Re: Tautologies Then and Now

From: Stephen Harris (cyberguard1048-usenet_at_yahoo.com)
Date: 12/07/04 

Date: Tue, 07 Dec 2004 00:17:46 GMT

"paul" <paul8801@on-ramp.nl> wrote in message
news:j1j9r0hnmc3kb0m15c7jss9n7n5dsm5pl9@4ax.com...
> On Mon, 06 Dec 2004 20:28:57 GMT, "robert j. kolker"
> <nowhere@nowhere.com> wrote:
>
>>
>>
>>paul wrote:
>>>
>>> They're called "universal truths," not tautologies, in the
>>> first-order predicate calculus.
>>
>>They are true under all standard interpretations. That is how truth
>>tables are extended to the first order logics.
>
>
> Can you cite a text that extends truth tables beyond propositional
> logic? My professors always said that doesn't happen, and it certainly
> didn't in any of my texts.
>
> - paul
>

These are the definitions:

http://www.earlham.edu/~peters/courses/logsys/glossary.htm
"This glossary is limited to basic set theory, basic recursive function
theory,
two branches of logic (truth-functional propositional logic and first-order
predicate logic) and their metatheory." ...

Tautology.
"A logically valid wff of truth-functional propositional logic. A compound
proposition that is true in every row of its truth table or in every
interpretation. See contingency; contradiction; logical validity; semantic
tautology; syntactic tautology.

Tautology schema (plural: schemata). A formula containing variables of the
metalanguage which becomes a tautology when the variables are instantiated
to wffs of the formal language.

Semantic tautology.
A wff of truth-functional propositional logic whose truth table column
contains
nothing but T's when these are interpreted as the truth-value Truth.
See syntactic tautology.

Syntactic tautology.
A wff of truth-functional logic whose truth table column contains nothing
but
T's when these T's are uninterpreted tokens rather than, say, truth-values.
The rules for generating the truth table column tell us to use one of these
uninterpreted T's in exactly those cases where semantic considerations would
have led us to use the truth-value Truth. See semantic tautology."

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