Syntactically free, semantically bound. A note on variables.
Hugues Leblanc · Notre Dame Journal of Formal Logic · 1968
Apparent variables.The symbol (i (x).φx" denotes one definite proposition, and there is no distinction in meaning between "(χ).φx" and "(y).φy"when they occur in the same context.Principia Mathematica, Introduction, Ch.I.The old distinction between an apparent variable and a real one was never too clearly drawn.Passages from Principia Mathematica and earlier logic treatises suggest, though, that an individual variable X apparently occurs in a formula A if X occurs in A just for form, i.e., if X can be replaced salvo sensu in A by some individual variable foreign to A, and that X really occurs in A if X does not apparently occur in A. Thus, ζ x' apparently occurs in Russell's '(VΛΓ)/W (= ζ {x).φx'), since V can be replaced salvo sensu in '(Vχ)f(x)' by ζ y\ whereas ζ x' really occurs in 'f(x)\ The distinction between a bound variable and a free one, which eventually displaced that between an apparent variable and a real one, does not match it, all assertions to the contrary notwithstanding.An individual variable may-by current standards-occur free in a given formula, and yet not really occur therein by the above critierion.ι x\ for example, though it occurs free in £ (VΛΓ)/(ΛΓ) & /(#)', does not really occur in '(Vx)f(x) & /(#)', since it can be replaced salvo sensu in ί (^x)f(x) & fix) 9 by any one of ζ y', 'z', V\ ζ y", 'z", and so on, or-as we prefer to put it-since '(Vχ)f(x) &/(#)' is semantically equivalent to any one of '(Vy)f(y) & f(y)', '(Vz)f(z) & /(*)', and so on.1 Because of this discrepancy we would urge that an individual variable X 9 when it occurs bound (free) in a formula A by current standards, be said to occur syntactically bound {syntactically free) in A, and that a fresh distinction be introduced according to which: (i) X is said to occur semantically bound in A if A is semantically equivalent to any (hence, to