We address the question:

Do the semantic and logical paradoxes reflect an identifiable ambiguity in our expression and subsequent interpretation of a formal mathematical language and, if so, is the ambiguity removable?

We presume familiarity with the semantic and logical paradoxes [1] which involve—either implicitly or explicitly—quantification over an infinitude.

Where such quantification is not, or cannot be, explicitly defined in formal logical terms—eg. the classical expression of the Liar paradox as `This sentence is a lie’—the paradoxes per se cannot be considered as posing serious linguistic or philosophical concerns.

Of course, it would be a matter of serious logical concern if the word `This’ in the English language sentence, `This sentence is a lie’, could be validly viewed as implicitly implying that:

(1) there is a constructive infinite enumeration of English language sentences;

(2) to each of which a truth-value can be constructively assigned by the rules of a two-valued logic; and,

(3) in which `This’ refers uniquely to a particular sentence in the enumeration.

In 1931, Kurt Gödel used the above perspective in his seminal paper on `undecidable’ arithmetical propositions:

(a) to show how the infinitude of formulas, in a formally defined Peano Arithmetic P [2], could be constructively enumerated and referenced uniquely by natural numbers [3];

(b) to show how P-provability values could be constructively assigned to P-formulas by the rules of a two-valued logic [4]; and,

(c) to construct a P-formula which interprets as an arithmetical proposition that could loosely be viewed—under the standard interpretation of the Peano Arithmetic P—as expressing the sentence, `This P-sentence is P-unprovable’ [5], without inviting a `Liar’ type of contradiction.

However, even where the quantification can be made explicit—eg. Russell’s paradox or Yablo’s paradox—the question arises whether such quantification is constructive or not.

Russell’s paradox: Define the set S by \{All \hspace{+.5ex} x: x \in S \leftrightarrow x \notin x\}; then S \in S \leftrightarrow S \notin S.

Yablo’s paradox: Defining the sentence S_{i} for all i \geq 0 as `For all j>i, S_{j} is not true’ seems to lead to a contradiction [6].

For instance:

(i) In Russell’s case it could be argued cogently that the contradiction itself establishes that S cannot be constructively defined over the range of the quantifier.

(ii) In Yablo’s case it could be argued that truth values cannot be constructively assigned to any sentence covered by the quantification since, in order to decide whether S_{i} is true or not for any given i \geq 0, we first need to decide whether S_{i+1} is true or not.

There are two issues involved here—not necessarily independent—reflected succintly in the following perspective offered by W. T. Gowers in his talk, Does mathematics need a philosophy?’, presented before the Cambridge University Society for the Philosophy of Mathematics and Mathematical Sciences, 2002, wherein he remarks that:

“If you ask a philosopher what the main problems are in the philosophy of mathematics, then the following two are likely to come up: what is the status of mathematical truth, and what is the nature of mathematical objects? That is, what gives mathematical statements their aura of infallibility, and what on earth are these statements about?”

We thus need to address the questions:

On Mathematical Truth Is quantification currently interpreted constructively over an infinite mathematical domain?

On Mathematical Objects When is the concept of a completed infinity consistent with a formal language?

Notes

Return to 1: Commonly referred to as the paradoxes of `self-reference’, even though not all of them involve self-reference, e.g., the paradox constructed by Stephen Yablo (see below).

Return to 2: Kurt Gödel. 1931. On formally undecidable propositions of Principia Mathematica and related systems I. Translated by Elliott Mendelson. In M. Davis (ed.). 1965. The Undecidable. Raven Press, New York., pp.9-13.

Return to 3: ibid., p.13-14.

Return to 4: ibid., p.13.

Return to 5: ibid, p.37, footnote 67.

Return to 6: Stephen Yablo. 1993. Paradox without self-reference. Analysis, 53(4), pp. 251-252.

Bhupinder Singh Anand

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