Three Paradoxes?

I've mentioned three kinds of paradoxes in these posts:

1. Semantic paradoxes (such as the liar)
2. Moorean 'paradoxes'
3. Open-question paradoxes (including Russell's paradox)

I wonder if all share the character of rendering themselves unintelligible if taken 'literally' (i.e. if the words used to utter them are interpreted in the 'usual' way, or canonically). I also wonder if this is interesting or, possibly, trivial: there are lots of ways of making no sense at all.

Semantic paradoxes arise when apparently 'grammatical' (i.e. 'well-formed') statements directly contradict themselves, which is immediately catastrophic. We imagine that 'well-formed' statements should be like formulae in a formal language, and should either be theorems or not - should be definitely true or false. (The scope of the 'formal language' metaphor is problematic, of course.)

A classic example of a liar paradox is:

(A): Sentence (B) is true.
(B): Sentence (A) is false.

Moorean paradoxes are not directly self-contradictory, but directly or indirectly undermine the grounds of their own intelligibility. They are usually constructed from a sentence function which produces true or false statements for most legitimate substitutions but becomes nonsensical (rather than simply false) when a variable is replaced with a first-person indexical.

A classic example of a Moorean paradox is:

"A believes it is raining, but it is not raining."

This is fine when A is replaced by most names or descriptive nouns, but not when it is replaced by a first-person indexical:

"I believe it is raining, but it is not raining."

Open question paradoxes arise from self-referential validity (or intelligibility) dependencies, where some normative concept must (directly or indirectly) include within its scope of adjudiation any account of how it may properly be applied.

A classic example of an open-question paradox is:

(C) "By adhering to theory X, we can reliably distinguish true statements from false statements in all circumstances."

(D) How do we show that (C) is true, without circularity or stipulation?

Open paradoxes unite the Analytic tradition in philosophy because they are generated by all of its main objects of study: goodness, knowledge, meaning, choice, validity etcetera.

Open question paradoxes can be generated in purely syntactical contexts (formal systems) as well as by semantic categories, as Kurt Gödel showed for theorem determination in arithmetic.

(Russell's paradox is an open question paradox because the normative adjudication 'is a member of' is rendered systematically ambiguous for certain cases.)

All of these paradoxes are relevant to enquiries into intelligibility. And they are tied up with the contextuality of language use - a contextuality which philosophers persist in thinking we can somehow 'cancel out', producing words without speakers, assertions without asserters, 'free-standing' statements that are not part of the world they describe ...

We can get some insight into this by considering the simple case of falsehood.

A false statement is the contrary of a statement that is true. We can construct a false statement by appending the 'not' operator to a true statement in whatever way is grammatically appropriate. If 'I am hungry' is true, then 'I am not hungry' is false. If P is true, then ~P is false.

It is easy to demonstrate that if we try to make 'P & ~P' - a statement and its negation - true, then we are committing ourselves to the truth of any random statement Q. This such a powerful logical principle that it can be used to test the consistency of an axiomatic system: to show that they do not lead to contradiction, we need only show that there is some well-formed formula that is not a theorem of the system.

Taken together, these criteria have the consequences that a false statement must 'make sense', but that asserting a false statement does not. Intelligibility and assertability must, in other words, be independent of one another. Trying to make sense of this has led to some bizarre metaphysical speculation (e.g. the 'reality' of 'possible' worlds ...)

Of course, re-engagement with context eliminates this problem: intelligibility and truth also re-engage. The context

 provides truth conditions, which, if necessary, can be explicitly introduced into the discussion. This is what Wittgenstein meant when he talked about 'language going on holiday'. Abstraction can lead us astray: grammar does not confer legitimacy, because we can find ourselves talking nonsense for reasons that we do not have the language to explicate. We think we understand the 'grammar' of a Moorean paradox because we have a picture of what grammatical conformity looks like: rules about how to move the puzzle pieces around, perhaps. But we can only state these rules in a language which we already presume to be intelligible.

Whenever we imagine we can do otherwise, we will find ourselves generating paradoxes that we don't know what to do with. We cannot specify the rules of valid rule generation. We generate rules by experiment, and we are occasionally deceived when we stumble on a rule whose contrary cannot make sense in any intelligible language (and so, also, cannot be the product of a legitimate translation schema for any prospective language).

An essential feature of an experiment is that its outcome can't be predicted (otherwise we wouldn't be undertaking it). When we try to say that a false statement 'makes sense' (even if it can't be seriously asserted), we are stipulating how the experiment must come out - even though we know that falsehood entails complete incongruence.

If we want to say something like 'the truth of X is intelligible, even though X is false', we have to explain what we mean by this. There will always be a contextual ambiguity behind such a statement. 'X is intelligible' and 'X is false' can only be entertained together if they do not share an alethial 'frame' (otherwise we have an ex contradictione explosion ...).

No competent interlocutor would make a paradoxical assertion, of course. There is no context outside philosophical abstraction in which these are intelligible. The fact that we can't write down a set of rules from which this can be deduced should tell us something about our conceptions of philosophical abstraction.