Wednesday Jun 06, 2007

Truth -- What's Consistency Got To Do With It?


Frege used a 2-D notation for his logic.

Many years ago, while with the Graduate Group in Logic and Methodology of Science at Berkeley, I had written a little paper with a tease of a title: "Consistency--What Is Logic Got To Do With It?"

After a recent discussion, I realized more clearly how that paper might have been making too far of a leap in logic.

Perhaps, it should have been named "Consistency--What Is Truth Got To Do With It?"  By "consistency," all along, I had meant "consistency" as understood by mathematicians and scientists, not what we understand "consistency" to be when we speak of, say, "consistent" behavior, which is a totally different concept when compared to a "consistent" theory.


In the mathematicians' definition of truth, as expounded by Alfred Tarski, one can only speak of "truth" within the confines of a mathematically "consistent" theory. Thus, one arrives, in mathematical logic, at incomplete theories that meet mathematical "truth" criteria even as they remain incomplete. In a sense, their incompleteness is more true about these theories than their "truth." Their "truth" exists within their limited, incomplete domain, which, always remains positively finite or countable or of much lower cardinality than the continuum of claims that they can neither prove nor disprove.

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