Maybe having numbers is an essential property of the universe. That is, the universe cannot exist without having numbers.

Again, definitions are sentences that give meanings to linguistic expressions. Definitions are what we use for meaning. So, definitions are not the same as meanings.

Some expressions are meaningful without synonyms. The term 'dog' is meaningful even without synonyms. 'Dog' can be paraphrased by some expressions, such as 'quadrupedal pet animal' and so on. When without such paraphrasing, 'dog' is still meaningful.
Synonyms are defined as 'linguistic expressions defined as the same thing'. Definitions are defined as 'sentences that give meanings to linguistic expressions'. Some synonyms are established by definitions. However, definitions are not established by synonyms. They are two distinct concepts.

Synonyms are the same expressions between linguistic expressions. Definitions are sentences that give meanings to linguistic expressions. Can you see the difference?

What you defend is merely synonymity. Synonymity is the sameness between expressions. However, meaning is not explained in that way. Meaning is broader than synonymity. Synonymity is one among which meaning includes.

Meaning is not equality. According to your theory, meaning is a kind of naming things, but it is not the case that naming things itself is equality. According to many others, meaning is not the same as equality.

There are lots of research results about 'means' in philosophical history. For example, Davidson interprets 'means' as 'is true'. So, Davidson suggests that 's means p' should be understood as 's is true if and only if p'. There are various attempts.
In your case, 'means' is a predicate. It's like other general predicates such as 'walk', 'have', 'hit', and so on. 'x means y' can be understood as ''x means y' is a two-place predicates'. Or 'x and y are in relation of 'means''. If you don't have a problem of 'x hits y', then you can accept easily 'x means y'.

It seems to be circular, but there's no trouble here. The form of 'definition of x' or 'meaning of x' is perfectly rational. We should distinguish definition from 'definition' (or meaning from 'meaning'). Definition is in the form 'definition of x' and 'definition' is the value of x in the form 'definition of x' (also the same as meaning and 'meaning').

The meaning of the word 'sense' is 'intension of linguistic expressions'. The meaning of the word 'intension' is 'what linguistic expressions mean'.

The meaning of 'all dogs are animals' is 'for every x, if x is a dog, then x is an animal.' The meaning of 'meaning' is 'sense of linguistic expressions'. I see no circularity here.

'Meaning' is defined as sense of linguistic expressions. 'Definition' is defined as a sentence giving a meaning to linguistic expressions.

Logic should be logical. We can make some language systems. Some languages may be logical, and others may be not. If your own language system is not logical, then it is not logic.

Two formulations are well-formed fomulae.

However, your analysis includes free variables x and y, which is not allowed in first-order predicate logic. Of course, if you used them as names, no troubles here.

Ah, you defined causation by biconditional (or necessary and sufficient condition) ... I got it.

1. Rxy
2. ~Cyx
3. ~(∃z)(Czx & Czy)
4. ~Nxy
5. Show Cxy ↔ (Rxy & ~Cyx & ~(∃z)(Czx & Czy) & ~Nxy)
6. Show Cxy → (Rxy & ~Cyx & ~(∃z)(Czx & Czy) & ~Nxy)
7. Cxy A
8. Show Rxy & ~Cyx & ~(∃z)(Czx & Czy) & ~Nxy
9. Rxy & ~Cyx & ~(∃z)(Czx & Czy) & ~Nxy 1, 2, 3, 4, Add
10. Show (Rxy & ~Cyx & ~(∃z)(Czx & Czy) & ~Nxy) → Cxy
11. Rxy & ~Cyx & ~(∃z)(Czx & Czy) & ~Nxy A
12. Show Cxy
13. (∀z)~(Czx & Czy) 3, QN
14. ~(Cxx & Cxy) 13, UI
15. ~Cxx ∨ ~Cxy 14, DM
INVALID
TheMadFool, your argument is not valid as shown above.

I am interested in logical forms of analyses. Or, I am interested in how to construct definitions in symbolic logic precisely.

I am examining correct formalization of Lewis' counterfactual analysis of causation. Lewis gives a semi-formal analysis, a combination of schematic letters (e.g. c, e, ...) and English expressions (e.g. causes, ...). What I am looking for is a formal analysis of causation symbolized completely down to schema.

Use theory of meaning is the view that meanings of linguistic expressions are controlled by uses of language. Suppose that you try to possess the meaning of the concept 'dog'. According to old-fashioned use theory of meaning, you can possess the concept 'dog' by using the sentence in which 'dog' is included, e.g. "All big dogs are dogs", and so on. Wilfrid Sellars says, "knowing a concept is knowing how to use the concept".
Animals can merely use sounds and signs to communicate. However, animals don't know how to use them as inferential roles. This follows that animals are not use theorists of meaning at all.

Here's an argument against faith:
(1) Anything dangerous is not good.
(2) Anything that is without evidence is dangerous.
(3) Faith is a thing without evidence.
Therefore, (4) faith is not good.

Suppose that you have no evidence that there is no bomb in Iraq. Suppose that you have merely faith in bombless in Iraq. You, then, will be able to go to 'dangerous trip' to Iraq. Evidence is safer than faith.