Can every proof calculus (e.g. natural deduction, Hilbert-style axiomatic systems, sequent calculus) be expressed as a generic 4-tuple {A, Ω, Z, I} consisting of: set alpha of proposition symbols, omega set of operator symbols, zeta set of inference rules and iota set of axioms?
If yes, does it mean that, in principle, all syntax can be abstracted away and all proofs can be expressed as syllogisms of the form?:
1. ...
2. ...
...
N. ...
∴ ...
How would e.g. Smullyan-style tableaux proofs ("truth trees") fit into that picture?
Indeed, how would Gentzen-style natural deduction with its non-linear proof trees fit into the picture?
And what on earth is meant by talking of "syllogisms" here?
It is very unclear what insights could be gained by over-abstraction of this kind.