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Wednesday, September 09, 2009

The Truth Machine (3)

Except:

The machine - as metaphor or as formal system - only appears in our theories as a description. In particular, it is described as following rules. (Specified rules, if it is a formal system).

Logicians distinguish between a formal system and its interpretation - we can have 'truth tables', uninterpreted (ones and zeros, Ts and Fs); and we can 'interpret' these as showing the circumstances under which 'Truth' is transmitted from one statement to another.

A difficulty with this distinction is that there is already some interpretation in the formal system. For instance, we reject 'P&~P' as contravening the law of non-contradiction if we regard both instances of P as 'meaning' the same thing. Even more fundamentally: Is 'P' the same symbol as 'P'? Obviously yes? Why?

We recognise them as the same, but they are in different places on the page, and are surrounded by different patterns of other symbols; and there are probably other 'differences'. Without 'same' and 'different' we don't even have a symbolism, and we only have 'same' and 'different' under interpretation.

And we only have interpretation if we already have a semantic framework.

And so we can only give sense to 'state of the machine', as well, within a semantic framework.

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