What is a predicate in first-order logic, formally?

Question

Recently, I am trying to gather together a consistent 'foundation of mathematics' from several sources/books/scripts which satisfies me as a 'working mathematician'.

When, for example, I want to start with set theory, I have to formulate the axioms of ZF (or ZFC). For this - in particular for the axiom schema of specification - I need what is apparently called First order logic (cf. also my question here on stack exchange).

When I want to write down what the definition of first order logic is, I do not understand what a predicate is, and by 'is' I mean what it is formally. There is the common example of

All humans are mortal
Socrates is human
therefore Socrates is mortal

where the used predicates $P$ are $\mathrm{IsMortal}$ an $\mathrm{IsHuman}$. So for example if you plug in $\mathrm{Socrates}$ then $\mathrm{IsMortal}(\mathrm{Socrates})$ is $\mathrm{true}$ - or say is $1$. Formally, a predicate (or perhaps better: the semantic of a predicate, see below) looks to me like a function $$ P\colon \mathrm{Everything}\to \{0,1\} $$ but there are two problems: $\mathrm{Everything}$ is not a set and even if it were, we are here BEFORE introducting ZFC and defining what a set is. If we already had the set theory of ZFC in place, we were able to formulate what a function $\varphi: A \to \{0,1\}$ is: A subset $\varphi\subseteq A\times\{0,1\}$ such that for all $a\in A$ there exists a $t\in \{0,1\}$ with $(a,t)\in \varphi$ and this $t$ is unique for the $a$.

In first-order logic there is a distinction between syntax and semantic, fine. Even though it feels a little uncomfortable to me to define the syntax of first-order logic without the notion of a set, I am fine with this. OK, now we know what a '(well-formed) formula' is but this has no meaning: Just a string of symbols. In particular, we can write $P$ for some predicate - but just as a symbol.

I guess, my question then is about the semantics:

What is a predicate (or the semantic of a predicate) in first-order logic, formally?

Do I really have to describe what (the semantic of) a predicate is by writing 'in english words' what a function $P:\mathrm{Everything}\to \{0,1\}$ would be if this concept already existed? For example the $\mathrm{IsMortal}$-predicate would be in a semantic (sorry, I struggle with the correct wording here) to specify for all $x\in \mathrm{Everything}$ whether $\mathrm{IsMortal}(x)$ is true or is false.

In particular, the notion of a 'function' $P:\mathrm{Everything}\to \{0,1\}$ (which even does not exist, strictly speaking) would be more fundamental than the notion of a set. This is very unsatisfactory to me. I hope I was able to describe my problem that it becomes understandable. Thanks for any help.

Answer 1

Posting as an answer because of formatting, but this is essentially a comment.

Here is a quote from the HoTT book that could be of some use to you:

Informally, a deductive system is a collection of rules for deriving things called judgments. If we think of a deductive system as a formal game, then the judgments are the “positions” in the game which we reach by following the game rules. We can also think of a deductive system as a sort of algebraic theory, in which case the judgments are the elements (like the elements of a group) and the deductive rules are the operations (like the group multiplication). From a logical point of view, the judgments can be considered to be the “external” statements, living in the metatheory, as opposed to the “internal” statements of the theory itself.

In the deductive system of first-order logic (on which set theory is based), there is only one kind of judgment: that a given proposition has a proof. That is, each proposition A gives rise to a judgment “A has a proof”, and all judgments are of this form. A rule of first-order logic such as “from A and B infer A ∧ B” is actually a rule of “proof construction” which says that given the judgments “A has a proof” and “B has a proof”, we may deduce that “A ∧ B has a proof”. Note that the judgment “A has a proof” exists at a different level from the proposition A itself, which is an internal statement of the theory.

If I was to agree with this then I'd come to the conclusion that this level "above" set theory truly is syntactic in nature. And this might as well be the answer to your question — you won't be able to find the semantics you want, or rather it might be best to settle for no semantics in this situation (cf. quote below). But I'm not speaking with any authority, just sharing my thoughts on this.

Another thought: just because something has its interpretation in ZF it doesn't mean it doesn't precede it. You can definitely capture the notion of a predicate defining it as a function but the notion already existed independently within the "deductive system" you're working in, which here is first-order logic.

Here is another quote from a really insightful answer to the question What is a predicate exactly in predicate logic

It is important to note that predicate logic's origins are deeply intertwined with the origins of the notion of set (or class, or collection). For this reason it can be difficult for modern mathematicians to appreciate that prior to the efforts of Skölem to base set theory on first-order logic, predicate logic sometimes referred to a semantic theory, sometimes a syntactic theory, sometimes a vague combination of syntax and semantics all of which were sometimes first-order, sometimes second-order, and sometimes something quite different. (a more detailed history of the development of first-order predicate logic can be found in Gregory H. Moore's The Emergence of First-Order Logic.)

Modern mathematicians often use predicate logic to refer to any of a number of equivalent Hilbert-like formalisms which include the axioms of a formalized sentence (propositional) logic together with axioms containing implications of quantified sentences (e.g. P(x)→(∃y)P(y) and (∀x)P(x)→P(y)) and inference rules containing implications of quantified sentences. The use of the term "predicate logic" to refer to a semantic theory is almost entirely a thing of the past, as we now use the term "Model Theory".

Finally, the most clear and exact description of predicate is to be found in a formal language the most common of which is a Hilbert type formalism where the word "predicate" is used as a classification word for signs (symbols) used according to specific rules of the formal language.

Answer 2

In mathematical proofs, you don't have to think of a predicate as a function with a range of $\{0, ~1\}$. It may even be counterproductive, as you seem to be discovering.

If $P$ is the name of a proposition, then the statement "$P$" by itself can be informally interpreted as stating that proposition $P$ is true, whereas "$\neg P$" by itself can be interpreted as stating that $P$ is false.

Similarly, if $P$ is the name of a unary predicate, then the statement "$P(x)$" by itself can be informally interpreted as stating that $P(x)$ is true, whereas "$\neg P(x)$" by itself can be interpreted as stating that $P(x)$ is false.

Answer 3

Attempt at an answer, elaborating on what I said in my answer-comment:

It seems that the fact that you can define a predicate within set theory causes the confusion, but what I understand from everything I quoted in my answer-comment, the predicate that "precedes" set theory is actually the predicate from the deductive system we used for stating the axioms of set theory, which is here first order logic. So if you want a satisfying notion of a predicate you need to choose the version of first order logic which defines/uses the notion of the predicate in a way that doesn't lead to circularity as the function example does. This great answer says that

the most clear and exact description of predicate is to be found in a formal language the most common of which is a Hilbert type formalism where the word "predicate" is used as a classification word for signs (symbols) used according to specific rules of the formal language

And if I'm not mistaken, this really would be a notion of a predicate that's syntactic in nature (cf. what I said and quoted in my other answer).

In summary: there's a notion of predicate that exists in terms of set theory but that's just an, loosely speaking, equivalent notion of the predicate from the deductive system/formal language we used for stating the ZF axioms. And if you want a "predicate" independent of set theory, you want to adopt the formal language that has it, and then state ZF axioms in it.

That being said, I'm curious to know what formal notion of the predicate did these authors of "foundations of mathematics" had in mind, is it too crazy to think that some didn't bother with it? I don't know, I'd like to see someone clarify this further.

I just looked up the first book on first order logic that popped up, and it's First-Order Logic by Raymond M. Smullyan and he really does use set theoretic concepts to introduce workings of the predicate.

By an interpretation I of E in a universe U is meant a function which assigns to each n-ary predicate P an n-place relation P* of elements of U. An atomic U-sentence P$\mathcal E_1' ... , \mathcal E_n$ is called true under I if the n-tuple $\mathcal E_1' ... , \mathcal E_n$ stands in the relation P*.

My guess is that this is a "rebirth" of the predicate using set theory as the formal language, as opposed to the primordial first order logic (this is the one you need to pick to be satisfied) which was actually had in mind when ZF was stated. Another guess is that the reintroduction of first order logic in set theoretic suit is because of practical reason, assuming we'll be using ZF as our framework, we may as well embed the meta-framework that is first order logic into the framework in which we work, so that we don't have to be switching between them all the time (or something like that).