User:Physis/Hilbert-style deduction system

Stratification
A functioning deduction system is a “big thing”, thus, it takes a complicated “machinery” to work. We can manage complexity in several ways,

One alternative: to use notion of axiom schema. In summary: as a kind of modular design, we build our machinery in several strata.
 * 1) First level: we have a “machine”, producing axioms for the second level. Substitution (more exactly, the kind of it used in logics) makes the picture more complex.
 * 2) At the second level, we have to ensure that the ensemble of the instances remain “well searchable”, i.e. form a recursive set.

Other alternatives slice bigger lumps out of the complicatedness of the machinery, transcending stratification, e.g. by usage of substitution axiom.




 * Syntax (itself a submachine, containing base terms (atoms) worked by formation rules)
 * Well-formed formulae


 * Axiom schemata
 * Axiom instances


 * Universal generalization
 * Hypotheses


 * Rules of inference
 * Theory (provable, deducible formulae)

Each box (but formation rules) is regarded as a set-set function, thus, the whole set is passed between them. Except for universal generalization, they cannot be reduced to pointwise.
 * formation rules can be regarded as an intensional definition of the set of well-formed formulae.
 * universal generalization is a set-set function that is "simpler", it can be reduced to pointwise in a sense. It can be deduced from a relation (by closure), or, alternatively, from a point-set function (by binding operation). But we may skip such reduction and define it directly via inductive definition.

In fact, the machinery illustration sacrifies factual practice for preview. The image suggests that the entire mass of all possible axiom instances are generated and pured into the feeding funnel of the submachine “rules of inference”. Instead, practically axioms are generated by demand, governed by rules of inference, in fact, axioms are rules of inference themselves. The image suggests strict evaluation instead. Thus, it differs from factual operating, but only in the evaluation strategy.

Shared with CL
The first two axioms use no connectives but that of implication. This minimality allows surprising connections. Curry-Howard correspondence will provide a connection to combinatory logic.


 * (K) $$\alpha \to \beta \to \alpha$$
 * (S) $$(\alpha \to \beta \to \gamma) \to (\alpha \to \gamma) \to \alpha \to \gamma$$

Negation

 * double neg
 * either red-ad-abs or a>b>nb>na.

Sent calculus can be based already, but even inside sent. calc. (let alone pred calc), two ways can be chosen. We are satisfied with our axioms (and and, or etc. are done by extension by definition), or we define them from scratch, thus, we do not get satisfied with the axioms till now.

Extension by definitions
Extension by definitions

Conservative extension from scratch
Calibration (strong enough vs weak enough). Confer this duality to that of natural deduction (introduction vs elimination).

Union (set theory) of two sets A and B is the smallest set being superset of both A and B. Intersection (set theory) of two sets A and B is the biggestest set being subset of both A and B. Of cause, this fact about sets andset operations can be formalized also for truth functions. (lattice, boolean algebra etc.). Let us express this fact in the languege we have at hand, an hope that we have managed to grasp a defining property!

Conjunction
Let us calibrate conjunction:
 * $$\alpha \land \beta $$ must be strong enough to imply both $$\alpha$$ and $$\beta$$:
 * $$\alpha \land \beta \to \alpha$$
 * $$\alpha \land \beta \to \beta$$


 * $$\alpha \land \beta $$ must be weak enough to be implied from the simultaneous presence of $$\alpha$$ and $$\beta$$:
 * $$\alpha \to \beta \to \alpha \land \beta$$

Disjunction
Let us calibrate disjunction:
 * $$\alpha \lor \beta$$ must be weak enough to be implied either from $$\alpha$$ or from $$\beta$$
 * $$\alpha \to \alpha \lor \beta$$
 * $$\beta \to \alpha \lor \beta$$


 * Provided that each of $$\alpha_1$$, $$\beta_1$$ imply $$\gamma$$, then $$\alpha_1 \lor \alpha_2$$ must be strong enough to imply $$\gamma$$
 * $$(\alpha_1 \to \beta) \to (\alpha_2 \to \beta) \to (\alpha_1 \lor \alpha_2) \to \beta$$

Binding operation
When using monads in functional programming, bind is an operation of monads. A simple example (enabled by the fact that sets can be regarded as monads as well): Let A be a set, $$f: A \to 2^B$$, $$A_0 \subseteq A$$, then $$A_0 >\!\!\!>\!\!\!= f = \bigcup \left\{\,f(a) \mid a \in A_0 \,\right\}$$, thus, f is applied pointwise and the union of the intermediate results is yield.