Does current foundation of first order logic need a fundamental change?

Question

Note the following (not too exact) correspondence between natural and formal languages.

---

a. In a natural language we begin with a set of alphabets.

a'. In a first order language we begin with a set of symbols.

---

b. In a natural language we construct (meaningful/legitimated) words from alphabets using particular rules. So an arbitrary finite sequence of alphabets is not necessarily a meaningful word.

b'. In a first order language we construct (meaningful/legitimated) terms from symbols using particular rules. So an arbitrary finite sequence of symbols is not necessarily a meaningful term.

---

c. In a natural language we construct (meaningful/legitimated) sentences from words using particular rules. So an arbitrary finite sequence of words is not necessarily a meaningful sentence.

c'. In a first order language we construct (meaningful/legitimated) sentences (formulas) from terms using particular rules. So an arbitrary finite sequence of terms is not necessarily a meaningful sentence (formula).

---

d. In a natural language we construct (meaningful/legitimated) texts from sentences using particular rules. So an arbitrary finite sequence of sentences is not necessarily a meaningful text.

d'. In a first order language we construct (meaningful/legitimated) theories from sentences without any rules. So an arbitrary (finite or infinite) set of sentences is a theory.

---

Question 1: Why the line of producing new legitimated objects using former and simpler legitimated objects is broken in theories of first order logic?

Question 2: Are there logics with particular rules for producing legitimated theories from sentences?

Question 3: Is there a reasonable criterion to determine which sequence of first order sentences is a legitimated first order theory?

---

Answer 1

(This is not quite an answer, but you might still find it useful enough.)

You need to discern between syntax and semantics. While the sentence "The dog programmed a cat to force a power set" is syntactically correct (I hope), it is semantically meaningless.

In first order logic, the theory $\{p,\lnot p\}$, while formally a theory (it is a set of well-formed sentences, given $p$ is such) it is inconsistent and therefore semantically meaningless.

We are not interested in every theory, we are interested in the theories which are not inconsistent, or at least not exhibiting obvious proofs of inconsistency¹ (assuming some reasonable foundational theory in the background, e.g. $\sf PA$ or $\sf ZFC$).

So your point, while correct, misses the point. Meaningfulness is semantic consistency, and in first-order logic we have the completeness theorem which tells us that a theory is consistent if and only if it has a meaning.

It seems to me, therefore, that all your questions are about consistency. That we should allow creating theories only when we can ensure they are consistent (and indeed in one model theory course that I took a theory was always assumed to be consistent within the definition).

For this the compactness theorem is wonderful. It tells us that a theory is consistent if and only if the conjunction every finite fragment is not a false sentence. Which gives us a wonderful criterion for meaningful theories.

---

Footnotes.

1. Much like humans, we are interested in information which sounds meaningful, but after some investigation we may conclude that it is pure nonsense, this is the analogy to theories which we cannot prove their consistency - but have not disproved them yet either.

Answer 2

I think that there is no reason to expect the analogy to continue with d and d'.

The intent of a formal theory is just to assert that the sentences it contains are all true. Even if we assume that the intent of uttering a natural language sentence is to assert its truth, the intent of a natural language text is usually not just to assert that the truth of all its component sentences. In particular, in a natural language text it is desirable to follow a kind of natural progression from one sentence to the next.

Probably a better analogy would be between formal proofs and natural language texts. Like a formal theory, a formal proof can be represented as a sequence of sentences, but unlike a formal theory there are rules and conventions for its formation, and it is supposed to follow a logical progression from one sentence to the next.