Re: Tautologies Then and Now

From: paul (paul8801_at_on-ramp.nl)
Date: 12/07/04 

Date: Mon, 06 Dec 2004 23:15:37 -0500

On Tue, 07 Dec 2004 00:17:46 GMT, "Stephen Harris"
<cyberguard1048-usenet@yahoo.com> wrote:

>
>"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:

  Right, it restricts tautologies to propositional logic, like I said.
OTOH, formulae in first-order predicate logic that are true on all
valuations are called "universally valid."

>
>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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