What’s the starting point for defining predicates, relations and functions?

Question

A function $f$ is defined in the following way:

Let $A, B$ be sets. A function $f$ from $A$ to $B$ is a subset of $A\times B$ ($\forall_x(x \in A \implies x \in B)$) that satisfies $$\forall x,y,y'\,\bigl((x,y)\in f \land (x,y')\in f\implies y=y'\bigr). $$

The truth value of the "statement" $x \in A$ depends on whether $x$ is in $A$ or not. We define a sentence as a statement that is either true or false. And here we have a statement that its truth value depends on x, so what is the formal definition of a statement that its value depends on a free variable $x$? You might say that it is a predicate, but the definition of a predicate is based on that of relation, which themselves are defined using a statement of the form $x \in A$ as a relation is a subset of a Cartesian product (and "inclusion/subset" is defined using the statement $x \in A$) - it seems that we define a predicate which is a relation, but the definition of a relation has a predicate in it ($x \in A$), so it seems that we have a loop. How do we resolve this (that's what I meant in the original question when I asked "where do we start from?")? Are we not defining what a predicate such as $x \in A$ means formally in the beginning?

Do we begin with an informal notion of a predicate, which we define as a statement that can be true or false depending on the value of a variable $x$, and then use this informal definition to define what relations and functions are? What is the informal starting point, and what is a consequence that we base our formal definitions and structures from? Do we start with an informal definition of a predicate, or do we even treat functions informally to define predicates formally? Or is there another way to explain or propose a solution?

I understand that there is a difference between a predicate and a predicate symbol. I also understand that we define an interpretation function $I$ that $I(Pred) \subset D^M$ - sends $Pred$ to a specific relation. But when we define a relation there is a predicate in the definition of a relation, $x \in D$. I understand that we define the functions and relations using set theory, and that we have to use a background/ambient language which is set theory since we use functions and relations ($I, Pred^M$). But we use logic to define these concepts (relations, functions, predicates) - we use $x \in D$, quantifiers and logical connectives.

By "intuition/informal" I don't mean something vague like a gut feeling. I mean that we understand, in an informal but "precise" (as far as we human can) way, what is meant when we say "$ x \in A $", or "if...then...", "and", "or", "for all", etc. We use these notions "informally" (as we start from them with our understanding of what they mean) — but clearly — to define what a relation or function is, and then use those to construct the formal framework of logic.

(I'm not talking here about the syntax — I understand that when we talk about syntax, we treat symbols purely as symbols, with no additional meaning, and focus on forming well-formed formulas.)

I read How to avoid perceived circularity when defining a formal language? and the linked questions and answers there, but I still don't understand. I tried to read in Enderton's or Mendelson's books, but it only got me more confused. I would be grateful if someone could clarify things. Thank you!

Answer 1

Adding to my comments, to address your question more specifically, you can think of there being "two" set theories. The first is a theory in the sense of first order logic; It's syntactically correct sentences will contain for example

$\forall x(\neg(x \in x))$

and also

$\forall x(x\in x)$

Even though the second is not a theorem of typical set theories, it is still a syntactically correct sentence. Then, using the inference rules and the axioms of your chosen set theory, you will be able to conclude that the following, for example, are valid sentences:

1.$\forall x(\neg(x \in x))$

2.$\neg\forall x(x\in x)$

3.$\forall x(\forall y(\forall z((\forall t(t\in x\implies t\in y)\land \forall t(t\in y\implies t\in z)) \implies \forall t(t\in x\implies t\in z))))$

Fine, now you have some theory, this a construct which is completely independent of the "idea" of a set, it uses only the basic notions of expressions, and simple rules for deducing that one sentence is valid from some other set of valid sentences.

This is the set theory people typically think about when they're dealing with sets. Then, there is a completely separate set theory, this one is obtained in a complicated way:

First, note that we can define functions and relations as being sets satisfy certain first order sentences in the language described above. So there is some definable relation (a formula of one free variable in the language $\in$ ), $rel(x)$ which we think of as meaning "$x$ is a relation" and similarly for functions. Then we can define a formula, $FOl(X)$ (lowercase l) which we think of as meaning "the set $X$ is a language of first order logic", which will roughly look something like

"$x$ is a tuple $(F,R)$ where $F$ is a set of pairs $(f,n_f)$ where $n$ is a natural number..."

where we think of the pair $(f,n_f)$ as representing a function symbol of arity $n_f$ (note that $n$ is a natural number is also defined as just $n$ satisfying some formula in the language $\in$)

Then you can define a formula $Strct(X,Y)$ which we think of as meaning "$Y$ is a structure in the language $X$", which will look something like

"$FOl(X)$ and $Y$ is a tuple $(y,\mathcal{F},\mathcal{R})$ such that $\mathcal{F}$ is a function from the function symbols of $X$ (since it satisfies $FOl$) to the set of functions from powers of $Y$ to itself..."

Note still, that all of this can be defined using simply the symbol $\in$ and the standard logical symbols. Then you can go on to define the set of valid sentences over a language $X$, $Sent(X)$, by identifying sentences with sequences, and employing some sort of recursive definition, and now you can define what it means for a structure to satisfy a sentence, again, this is will just be some formula, $\models(S,Y)$, which we'll think of as meaning "$S$ is a subset of some $Sent(X)$ and $Y$ satisfies $Strct(X,Y)$ and ..." all of these, are simply formulas in the language $\in$.

Then finally you can define a "language of set theory" to be a set $X$, satisfying the formula in the language $\in$, which says that $X$ consistents if a single binary relation. Then using our encoding of sentences as sequences, we can define the set of axioms of some chosen set theory, over some "language of set theory", and using our formulas $Strct, \models, Sent, rel$ etc. We can define what it means for a set $X$ to be a model of the axioms of our chosen set theory. This is the second kind of "set theory", and you can see just from the definitions that this is something entirely different (and more complicated) than the first type, which was just a first order theory. The point is that model theory can be developed only on the weak assumption that we will agree on the syntactical correctness of expressions in the first order language $\in$. The rules for this correctness are very simple, and we've never encountered a situation where two mathematicians disagree on whether they've been applied correctly, so it seems like a very harmless assumption.

I've chosen to write the language as $\in$ throughout, instead of $\{\in\}$ to really emphasise that one doesn't need "set theory" to talk about first order logic. Of course, almost all of the detail has been left out, but any book on model theory will contain a decent description, which you can turn into a description in only the language $\in$, if you really want to. I can recommend the lecture notes https://akruckman.faculty.wesleyan.edu by Alex Kruckman

Answer 2

Regarding the new version of the question, and specifically wrt Edit #2:"do we start with "$\in$", the connectives and quantifiers, or do we start with something else?"

mathematical logic starts with the logical machinery: connective and quantifiers, and with the intuitive notions of relation (in general) and function (specifically mathematical concept).

In general, we have an intuitive understanding of basic logical notions (starting at least with Aristotle) involving "formal" patterns of reasoning.

And we have obviously an intuititive notion of relation: "x is father of y".

Less intuitive is the mathematical notion of function, that slowly evolved since 17th Century. See also Leibniz on functions.

Then we had a complex evolution, starting with Frege's analysis of language, based on functions and objects.

See at least Gottlob Frege, Philosophical Writings (Geach & Black edition - 1952) : Function and Concept (Über Funktion und Begriff - 1891), page 21-on.

He start his analysis from the "typical" mathematical fuctions, like :

$x + 1$

which get a number as "input" and produce as "output" a number, but then he generalize them to concepts, i.e. functions from objects to truth-value, like :

$x > 1$.

In this setting, a binary relation is a function from objects to truth-value with two argument places, like :

$x > y$.

See page 39 :

We have here a function whose value is always a truth-value. We called such functions of one argument concepts; we call such functions of two arguments relations.

In modern mathematical logic, the first-order language is based on individual variables and predicate letters.

An (atomic) expression is : $R(x,y)$, where $R$ is a binary predicate letter. In the language of arithmetic, we can use instead of the predicate letter $R$ the symbol ">" standing for the (binary) relation "greater than"; thus, we have the expression : $x > y$.

The basic difference between modern semantic for a first-order language an that of Frege is that today an expression like $x > y$ it is not interpreted as a function from couples of objects to truth-value but as the set of all couples satisying the relation.

If $\mathbb N$ is set of natural numbers which is the domain of our interpretation, we have that, writing $R^N$ for the interpretation of $R$, in the standard mathematical logic semantics :

$>^N \subseteq \mathbb N \times \mathbb N$

while for Frege :

$>^N : \mathbb N \times \mathbb N \rightarrow \{ \text {TRUE, FALSE} \}$.

The "shift" occurred progressively; already in Alfred North Whitehead and Bertrand Russell, Principia Mathematica (1910-1927) relations was a primitive notion (and functions only "special" relations : the "functional" ones).

In 1914 Norbert Wiener (A Simplification of the logic of relations) and in 1921 Kazimierz Kuratowski (Sur la notion de l'ordre dans la Théorie des Ensembles) find a way to define in the language of set theory the concept of ordered pair thus reducing the notion of relation to that of set.

See also Origin of modern definition of a function as a graph.

Thus, simplifying a lot, while for Frege a relation was a kind of function, in current mathematical logic a function is a type of relation which, in turn, is a special kind of set (which, for Frege, was the extension of a concept).

The subsequent evolution was due to an inteplay between logic and mathematics.

Mathematics used as a tool necessary to formalize logic: syntax and semantics. This has been done using basic mathematical machinery: number, function, set.

And logic used as a mathematical tool to investigate specific features of mathematical theories: model theory, proof theory.

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What about Edit #3: "The truth value of the "statement" $x \in A$ depends on whether $x$ is in $A$ or not", we have to consider the way formulas with free variables are evaluated in an interpretation. See e.g. Sentence ( closed formulas) vs formulas with free variables.

But specifically regarding the "definition of a predicate" we have again to disentangle the syntax: w ehave symbols for connectives and quantifiers, we have symbols for variables: $x_1,x_2,\ldots$, and we have symbols for predicates: $P_1,P_2,\ldots$.

All this is parte of the language whose syntax is defined mathematically with "basic" tools: numbers, functions, the language of sets, inductive definitions.

When we interpret a specific language, e.g. that used for a formula like $\forall x (x \ge 0)$, we choose a domain: e.g. the set $\mathbb N$ of naturals, a reference for the individual constants of the language: in this case the numebr zero for the symbol $0$, and the meaning of the binary predicate $\ge$.

In this case, the interpretation for the symbol "$\ge$" will be a relation on $\mathbb N$.