SUO: Re: Critique Of Non-Functional Reason
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| A. Automated Reasoning (AR)
|
| The standard will be suitable for automated logical inference
| to support knowledge-based reasoning applications.
| B. Inter-Operability (IO)
|
| The standard will provide a basis for achieving Inter-Operability
| among various software and database applications.
[A]
Examples to be used again:
| 9. "All men are mortal" implies "all white men are mortal",
|
| 13. All men are mortal => all white men are mortal.
And now, back to our story:
| The relation of implication in one fairly natural sense of the term,
| viz. 'logical implication', is readily described with help of the
| auxiliary notion of 'logical truth'. A statement is logically
| true if it is not only true but remains true when all but its
| logical skeleton is varied at will; in other words, if it is
| true and contains only logical expressions essentially, any
| others vacuously (cf. Introduction). Now one statement may
| be said logically to imply another when the truth-functional
| conditional which has the one statement as antecedent and the
| other as consequent is logically true. Thus (9), so construed,
| is equivalent to:
|
| (13) is logically true.
|
| Quine, 'Mathematical Logic', page 28.
|
| Willard Van Orman Quine,
|'Mathematical Logic', Revised Edition,
| Harvard University Press, Cambridge, MA,
| 1940, 1951, 1981.
Notes On Notation, Translation, Transcription.
Quine is drawing a significant distinction
between "connectives" and "predicates":
1. The most commonly employed connectives
are symbolized in this transcription as:
"~", "&", "v", "=>", "<=>".
These are the "truth-functional connectives" that Quine refers to as:
"denial", "conjunction", "alternation", "conditional", "biconditional",
respectively.
In English, the statement connectives can be used to bind
a list of statements into a new statement by interweaving
the list with a suitable sequence of connective particles:
Denial. Ply "~" as "not ...", or "it is not the case that ..."
Conjunction. Ply "&" as "... and ..."
Alternation. Ply "v" as "... or ... ", or "either ... or ..."
Conditional. Ply "=>" as "... only if ...", or "if ... then ..."
Biconditional. Ply "<=>" as "... if and only if ..."
2. Predicates that make statements about statements are typically
expressed by interweaving a VP in the appropriate way through
a list of NP's that happen to name the statements in question.
For instance, running parallel to, but entirely distinct from,
the above connectives, we have the following k-ary predicates:
Negation. "... is false"
Compatibility. "... and ..."
Implication. " ... implies ..."
Equivalence. " ... is equivalent to ..."
In the present ASCII text, I am having to use the symbol "=>" for
Quine's (actually, Peano's rotated "C" for "converse consequence")
horseshoe symbol, and the symbol "<=>" for his underscored equals.
Some writers use "->" and "<->" for the corresponding connectives,
reserving the double-arrowed "=>" and "<=>" for the corresponding
predicates, commonly understood as the assertion of conditionals.
But I am double-barred from this convenient compact of practice
because my heavy deployment of functional arrows outweighs it.
Jon Awbrey
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