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

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?

• I would actually argue it's a strength of first order logic that it places practically no constraints on what counts as a theory. Perhaps even the primary strength. – Malice Vidrine Nov 24 '13 at 1:59
• I don't understand your a--a' and b--b correspondences. All written natural languages consist of one- or two-dimensional arrays of symbols from a finite alphabet ("alphabet" being a set of symbols broadly interpreted to include boundary indicators, punctuation, diacritics, typographic distinctions, Chinese characters, etc., as required). The atoms of a natural written language, just as with a formal language, are thus symbols, not sets of symbols (or alphabets). – John Bentin Nov 24 '13 at 14:51
• Most natural languages have never been written. The natural form of every natural language is either speech or, in the case of Nicaraguan Sign Language, gesture. It isn’t even true that the atomic elements of natural languages are words, since there is no satisfactory cross-linguistic definition of word. You might be able to justify a rough equivalence between mathematical symbols and morphemes, though even that is pretty shaky. That said, I agree with Trevor that proofs are a better analogue of texts. – Brian M. Scott Nov 25 '13 at 1:26

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.

• Your correspondence between formal proofs in first order language and texts in natural languages is interesting. Is there any similar structure in natural languages for first order theories? – user108850 Nov 24 '13 at 3:58

(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.
• The not too exact phrase in my first sentence refers to this distinction between "syntax" and "semantics" which you correctly expressed. But the soul of my question is about the method of constructing new complicated syntactically legitimated objects from simple ones in first order logic which is suddenly broken when we want to produce theories. The way which we define theories in first order logic shows a deep difference with the ways which we define other lingual objects like "symbols", "terms" and "formulas". – user108850 Nov 24 '13 at 2:29
• But the point is that in natural language we may construct syntactically correct, but meaningless texts. It seems that the tool you want is consistency. Again... you can merge two theories if the union is consistent. – Asaf Karagila Nov 24 '13 at 2:34
• I infer from item d in the question that you do not consider poetry to be meaningful/legitimated texts, because poets would undoubtedly object to the idea that there are rules governing how they can assemble sentences (or even requiring them to use sentences). Even apart from poetry, I would be interested to see the rules governing how texts can be assembled in, say, postmodernist philosophy. (I won't object if you declare those texts meaningless, but declaring them illegitimate might be going a bit too far.) – Andreas Blass Nov 24 '13 at 2:53
• I see the point you said. But I feel something is wrong here. Based on the natural meaning of the word "theory", a theory/story/poem ... is not just an accumulation of some sentences without any discipline. There is a structure on the sentences in a particular text of the natural language like a theory/story/poem. – user108850 Nov 24 '13 at 3:04
• @SaintGeorg: Nothing is wrong here. All you’re saying is that in first order logic the word theory has a precise meaning that does not include all of the connotations of the everyday sense(s) of the word. That’s not at all unusual when everyday words are given precise technical meanings in some discipline. – Brian M. Scott Nov 25 '13 at 1:32