September 21, 2006

Löb's Legacy

Richard Zach's Log Blog has brought to my attention the recent death of Martin Löb. It is fitting that we think here about his work. The famous Löb Theorem (in "Solution of a Problem of Leon Henkin" JSL, 1955) generates Löb's paradox (ibid.), which goes something like this:

(*) If this sentence is true, then so is A.

Notice that (*) is provable for an arbitrary proposition A. Here's the proof. Suppose (*) is true. Then it satisfies its own antecedent. So it follows that A. By conditional proof, if (*) is true then so is A. And that is just to say that (*) is true.

Notice also, that the provability of (*) underwrites the truth of any proposition A. For this reason and since both (*) and its proof are negation-free, Löb offers (*) as a test for inconsistency of negation-free languages (that allow self-reference).

Löb credits an anonymous referee for extracting the paradox and the insight about how to test for inconsistency without negation. Curry (1942), and not Löb (1955), usually gets credit for the above insights. Johan van Benthem ("FourParadoxes" JPL 1978), however, argues that the Löb+Referee insights were developed independently of Curry's work. Moreover, the Curry paradox is treated by Curry and his students as a feature of formal systems only, whereas Löb's paradox is a natural language paradox.

An interesting loose-end is the identity of the 1955 anonymous referee that extracted the paradox from Löb's Theorem.

September 18, 2006

Call for Papers: Knowability and Beyond

CALL FOR PAPERS (May 1, 2007)

Knowability & Beyond
Special issue of Synthese
  • Can there be non-actual knowledge of what is actually the case?
  • Is the concept of knowability basic or is it semantically decomposable into knowledge and (alethic) possibility?
  • Should an intuitionist find a way to express an existential commitment to some ignorance and undecidedness?
  • Are there more truths than knowables?
These are possible topics for a special issue of Synthese that I will be editing. For a description of the issue and further details please go here.

September 11, 2006

Visitors at Knowability

Sporadically one will find here at Knowability a post authored by someone not identical to me. The explanation is that, although this is not currently a guest blog, I will be hosting visitors now and again. I look forward to these visits.

September 06, 2006

The Logical Necessity of Some Ignorance

Modest modal epistemic reasoning reveals the equivalence of the following two principles:

(1) Any truth can be known.

(2) All truths are known.

Jon Kvanvig, in his latest book The Knowability Paradox (2006), poses a challenge to anyone who accepts the validity of the reasoning---viz., explain the loss of the apparent logical distinction between these two principles. Here's one way to go.

It is logically necessary that each of the above propositions is false. Logically necessary propositions often appear to express different thoughts, especially if one involves a concept not involved in the other.

Why think that the above propositions are necessarily false? An appendix to Nicholas Rescher's Epistemic Logic (2005) inspires an answer. It is a logical fact that there are more truths than knowables. Knowledge requires thought, and we could at most think a countable number of propositions. However, the true propositions themselves are uncountable. A diagonalization argument is required to make this stick, but it shouldn't be difficult to construct one for, say, a class of truths about the rational numbers. So if (i) there are more truths about the rational numbers than things that can be known about the rational numbers and (ii) the proof of this rests on no non-logical facts, then it is logically necessary that there is an unknown truth. (2) is logically false. And by the same reasoning so is (1).

The apparent logical distinction is explained by the fact that (2) seems on first glance to be stronger than (1). But the appearance is the result of not immediately recognizing that both propositions are logical falsehoods.