Bill Vallicella comments: "One cannot eat without eating something, and indeed something that exists. And one cannot desire without desiring something -- but in this case the thing desired needn't exist." But then we have the problem of 'something desired which doesn't exist', which seems contradictory. Is the problem about ontology at all?
Consider
(1) Jake is searching for a gold mine near Cripple Crow Creek.
(2) There are no gold mines near Cripple Crow Creek.
There is not even even a whiff of contradiction or paradox here, and we are not tempted to posit 'non-existent objects' or suchlike. Of course, if we try to analyse them in terms of predicate calculus, we do get a contradiction:
(1a) For some x, [x is a gold mine near Cripple Crow Creek, and Jake is searching for x].
(2a) Not for some x, [x is a gold mine near Cripple Crow Creek].
The first sentence logically implies 'for some x, x is a gold mine near Cripple Crow Creek', which directly contradicts the second. But that suggests a problem with the analysis of (1) into an inappropriate formalisation such as (1a), rather than any question of 'ontology'. As soon as we even ask the question about the 'ontological status' of the sought-for gold mine, we are already on the metaphysical sandbank. E.g. we might try resolve the contradiction between (1a) and (2a) by replacing (2a) with
(2b) Not for some x, [x is an existent gold mine near Cripple Crow Creek].
This resolves the contradiction. But by now we are well into the Meinongian jungle. This is entirely a problem of language and logic, as I see it. What is the deep structure or logical analysis of sentence (1) which makes it transparently clear that it is not inconsistent with (2)? Note that we do get inconsistency if we turn the grammatically active sentence (1) into a passive without getting 'existential implication'.
(1b) Some gold mine near Cripple Crow Creek is sought for by Jake.
In its most natural reading, this contradicts (2). So, what is the true semantic structure of sentence (1)? That is the real question, and it has nothing to do with metaphysics. As a suggestion, consider
(3) Jake says that there is a gold mine near Cripple Crow Creek.
which we can analyse into
(3a) Jake says that for some x, x is a gold mine and x is near Cripple Crow Creek.
This, unlike (1a) above, is not inconsistent with (2a) above, since it does not imply that for some x, etc. Could there not be some analogous analysis of (1) into
(1c) Jake is searching-that-there-is a gold mine near Cripple Crow Creek ?
Could it be that 'searching for' has an embedded that-clause which invalidates the inference to 'for some x there is ...', but which is not visible at the surface level of the sentence? That seems a much cleaner way to resolving the difficulty than all this intentional objects nonsense. For nonsense it is.
Showing posts with label analysis. Show all posts
Showing posts with label analysis. Show all posts
Tuesday, December 28, 2010
Monday, September 13, 2010
'Anybody who knows ...'
One of the comments at the discussion raging at Vallicella's site illustrates perfectly the 'translation problem' that I mentioned in an earlier post. The problem is that any attempt at 'proving' or 'disproving' an ordinary language statement by using the well-defined proof procedure of the modern predicate calculus is highly vulnerable to the process of interpreting the ordinary language statement in the calculus. If the interpretation is not correct, then the proof, though perfectly valid, may be proving the wrong thing.
Suppose we want to prove the validity of an ordinary language consequence having the following form.
(*) If it was the case that A was identical with B then it is the case that A is identical with B
We can try to do this by translating the placeholders A and B (which substitute for grammatically singular OL terms) into the 'a' and 'b' of the predicate calculus (which substitute for logically singular terms), and translating the tensed statements of OL into the 'nec' or 'necessary' of predicate calculus, as follows:
(1) a=b (Assumption for conditional proof)
(2) a=b -> (Fa -> Fb)
(3) Nec(a=a)
(4) a=b -> (Nec(a=a) -> Nec(a=b)) (Substitution Instance of (2))
(5) Nec(a=a) -> Nec(a=b) (Modus Ponens, 1&4)
(6) Nec(a=b) (Modus Ponens, 3&5)
(7) a=b -> Nec(a=b) (Conditional Proof, 1-6)
The problem is that the 'proof', if understood as a proof of the ordinary language consequence, can't possibly be valid. Substitute 'the president of the US' for 'A' and 'John F. Kennedy' for 'B' to give
(*) If it was the case that the president of the US was identical with John F. Kennedy then it is the case the president of the US is identical with John F. Kennedy
But ex vero nunquam sequitur falsum: the false cannot follow from the true. Whenever the antecedent is true and the consequent false, consequentia non valet, the consequence is not valid. But the antecedent is true - the president of the US was (in September 1963) identical with John F. Kennedy, and the consequent false - the president of the US is (in September 2010) not identical with John F. Kennedy. So the consequence is not valid. If the formalised part of the proof is valid (which I am not denying), it follows that the translation of our ordinary language consequence into the formal consequence (i.e. (7) above) is wrong. But that is just the place we forgot to look.
At this point, the formalist will object that the translation "would be accepted by just about anybody who knows how to translate from ordinary language to formal logic". And that is another problem: a cultural problem, not a logical or philosophical one. We were taught as students the 'correct' way to translate awkward and messy ordinary language statements into the clean language of MPC. I too was taught this (using what was only 10 years old then, but has since become a classic text) quite some time ago. Having learnt this, we 'know' how to translate from ordinary language to formal logic. And, proud of this knowledge, we are now 'anybody who knows', and we can put down anyone who does not know.
What a formidable barrier to progress.
Suppose we want to prove the validity of an ordinary language consequence having the following form.
(*) If it was the case that A was identical with B then it is the case that A is identical with B
We can try to do this by translating the placeholders A and B (which substitute for grammatically singular OL terms) into the 'a' and 'b' of the predicate calculus (which substitute for logically singular terms), and translating the tensed statements of OL into the 'nec' or 'necessary' of predicate calculus, as follows:
(1) a=b (Assumption for conditional proof)
(2) a=b -> (Fa -> Fb)
(3) Nec(a=a)
(4) a=b -> (Nec(a=a) -> Nec(a=b)) (Substitution Instance of (2))
(5) Nec(a=a) -> Nec(a=b) (Modus Ponens, 1&4)
(6) Nec(a=b) (Modus Ponens, 3&5)
(7) a=b -> Nec(a=b) (Conditional Proof, 1-6)
The problem is that the 'proof', if understood as a proof of the ordinary language consequence, can't possibly be valid. Substitute 'the president of the US' for 'A' and 'John F. Kennedy' for 'B' to give
(*) If it was the case that the president of the US was identical with John F. Kennedy then it is the case the president of the US is identical with John F. Kennedy
But ex vero nunquam sequitur falsum: the false cannot follow from the true. Whenever the antecedent is true and the consequent false, consequentia non valet, the consequence is not valid. But the antecedent is true - the president of the US was (in September 1963) identical with John F. Kennedy, and the consequent false - the president of the US is (in September 2010) not identical with John F. Kennedy. So the consequence is not valid. If the formalised part of the proof is valid (which I am not denying), it follows that the translation of our ordinary language consequence into the formal consequence (i.e. (7) above) is wrong. But that is just the place we forgot to look.
At this point, the formalist will object that the translation "would be accepted by just about anybody who knows how to translate from ordinary language to formal logic". And that is another problem: a cultural problem, not a logical or philosophical one. We were taught as students the 'correct' way to translate awkward and messy ordinary language statements into the clean language of MPC. I too was taught this (using what was only 10 years old then, but has since become a classic text) quite some time ago. Having learnt this, we 'know' how to translate from ordinary language to formal logic. And, proud of this knowledge, we are now 'anybody who knows', and we can put down anyone who does not know.
What a formidable barrier to progress.
Wednesday, September 08, 2010
The Perils of Analysis
An argument mentioned by Bill Vallicella here neatly illustrates the danger of using modern predicate calculus to penetrate the logic of natural language. He cites an argument of Peter Van Inwagen, as follows.
(A) There is a large parcel of land having two smaller parcels as proper parts.
(B) There are three things (the large parcel of land and the two smaller parcels)
If the inference is valid, then there is a third thing 'over and above' the two smaller things, and there is an 'ontological distinction' between the large parcel of land and its parts. Otherwise there are only two things, and the existence of the 'large parcel of land' simply reduces to the existence of the parts. It is not 'ontologically distinct'. This is an important philosophical conclusion, if we can establish it.
The procedure is to translate both statements into the language of predicate calculus, which has a determinate proof procedure, and see whether the inference holds. Thus
(A*) For some x, x is a large parcel of land having proper part x1 and proper part x2 and x1 /= x2.
(B*) For some x, for some y, for some z, x/=y and x/=z and y/=z
Clearly, given the additional premiss that no object x is identical to any of its proper parts (i.e. x /= x1 and x /= x2) we can establish that B* follows from A*. Thus the apparently simple translation from a natural language statement into the language of modern predicate calculus apparently leads to a philosophical conclusion. And a lot of modern analytic philosophy is like that. We are worried about whether the ordinary language statement A implies the ordinary language statement B. For ordinary language has no agreed and determinate proof procedure. So we translate A into a statement A* of the predicate calculus, which does have an agreed and determinate proof procedure, and we translate B into B*. Then we determine whether A* implies B*, which seems to solve the problem. But of course it doesn't, for the real question is whether the translation is correct. If we are unsure whether A implies B, how can we be sure that either of the translations (of A onto A* and B into B*) are correct? If the translation is obvious, how is it we were unsure of the implication in the first place?
The 'method of analysis' is not fundamentally unsound. If we are certain of the 'logical form' of an ordinary language statement - i.e. a form that makes inferences to other statements determinate and certain, then analysis is a useful technique. Otherwise it is not. What is the logical form of 'this big parcel consists of two small parcels'? If it is the same as 'This pair of shoes consists of 2 shoes', then we should proceed with caution.
Suppose that there exists nothing but my big parcel of land and such partsVan Inwagen's argument employs a method that is fundamental to all analytic philosophy. We have two ordinary language statements A and B below, and we want to decide whether B follows from A.
as it may have. And suppose it has no proper parts but the six small parcels. .
. . Suppose that we have a bunch of sentences containing quantifiers, and that
we want to determine their truth-values: 'ExEyEz(y is a part of x & z is a
part of x & y is not the same size as z)'; that sort of thing. How many
items in our domain of quantification? Seven, right? That is, there are seven
objects, and not six objects or one object, that are possible values of our
variables, and that we must take account of when we are determining the
truth-value of our sentences. ("Composition as Identity," Philosophical
Perspectives 8 (1994), p. 213)
(A) There is a large parcel of land having two smaller parcels as proper parts.
(B) There are three things (the large parcel of land and the two smaller parcels)
If the inference is valid, then there is a third thing 'over and above' the two smaller things, and there is an 'ontological distinction' between the large parcel of land and its parts. Otherwise there are only two things, and the existence of the 'large parcel of land' simply reduces to the existence of the parts. It is not 'ontologically distinct'. This is an important philosophical conclusion, if we can establish it.
The procedure is to translate both statements into the language of predicate calculus, which has a determinate proof procedure, and see whether the inference holds. Thus
(A*) For some x, x is a large parcel of land having proper part x1 and proper part x2 and x1 /= x2.
(B*) For some x, for some y, for some z, x/=y and x/=z and y/=z
Clearly, given the additional premiss that no object x is identical to any of its proper parts (i.e. x /= x1 and x /= x2) we can establish that B* follows from A*. Thus the apparently simple translation from a natural language statement into the language of modern predicate calculus apparently leads to a philosophical conclusion. And a lot of modern analytic philosophy is like that. We are worried about whether the ordinary language statement A implies the ordinary language statement B. For ordinary language has no agreed and determinate proof procedure. So we translate A into a statement A* of the predicate calculus, which does have an agreed and determinate proof procedure, and we translate B into B*. Then we determine whether A* implies B*, which seems to solve the problem. But of course it doesn't, for the real question is whether the translation is correct. If we are unsure whether A implies B, how can we be sure that either of the translations (of A onto A* and B into B*) are correct? If the translation is obvious, how is it we were unsure of the implication in the first place?
The 'method of analysis' is not fundamentally unsound. If we are certain of the 'logical form' of an ordinary language statement - i.e. a form that makes inferences to other statements determinate and certain, then analysis is a useful technique. Otherwise it is not. What is the logical form of 'this big parcel consists of two small parcels'? If it is the same as 'This pair of shoes consists of 2 shoes', then we should proceed with caution.
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