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                                    Language Paradoxes

If we take such apparently paradoxical statements as ‘I am lying’, the first step is to translate it as ‘this statement is false’, or ‘deliberately false’ and then to set out the paradox and try to resolve it. Consider A: ‘this statement is false’. If A is true, it is false; if false, true. Let us now try to resolve such paradoxes, starting with an examination of B: ‘this statement is short’. Since B is indeed short, it can hardly be denied that it makes sense; therefore, not all apparent self-reference is problematic, though in the present case we are left with a problem of analysis or translation. I think that B should be translated as: ‘the sentence ‘this statement is short’ is short’. Why should we accept this translation? Because it follows from the rule, hidden in this case, that the length of a statement is to be measured by the number of words in the sentence expressing it. But now that this rule has been made explicit, the apparent self-reference is shown to be illusory, the statement referring to itself only as a form of shorthand.

            Similarly, the statement: ‘I am writing these words of my own free-will’ may be translated as: ‘I am writing the words I am writing these words of my own free-will of my own free-will, as in the remainder of this confession or account.’

            To take another example, consider an otherwise empty sheet of paper on which are displayed the statements C: ‘2+2 = 4’ and D: ‘all statements on this page between noon and 1pm are false’. Clearly there is no paradox about D if the page is read before noon, since it is possible that D will disappear between then and 1pm, in which case it will prove to be false. But suppose that at noon it remains on the page, strange to say, and now we declare that D has become paradoxical by referring to itself. Then what we need is a linguistic rule that will remove the air of paradox, and this is to be found in the conventional use of such words as ‘statement’ and ‘proposition’.

            When I assert E: ‘this statement is false’, or ‘true’, then by convention I refer to a previous or future statement, the existence of which I imply, and which I state to be false – or true. It follows that E may be true or false on two counts: if it successfully refers to another statement, that statement may be true or false; if it does not, then it falsely implies that it does and is therefore false. To make E appear more self-referential, consider F: ‘this present statement is false’. It could still refer to another statement, in which case it is unobjectionable, but if not then it is false, because the words ‘this present statement’ do not constitute a statement; therefore, if at the same time they fail to denote another statement, as they imply that they are able to do, then F is false.

            Using this analysis, let us return to the sheet of paper on which is written C: ‘2+2 = 4’ and D: ‘all statements on this page between noon and 1pm are false’. Ignoring the time restriction, which no longer matters, D refers to C, so that D is false.

            Now consider the proposition G: ‘all statements are false’. On our analysis there is no paradox, since G refers to all statements H, I, J, …. So, we are free to ask not whether G is true, since we know it is not, but whether it is self-consistent. Well, no it isn’t, and the reason is that all statements need a system of language and reasoning in which to make sense. Thus, if G is true, then no statement can be verified, both because ‘some statements are verifiable’ is itself a statement, thereby condemned as false, and because no statement can be verified without generating other statements, also condemned if G is true. One does not need to be a verificationist to conclude that G is incoherent.

            If this is correct, does it follow that at least one statement is true? Not formally, since G being incoherent is not the same as its being false, from which, of course, it would indeed follow that at least one statement is true. Nevertheless, it does follow, and the reason is that one statement depends on others for its meaning, which themselves depend on others, all forming a network in which the original statement finds a place. But the question whether at least one statement is true is a question within that system, so that the conditions for asking the question are such that the answer must be ‘yes’.

            Descartes said that he thinks, therefore he exists; instead, perhaps he should have said that he forms propositions; therefore, it cannot be denied that at least one of them is true. In that case it cannot be denied, either, that the statement ‘all memories are false’ is false, since we must agree that we remember it. But then, there is nothing especially secure about our memory of language events; so on what grounds could we be sceptical about memory, as against particular memories? And so on.