On Apr 16, 2009, at 6:32 PM, Bart Gajderowicz wrote:
> Here's the way I see the argument whether syllogisms, recursive or
> not, are valid, and why we can continue using programming languages
> based on the principles of turing machines.
>
> As with any proper recursive function, we need a stop condition. We
> of course can't use a turing machine to figure this out, as it will
> not stop if the program does not terminate. That's a theoretical
> issue. (01)
I am not understanding some of your terminology here. As standardly
(and pretty much universally) defined in logic, syllogisms are
arguments with two premises and a conclusion satisfying a certain
general form in which the notion of recursion plays no role whatever
(understandably, since the notion of a syllogism is largely unchanged
since it was first introduced by Aristotle). Could you say what you
mean by a recursive syllogism and perhaps give an example? (02)
Second, what do you mean by a "proper recursive function"? A
recursive function is simply a mathematical function from (n-tuples
of) natural numbers to natural numbers. (Through the magic of
encoding, of course, we can extend the notion to functions on, e.g.,
real numbers, strings, database records, etc.) A "stop condition" (if
I'm understanding you) is a programming construct that has nothing to
do with functions per se. Of course, it is possible to *characterize*
the class of recursive functions in terms of programming languages --
a recursive function is any function (from N^m into N) calculated by a
program in a language meeting certain conditions. But even then I am
confused by your comment that "any proper recursive function" --
understood to mean any proper *program* that calculates a recursive
function -- needs a "stop condition". For this appears to mean that,
for any recursive function, a program that calculates that function
has to terminate when executed. And that, of course, is false. (03)
Perhaps you could clarify. (04)
Chris Menzel (05)
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