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I am completely new to Quine's theories and thank you for your patience in answering my novice questions.

In his paper "Ontological Relativity", Quine says:

Consider the case of a thoughtful protosyntactician. He has a formalized system of first-order proof theory, or protosyntax, whose universe comprises just expressions, that is, strings of signs of a specified alphabet. Now just what sorts of things, more specifically, are these expressions? They are types, not tokens. So, one might suppose, each of them is the set of all its tokens. That is, each expression is a set of inscriptions which are variously situated in space-time but are classed together by virtue of a certain similarity in shape. The concatenate x~y of two expressions x and y, in a given order, will be the set of all inscriptions each of which has two parts which are tokens respectively of x and y and follow one upon the other in that order. But x ~ y may then be the null set, though x and yare not null; for it may be that inscriptions belonging to x and y happen to turn up head to tail nowhere, in the past, present, or future. This danger increases with the lengths of x and y. But it is easily seen to violate a law of proto syntax which says that x = z whenever x ~ y = z ~ y.

Why does x ~ y being a null set violate the law of protosyntax?

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Frankly speaking, what I can imagine is this...

With concatenation we can form an expression x⌢y that has never been used (and never will be), and thus it is a type with no tokens, i.e. the null set.

If the same happens with z⌢y, we have that x⌢y=z⌢y, because both are the null set.

But x and z may be two different expressions, i.e. x ≠ z.

This violates the laws of syntax because if two expressions are equal and they have the same tail, their heads must be equal.

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