From owner-cse663-fa08-list@LISTSERV.BUFFALO.EDU  Sun Sep  7 13:41:14 2008
Date:         Sun, 7 Sep 2008 13:41:06 -0400
From: "William J. Rapaport" <rapaport@cse.Buffalo.EDU>
Subject: 663: SCOPE, FREEDOM, AND BONDAGE
To: CSE663-FA08-LIST@LISTSERV.BUFFALO.EDU

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Subject: SCOPE, FREEDOM, AND BONDAGE
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Here are more precise definitions of "scope" and of "free" and "bound"
occurrences of variables in quantified wffs of FOL.

	Digression:  In what follows,

	(1) "Q" will be any quantifier, either the universal quantifier
	(here written as "A") or the existential quantifier
	(here written as "E") (and I'll use brackets to delimit the wff
	that follows the quantifier+variable phrase, instead of B&L's
	"."),

	(2) "v" will be a metavariable ranging over variables, and

	(3) "Z" and "Y" will be a metavariables ranging over predicates
	(I should use "alpha" and "beta", but I can't in this font).

	(4) "is(df)" means:  is by definition

	(5) "=df" means: is defined as

First, there are two slightly different definitions of "scope" floating
around in the literature.

        S1:     Let Qv[Z] be a wff.
                Then Z is(df) the scope of (that occurrence of) Q.

        S2:     Let Qv[Z] be a wff.
                Then Qv[Z] is the scope of (that occurrence of) Q.

B&L do not define "scope"!  But their definition is consistent with S2
(whereas I had given you S1 in lecture).

Following S2, we can then define freedom and bondage of variable
occurrences as follows (these definitions are based on:
Kalish, Donald; Montague, Richard; & Mar, Gary (1980), _Logic:
Techniques of Formal Reasoning, 2nd Edition_ (New York:  Harcourt Brace
Jovanovich).

Def:    Let Y,Z be wffs.
        Then an occurrence of a variable v is bound in Z =df
             that occurrence stands within an occurrence in Z of a
             wff of the form Qv[Y].

Def:    Let Z be a wff.
        Then an occurrence of a variable v is free in Z =df
             that occurrence stands within Z but is not bound in Z.