of Forms) is incoherent because certain terms do not have a clear sense. Well, then, you proceed something like the following:
Suppose the terms in the theory in question have the following sense:
....(here give something like a definition). Then if the terms mean such-and-such, then the theory would entail unacceptable consequences (a contradiction, an obvious falsehood). Therefore,given these meanings of the terms in question, the theory is either inconsistent or false. The proponent of the theory then would have the opportunity to revise the meaning-assignment to the terms; the opponent then goes through the same steps...and so on.
Is there anything wrong with this?
b) Your point about the possibility of converting every reductio into a direct argument is unclear to me. Take your example. I take that the point of this example is that from premises that appear to be reasonable and true a certain conclusion follows that is puzzling. But, now, what exactly is puzzling about the conclusion here? I suppose the puzzle here is that the conclusion contradicts something we think is true, namely, that there is motion. So since the "direct" argument entails a conclusion that contradicts what we take to be true; i.e, that there is motion, it follows that not all the premises of the argument can be true. But they all seem to be quite reasonable.
So, what i see here is that you simply failed to state one of the premises in this argument, namely, that there is motion. OW i do not see wherein lies the puzzle. Hence, your example is a reductio argument after all.
peter
P.S. sorry for the lenght of this note.
Is there anything wrong with this?
Heh. Other than that it describes an infinite process?
Of course where would the blogosphere be if it weren't possible to infinitely argue about nothing? The halting problem isn't a bug, it is a feature. :-)
zippy,
"Other than that it describes an infinite process?"
I do not see how that follows. It may describe a lengthy process, perhaps; but, infinite? nope!
...unless we are facing here a very stubborn fellow who will go through every possible sense (or meaning) no matter how unrelated and irrelevant to the main point it may be; and if there are infinitely many senses (something i do not know, but willing to grant), then the process could turn out to be potentially infinite (or, perhaps, you meant, indefinite).
But, then, so what? so is life when facing the property of "stubborness ".
In practice, however, several rounds will suffice to convince one side or the other that some definite conclusion follows here.
peter
Peter:
If you look at my argument, you'll notice that "There is motion" is not a premise in it. It's converted from a reductio into a direct argument for the conclusion that there is no motion.
Alex,
I did noticed that your explicit formulation of the "direct argument" does not feature the premise "there is motion", which is exactly the reason that I do not think that it is a plausible conversion of the corresponding reductio type argument.
Here is how to look at this.
In order for your direct argument to be a legitimate conversion of the original reductio form it has to preserve some central feature of the original. The most salient feature of the reductio original is that it presents a puzzle about motion. What is the puzzle? Well, the reductio argument entails a contradiction from reasonable premises each of which we take to be true and one of which is that motion exists.
Question: Does your direct argument present us with a similar puzzle about motion? It does only if there is a hidden premises not explicitly stated in the argument; namely, that motion exists. For suppose that there are no hidden premises. Then your argument entails that there is no motion. But, now, why should we think that this conclusion is puzzling? It is puzzling only because it contradicts a proposition we think is true, namely, the proposition that there is motion.
So in order for your direct argument to be a legitimate conversion of the reductio original, it has to preserve the same sense of puzzlement engendered by the original. If it does so, then it is nothing but an implicit form of a reductio argument; if it does not, then it fails to preserve the puzzling element of the original and, therefore, I claim it cannot be a conversion of the reductio argument.
peter
P.S. I have read your note "On Reduction and the argument from evil", but i am unable to respond on that site, so i might just respond here, if that is ok with you.
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