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Let $M$ be a model we want to study. The first order language we plan to use is $L$. Now we extend the language $L$ by adding a few constant symbols. Does the supplement of extra constant symbols strengthen our ability to talk about that model $M$?

I'm a newbie to logic, so I'm not sure about it, but I think that depends on the interpretation of the new constant symbols. There are 2 cases:

  1. The interpretation of the new constant symbols are all definable elements of the original language $L$. In this case, even if the new symbols are taken away, we can still make do with the formulas used to define those definable elements, so it makes no difference whether we are supplied these new symbols.

  2. The interpretation of the new constant symbols are not all definable elements of the original language $L$. In this case, we need those new symbols to talk about the undefinable elements, so they strengthens our ability to talk about the model $M$.

Am I correct?

In writing the above question, another problem hit me. What is the model $M$? Where does the model $M$ come from? Must it be something we can get/describe earlier in some language $\Bbb{L}$? Does that have something to do with the above question?

Michael
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1 Answers1

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I'd like to take up a conceptual aspect of the question; for a model-theoretic treatment at at the level of theorems, one may refer to the chapter Chang and Keisler have devoted in their book Model Theory, "Models Constructed from Constants".

In a formal language $\mathcal{L}$, individual constants are items of non-logical vocabulary and designate specific members of the structure's domain of discourse. Variables are items of logical vocabulary and either stands for an indeterminate member in the domain as a constituent of an open formula or sweeps over the domain (again actually, indeterminately) as a constituent of quantification. A view to better grasp the variables is due to Quine. According to Quine (Mathematical Logic revised edition, p.70):

The variables have no meaning beyond the pronominal sort of meaning which is reflected in translations [. . .]; they serve merely to indicate cross-references to various positions of quantification.

Hence, variables fulfil a cataphoric role that can be illustrated (indeed, replaced) by a diagram. I borrow an example from Button and Walsh's Philosophy and Model Theory (p. 14), since it is plainer and clearer than Quine's (I'd be grateful if anyone supplies MathJax code for it). Thus,

$$\exists x\forall y((\phi(x, y)\wedge\exists z\phi(x, z))\rightarrow\phi(y, x))$$

can be represented as

enter image description here

Consequently, in case that we need to designate a specific member of the domain, we have to conjure up a formula that defines it (see my post). However, if the member is not definable in the language, then we have to appeal to the extra-logical method and add the member among the individual constants.

Consider a familiar structural representation of the system of real numbers:

$$\mathfrak{R} =\langle\mathbb{R}, <, +,\cdot, 0, 1\rangle$$

The transcendental number $\pi$ is not definable by a first-order formula. We can expand the language of the structure $\mathfrak{R}$ by adding a constant symbol that represents and obtain $\mathfrak{R}^{+}$:

$$\mathfrak{R}^{+}=\langle\mathbb{R}, <, +,\cdot, 0, 1, \pi\rangle$$

In other words, individual constants act like nouns and variables act like pronouns of natural language. In case that pronouns do not suffice to determinately point to some specific object, then we can name it explicitly and include it into the "talk".

Tankut Beygu
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