Simple question: is this a correct symbolic formulation of the idea that one symbol can't mean two different things at the same time?
((x = y) ∧ (x = z)) → y = z
If not, what would be a correct one?
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4You can not express what you say symbolically in first order predicate logic. The idea that symbols retain their referents in different places (or that they even have referents) is meta-theoretical, and first order logic simply lacks means to express it. In your formula x,y,z refer to objects, not symbols, that would require writing something like "x", etc. You can not even express that the relation symbol "=" means what you want it to mean, your formula simply says that the relation so symbolized is transitive.– ConifoldApr 27, 2017 at 20:55
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2 Answers
The answer is a bit complicated.
First, let's start with the symbolic formulation you wrote:
((x = y) ∧ (x = z)) → y = z
I assume that this is intended to be a formula in standard (first-order) logic. In words, this formula says: "If x is identical to y, and x is identical to z, then y and z are identical to each other." This is true. If y and z are identical to the same thing, they are identical to each other.
Identity is actually a fairly subtle concept, so we need to be careful in interpreting this formula. First of all, this formula neither asserts nor presupposes the existence of more than one object. In other words, the formula would remain true even in a world that had exactly one object. This is important because it's easy to get the sense that we are talking about three different things here: x, y, and z. But if the formula is true, we're actually only talking about one thing: the thing to which x, y and z are all identical.
A more helpful way of thinking about the formula 'x=y' is as follows. In order to tell whether this formula is true, we first ask: what does the symbol 'x' refer to? And what does the symbol 'y' refer to? If we find that these two symbols refer to the same individual object, then 'x=y' is true. However, there is a subtlety here also. What I just said is a description of what we do to figure out whether the formula is true. It is not what the formula itself says. The formula 'x=y' is not even talking about symbols or reference. The symbol 'x' is used in the formula, but it is not mentioned.
You asked: is this the correct symbolic formulation for the idea that one symbol can't refer to two different things? (I made one change to your question; I'll get back to that.) Well, it is true that one symbol cannot refer to two different things (in first order logic) 1. That is a fact about the semantics of first-order logic. And ultimately, it is this fact about the semantics of first-order logic that makes your symbolic formulation true. Because 'x' cannot refer to two different things, if it is coreferential with 'y' and 'z' then those terms are coreferential themselves. But your symbolic formulation does not say this.
I changed your question to be about reference rather than meaning. That's because, in first-order logic, formulas and symbols are not explicitly associated with meanings. A possible formulation for your original idea (in first order logic), then, is:
S(x) & M(x,a) & M(x,b) → a=b
Where 'S(x)' means 'x is a symbol," and 'M(x,a)' means 'x has the meaning a.' This formulation regards both symbols and meanings as first-order entities.
1 More precisely: one symbol cannot refer to more than one thing, given an assignment function.
You can denote the symbol or name of x by f(x) and the symbol or name of y by f(y). Then the implication f(x) = f(y) → x = y excludes that different objects have same name: If the names are equal, then the objects are equal.
