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I do not get the intuition behind the structure (I think it's called "interpretation" in other texts) of a first order language. Here I use the following definition, given in Shoenfield's Introduction to Mathematical Logic:

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I do get the definitions, but I am at loss to why such definitions were invented in the first place, and some things are unclear to me. Specifically:

  1. I do not that the paragraph "We want to define a formula $A$ [called "well-closed formulas", or "wfs," in some texts"] to be valid... this leads us to the followong definitions." Could you provide a specific example that explains the intuition behind the definitions?

  2. It's written "for each individual $a$ of $\alpha$, we choose a new constant, called the name of $a$." Where does the new constant come from? Are we assuming descriptive set theory and choose an elment not in the $0$-ary function symbol and adding it to the language being considered?

  3. If $A$ is $p$ with $p$ a $0$-ary predicate symbol, then $\alpha(A)= T$ iff $\alpha(\emptyset)$ belongs to the predicate. However, we haven't defined what $\alpha(\emptyset)$ is,unless we are making another assumption that there exist an individual of $|\alpha|$ that has $\emptyset$ as its name. Am I mistaken?

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  • $\begingroup$ The language gives us syntax and the structure gives us the semantics of the formulas in the language. Therefore, the structure must provide interpretations for each of the symbols in the language. This allows us, by induction, to define the truth values of arbitrary formulas in the language. $\endgroup$ – John Douma Mar 1 at 17:33
  • $\begingroup$ Question 3 - usually (1) 0-ary function symbols are interpreted as constants and (2) 0-ary predicate symbols are interpreted as propositions, i.e. true or false. .....Question 2 - I am not sure I get the meaning of that question. $\endgroup$ – FWE Mar 1 at 21:30
  • $\begingroup$ @FWE thank you or your answer and comment. For (2), what I mean is that the mentioned statement seems to suggest that, assuming that the new constant is a $0$-ary function from the first order language $L$ we are considering, there actually exist a $0$-ary function. For example, if $L$ has no predicate symbols and no function symbols and $|\alpha|$ is the set $\{a,b,c\}$ then we do not have any constants to assign to $a$, say. The problem is that the definition allows us to have more elements in $\alpha$ than the number of constants. Also, since "different names are chosen for different indivi $\endgroup$ – Cute Brownie Mar 2 at 5:08
  • $\begingroup$ individuals," it seems like this implies that $|\alpha|$ cannot be of larger cardinality than the set of $0$-ary constants of $L$, unless we are allowed to assume that for any given set $A$ there exist an element not in $A$. If it's the latter, we are actually allowed to have, for example, $\{p,q\}$ as the set of $0$-ary functions and $\{a,b,c\}$ as $|\alpha|$ (I forgot to mention, in Shoenfield's book, a "constant" is defined to be a $0$-ary function symbol.) $\endgroup$ – Cute Brownie Mar 2 at 5:11
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1) "We want to define a formula $A$ to be valid... this leads us to the followong definitions." Could you provide a specific example that explains the intuition behind the definitions?

The intuition behind the definition of a formula $A$ being valid in a structure $\mathfrak A$ is quite "natural".

A structure is a piece of the "mathematical world" made of objects (e.g. natural numbers), properties (e.g. odd and even) and relations (e.g. less than) between them.

Thus, to interpret a language is to link the symbols of the language to objects and relations of the structure.

In this way, expressions (terms and formulas) of the language, when interpreted, have meaning: terms are names for objects, and formulas are statements expressing facts about objects.

To be valid in $\mathfrak A$ means that, according to the way we have chosen to interpret in $\mathfrak A$ the symbols of the language, the interpeted formula will express a fact that is true in the structure.


2) It's written "for each individual $a$ of $\mathfrak A$, we choose a new constant, called the name of $a$." Where does the new constant come from? Are we assuming descriptive set theory and choose an elment not in the $0$-ary function symbol and adding it to the language being considered?

Correct; for every "object" of the "universe of discourse", i.e. for every element $a$ of the domain of the structure $\mathfrak A$, we add to the language a new constant symbols a whose reference is the object $a$: thus, the symbol a is the "name" in the expanded language $L (\mathfrak A)$ of the object $a$.


3) If $A$ is $p$ with $p$ a $0$-ary predicate symbol, then $\mathfrak A(A)= \text T$ iff $\mathfrak A(\emptyset)$ belongs to the predicate. However, we haven't defined what $\mathfrak A(\emptyset)$ is,unless we are making another assumption that there exist an individual of $|\mathfrak A|$ that has $\emptyset$ its name.

If my memery is sound, the case for $0$-ary predicate symbols is not explicitly discussed in Shoenfield's textbook...

Having said, that, a $0$-ary predicate symbol is a propositional symbol, like those of propositional logic. Thus, the "natural" interpretation is trough truth-values: $\text T, \text F$.

We may choose to map $\text T$ on $|\mathfrak A|$ and $\text F$ on $\emptyset$, and this is consistent with the fact that the interpretation of unary predicate symbols of the language are subsets of the domain of the structure.

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