Formal reference shifting or how to have formulas without variables

We must not forget that variables are a very recent development within modern mathematics (from late XVI century). We are so used to them that it is easy to forget how artificial they are. Natural language does not have variables. How does it manage? Most people respond with “pronouns”. But there are also reference shifting markers. Traditionally, reference shifting markers track whether and how the referents of two related clauses are coordinated. The most basic sort of coordination is plain coreference, but there are reference shifting markers for other sort of relations, like disjointness, simultaneity, etc. In any case, markers are linguistic resources we have not exploited in formal languages, but we could, i.e., we could use them instead of variables. So, for example, instead of the formula (P→Q) → (R→P), we could just have the sequence →2→→  where that 2 is a reference shifting marking that tells us how the arguments of the basic operators are coordinated, and in particular, that the antecedent of the antecedent implication is the same as the consequent of the consequent implication.

In strict sense, we only need four switch reference markers/constants for coreference/coordination of formulas with up to four argument spaces. Why, you might ask, since there are actually nine ways formulas with up to four argument spaces might have two of them coordinated? Well, the answer is to exploit other syntactic information in the formula:

	Matrix	Marked Clause	No repeat	No variables	Sets	No variables	WIth variables	Matrix	Marked Clause
1	P	P	A1B	V1O1V1	A(B)1	<V1, V2>, RS1	O(V1a, V2a)	Intransitive	Intransitive
2	P	P.R	A1BC	V1O1R1	A(BC)1	<V1, R1>, RS1	O(V1a, R1(a, b))	Intransitive	Subject
3	P	R.P	A2BC	V1O2R1	A(BC)2	<V1, R1>, RS2	O(V1a, R1(b, a))	Intransitive	Object
4	P.Q	P	AB1C	R1O1V1	(AB)C1	<V1, R1>,  RS3	O(R1(a, b), V1a)	Subject	Intransitive
5	P.Q	P.R	AB1CD	R1O1R2	AB(CD)1	<R1, R2>, RS1	O(R1(a, b), R2(a, c))	Subject	Subject
6	P.Q	R.P	AB2CD	R1O2R2	AB(CD)2	<R1, R2>, RS2	O(R1(a, b), R2(c, a))	Subject	Object
7	Q.P	P	AB3C	R1O3V1	(AB)C3	<R1, V1>, RS3	O(R1(b, a), V1a))	Object	Intransitive
8	Q.P	P.R	AB3CD	R1O1R2	AB(CD)3	<R1, R2>, RS3	O(R1(b, a), R2(a, c))	Object	Subject
9	Q.P	R.P	AB4CD	R1O4R2	AB(CD)4	<R1, R2>, RS4	O(R1(b, a), R2(c, a))	Object	Object

RS1: subject = subject

RS2: subject = object

RS3: object = subject

RS4: object = object

or to be more explicit

RS1: subject of the matrix = subject of the marked clause

RS2: subject of the matrix = object of the marked clause

RS3: object of the matrix = subject of the marked clause

RS4: object of the matrix = object of the marked clause

i.e.

RS1: first argument of the operation in the first argument of the main operator = first argument of the operation in the second argument of the main operator 

RS2: first argument of the operation in the first argument of the main operator = second argument of the operation in the second argument of the main operator 

RS3: second argument of the operation in the first argument of the main operator = first argument of the operation in the second argument of the main operator 

RS4: second argument of the operation in the first argument of the main operator = second argument of the operation in the second argument of the main operator