Let's suppose that the set of English assertive sentences of less than n (for n>6) words is finite. We can then associate a physical token (a sound, a scribble, an object) with each of these sentences.
We will now make two piles of these objects - Pile A and Pile B. We will put all of the objects associated with true assertions in Pile A and all of the objects associated with false assertions in Pile B.
All statements of the form "Object X is in Pile B" are, of course, associated with an object in one of the piles.
Now let Z = "Object P is in Pile B", and let Object P be the token for Z.
If Object P is in Pile B, then Z is true. However Pile A contains the tokens for true statements. However, if Object P is in Pile A, then Z is false, and Pile B contains the tokens for false statements.
So far, this is looks like a re-statement of a traditional paradox.
However, this construction of it is particularly awkward for any kind of correspodence semantics, because the paradox is a direct consequence of the kind of token/fact relationship such a semantics would require.
Saturday, November 22, 2014
Wednesday, October 01, 2014
Probability
Let's say that we need reasons to attribute a probability p to an event e. Without the possibility of reasons, attribution of probability is unintelligible. This might be taken to be a way of saying that there are no 'real' probabilities, but since 'real' and 'can talk as if real' work in the same way in all discourse, we don't need to worry about this.
If I say 'e is more probable than ~e', I am saying that p > 1/2. All probability attributions are at least strict inequalities, if not specific values.
My reasons for attributing probability p to event e must, then, include a computation that demonstrates the vailidity of the strict inequality. In Baysian computations, we show how some probabilties are derived from others - the results of statistical tests are an example. At the basis of these are appeals to theories (e.g. combinatorial considerations, engineering descriptions, or wave and field equations) from which basic probabilities can be directly derived.
These computations, and the theories which provide the primitive probabilities, must be articulated in a shared language. There is no way of attaching a meaning to questions (in the same language) about the probability that the way we speak the language, or the language itself, may be unreliable here (whether or not we can attach a meaning to the possibility of this). We cannot say 'it is more likely than not that our language works' because no conceivable computation could validate the relevant inequality.
(Equally, we cannot say 'It is quite certain that our language works' if we read this in the sense of the probability of its not working being zero. Our ability to attribute probabilities depends upon our ability to speak, so attributing a probability to our ability to speak is circular.)
We can make the classical distinction between risk and uncertainty by saying that we can attribute probabilities to risky outcomes, but uncertain ones may only be partly tabulatable. We can make a list of some of them, but neither attribute probabilities to them nor be sure that the list is exhaustive.
To say that there is a definite, but unknown p of e is to promise a computation that produces p. The Born rule can be accepted in the 'world' in which we speak, because - in that world - it is a testable empirical theory. Maybe if we want to talk about many worlds then we cannot talk - and we do not need to talk - about Born probabilties. Only the illusion of 'real' probability makes this seem puzzling.
If I say 'e is more probable than ~e', I am saying that p > 1/2. All probability attributions are at least strict inequalities, if not specific values.
My reasons for attributing probability p to event e must, then, include a computation that demonstrates the vailidity of the strict inequality. In Baysian computations, we show how some probabilties are derived from others - the results of statistical tests are an example. At the basis of these are appeals to theories (e.g. combinatorial considerations, engineering descriptions, or wave and field equations) from which basic probabilities can be directly derived.
These computations, and the theories which provide the primitive probabilities, must be articulated in a shared language. There is no way of attaching a meaning to questions (in the same language) about the probability that the way we speak the language, or the language itself, may be unreliable here (whether or not we can attach a meaning to the possibility of this). We cannot say 'it is more likely than not that our language works' because no conceivable computation could validate the relevant inequality.
(Equally, we cannot say 'It is quite certain that our language works' if we read this in the sense of the probability of its not working being zero. Our ability to attribute probabilities depends upon our ability to speak, so attributing a probability to our ability to speak is circular.)
We can make the classical distinction between risk and uncertainty by saying that we can attribute probabilities to risky outcomes, but uncertain ones may only be partly tabulatable. We can make a list of some of them, but neither attribute probabilities to them nor be sure that the list is exhaustive.
To say that there is a definite, but unknown p of e is to promise a computation that produces p. The Born rule can be accepted in the 'world' in which we speak, because - in that world - it is a testable empirical theory. Maybe if we want to talk about many worlds then we cannot talk - and we do not need to talk - about Born probabilties. Only the illusion of 'real' probability makes this seem puzzling.
Tuesday, May 27, 2014
Consequential intelligibility
It is a profound mistake to think that a process which generates intelligibility must itself be intelligible.
Tuesday, May 13, 2014
Paradox and method
No method can be shown to parsimoniously select methods which do not generate paradoxes, since it, itself, could not be independently shown not to generate paradoxes without generating a regress of such methods. (Apart from all the halting problem etc. issues ...)
On the other hand, we need to assume that whatever method we are using to construct this argument does not lead to paradox, since that would render it unintelligible. And, ultimately, we can't constructively substantiate this assumption without generating a regress.
However, we have the comfort that 'nothing can be shown to be intelligible' must, at least, be intelligible - and so false. This gives us a recursive solution to the 'constructive substantiation' problem, and possibly also the problem of paradoxes - although I think these will turn out to be necessary, rather than inconvenient.
The reason for this is that we give reasons. Any attempt to give complete reasons that doesn't point to a recursive root will generate paradoxes, and recursive roots are (at least tacitly) agreed rather demonstrated. (Persistent denial of candidates has the consequence of rendering even the denial unintelligible.)
On the other hand, we need to assume that whatever method we are using to construct this argument does not lead to paradox, since that would render it unintelligible. And, ultimately, we can't constructively substantiate this assumption without generating a regress.
However, we have the comfort that 'nothing can be shown to be intelligible' must, at least, be intelligible - and so false. This gives us a recursive solution to the 'constructive substantiation' problem, and possibly also the problem of paradoxes - although I think these will turn out to be necessary, rather than inconvenient.
The reason for this is that we give reasons. Any attempt to give complete reasons that doesn't point to a recursive root will generate paradoxes, and recursive roots are (at least tacitly) agreed rather demonstrated. (Persistent denial of candidates has the consequence of rendering even the denial unintelligible.)
Monday, February 10, 2014
Emergence
Emergence is a process required to save a fundamental theory which 'must' be true but which cannot provide the explanations it promises.
It is an arbitrary device for avoiding counter-examples, and any theory which must resort to it is false.
I don't really know why this isn't obvious.
A tiger is not a bounded sub-space of some set of physical properties, however defined. A tiger is a tiger even without it's whiskers, and sometimes without it's tail. It may also be a tiger when it is the wrong colour, or has some deformity. 'Tiger' is not a complete concept (Waismann). We haven't made up our minds yet about all the things that might count as tigers.
Well. It's late ...
It is an arbitrary device for avoiding counter-examples, and any theory which must resort to it is false.
I don't really know why this isn't obvious.
A tiger is not a bounded sub-space of some set of physical properties, however defined. A tiger is a tiger even without it's whiskers, and sometimes without it's tail. It may also be a tiger when it is the wrong colour, or has some deformity. 'Tiger' is not a complete concept (Waismann). We haven't made up our minds yet about all the things that might count as tigers.
Well. It's late ...
Sunday, February 09, 2014
Languages and Truth - a clearer exposition ... maybe ...
The 'brick slab' language, and Tarski's object languages (which avoid paradox?) are only languages by stipulation. There is no behvioural or reductive test for something being a language that doesn't fall foul of Kripke's paradox.
If we want to be able to say 'these people are speaking a language', then we need to have a concept of truth, because we need to be able to say 'these people are speaking a language properly'.
If we find someone speaking a language 'improperly' it is because they can do otherwise. A noisy dog is not a deficient speaker of English. Only an English speaker can lie or make mistakes in English.
For someone to be a speaker of a language they need to be able to tell the truth in the language (whether or not they actually do). When we attribute the capacity to speak to someone, we also attribute the capacity to tell the truth.
When we agree that something counts as a language, we are saying (with Davidson) at least that it is translatable. When we say that what we are using now (in this conversation) counts as language, we are saying something that, if it can be speculated explicitly (within this conversation), must be true. It's falsehood is inconsistent with its explicit speculability.
So:
The statement 'we are using a language' is reliably true if it occurs in a conversation in that language, because its contrary (in the context of that conversation) would be unintelligible. Kripke's problem makes any behavioural or material ground for the judgment that we (or others) are using a language radically unreliable.
We might judge of others that they are using a language, but until we share that judgement with them in a conversation it remains subject to Kripke's paradox.
Also, since any judgement that a language is being used includes a judgement about it's being used properly, such a judgement depends upon a judgement that it is possible to tell the truth (since this is part of the judgement that the language is being used properly).
The judgement that we have correctly translated a language used by third parties is (Davidson and Kripke) radically unreliable without some 'Principle of Charity'. However in the context of any shared conversation with the natives - using either or both languages - this radical unreliability is ruled out a priori since 'we may have radically failed to interpret your language' (expressed in either language) would not, if it were true, mean what it appeared to mean to all participants in the conversation. It would entail that there was no conversation of which it could be a part.
The 'slab and brick' game, and all games in which paradoxes have been excised (languages without truth predicates), the judgement 'we are speaking a language' cannot be formed, because they cannot contain their own truth predicates. Because of this, they can only be identified as languages by stipulation or on material or behavioural grounds - and so these identifiecations are vulnerable to Kripke's problem, or are arbitrary.
Solving the problem of paradox this way comes at the price of rendering any judgement that we are using a language radically unreliable.
On the other hand, living with paradox allows us to make a priori, and incorrigible, judgements that we are using a language.
And worse: it is the possibility of these judgements that gives sense to the word 'language', whether we use it to refer to the tools of our present conversation, or whether we use it to speculate about other possible conversations - corrigibly or otherwise. The meaning of 'language' depends, necessarily, on being able to point to 'what we are doing now, in speaking to one another'. To call something a language, corrigibly or incorrigibly, is to rely on a concept which can only be rendered recursively, with a root in what we are presently doing when we speak to one another.
If we want to be able to say 'these people are speaking a language', then we need to have a concept of truth, because we need to be able to say 'these people are speaking a language properly'.
If we find someone speaking a language 'improperly' it is because they can do otherwise. A noisy dog is not a deficient speaker of English. Only an English speaker can lie or make mistakes in English.
For someone to be a speaker of a language they need to be able to tell the truth in the language (whether or not they actually do). When we attribute the capacity to speak to someone, we also attribute the capacity to tell the truth.
When we agree that something counts as a language, we are saying (with Davidson) at least that it is translatable. When we say that what we are using now (in this conversation) counts as language, we are saying something that, if it can be speculated explicitly (within this conversation), must be true. It's falsehood is inconsistent with its explicit speculability.
So:
The statement 'we are using a language' is reliably true if it occurs in a conversation in that language, because its contrary (in the context of that conversation) would be unintelligible. Kripke's problem makes any behavioural or material ground for the judgment that we (or others) are using a language radically unreliable.
We might judge of others that they are using a language, but until we share that judgement with them in a conversation it remains subject to Kripke's paradox.
Also, since any judgement that a language is being used includes a judgement about it's being used properly, such a judgement depends upon a judgement that it is possible to tell the truth (since this is part of the judgement that the language is being used properly).
The judgement that we have correctly translated a language used by third parties is (Davidson and Kripke) radically unreliable without some 'Principle of Charity'. However in the context of any shared conversation with the natives - using either or both languages - this radical unreliability is ruled out a priori since 'we may have radically failed to interpret your language' (expressed in either language) would not, if it were true, mean what it appeared to mean to all participants in the conversation. It would entail that there was no conversation of which it could be a part.
The 'slab and brick' game, and all games in which paradoxes have been excised (languages without truth predicates), the judgement 'we are speaking a language' cannot be formed, because they cannot contain their own truth predicates. Because of this, they can only be identified as languages by stipulation or on material or behavioural grounds - and so these identifiecations are vulnerable to Kripke's problem, or are arbitrary.
Solving the problem of paradox this way comes at the price of rendering any judgement that we are using a language radically unreliable.
On the other hand, living with paradox allows us to make a priori, and incorrigible, judgements that we are using a language.
And worse: it is the possibility of these judgements that gives sense to the word 'language', whether we use it to refer to the tools of our present conversation, or whether we use it to speculate about other possible conversations - corrigibly or otherwise. The meaning of 'language' depends, necessarily, on being able to point to 'what we are doing now, in speaking to one another'. To call something a language, corrigibly or incorrigibly, is to rely on a concept which can only be rendered recursively, with a root in what we are presently doing when we speak to one another.
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