Truth

Unfortunately, this somewhat vague definition, natural as it may seem, cannot be more than an approximate account of truth. As any lawyer can tell you, truth is a problematic concept and difficult to define. The problems start with the notorious ►self-referential sentence "This sentence is not true". The problem here is that however we define truth, this sentence's being false logically entails that it is true, while its being true entails that it is false, suggesting a logical contradiction in the heart of truth itself.

A related puzzle concerns the limits of our ability to recognize whether something is true. Certainly, truth is not an independent object but a property — more specifically, a property of statements, claims, or beliefs. (We ignore here related uses of "true" by which one may speak of, for example, a true friend, or of a gun's having true aim.) But how can we recognize this property? Can we establish a set of general rules determining when something is true and when it is false and write these rules down on a finite sheet of paper? Or do we need an infinite amount of time to find out about the truth of a statement? Let us explore this problem right away.

Let us assume that there is a truth machine — a powerful computer, say — into which we can feed this or that statement (where a statement may consist of a sentence, a larger document, or even an entire book). This computer would then analyze the statement introduced and infallibly recognize whether it is true or false. If the statement is true, the word "TRUE" flashes up on a display panel; if it is false, the word "FALSE" flashes up instead.

Obviously, the computer program must not only be able to read and understand sentences and books; it must also contain a complete definition of truth. If the definition is finite, we can print it out on a large sheet of continuous paper.* Mischievous as we are, we add to this printout the following line:

"A computer running the above program will not judge this document to be true."

A Mischievous Experiment

Now we feed the augmented document back into the computer and wait for the output. What will it be? "True"? "False"? Black smoke? The computer cannot judge the document to be false, since by doing so it would confirm the document (establish it to be true), disqualifying the computer as a truth machine. Conversely, the computer cannot judge the document to be true, either, since that would obviously show the document to be false, contrary to the computer's evaluation. Thus the truth machine will be wavering between true and false until it blows a fuse.

And that eventually confirms our document: The computer has failed to evaluate it as true. Thus, the document was true and the truth machine was unable to detect this — a paradox of truth. Or does our truth program merely have a ►bug? If so, we can solve this problem by building an even larger computer that runs an improved program and feeding this computer with our document. Since the program print-out now no longer refers to the computer itself, the new computer has no problem confirming the truth of the document. The situation changes again, however, if we compile a new document referring to the print-out of the new, improved truth program.

Clearly we need an infinite series of computers and truth programs to discover the truth of each and every document. No finite program is ever capable of fully covering the seemingly simple concept of truth. The sentence "This sentence is not true", followed by a definition of truth, would together constitute an infinitely long construction.

Now, we may be tempted to conclude that truth machines fail to distinguish between truth and falsity only in those cases in which their own blueprint or program is involved. But this is incorrect. In the 1930s Kurt Gödel proved that in all complex systems there are statements whose truth or falsity cannot be decided within the respective system. An example of such an undecidable statement in mathematics is the ►continuum hypothesis. Hence, the paradox of truth is no mere sophistical curio, but is one of the roots of Gödel's incompleteness theorem.

* Continuous paper is not really continuous; otherwise it needed to be infinite, or at least unbounded.