is a kind of proof calculus in which logical reasoning is expressed by inference
rules closely related to the natural way of reasoning.
We can also define
derivation in terms of form. It can be taken as game but there is no connection
between derivation and semantic. Axiomatic and natural derivation can be seen
as a kind of game.
We do not need a
strategy about what we have to do, everything is to know how does the rules
work and how we can play this game. To start this game we need to begin with a
premises in the top and a goal on the bottom. We also need to know how to apply
the rules we need to follow each step.
We need to know
that is the premises are true the goal must be true as well. There is no row
where the premises are true and the conclusion false. How we can know that we
win the game? Because the argument must be valid.
Assumptions are The premises are against a line which indicates the range or
scope over which the premises apply, in each case the line extends from the
premises to the conclusion indicating that the conclusion is derived from them.
It is always our
aim to derive the conclusion under the scope of the premises alone. But our
officially derivation system will allow appeal to certain auxiliary assumptions
in addition to premises.
without auxiliary assumptions we have been able freely to appeal to any formula
already in our hand.
It may be helpful
to think of a completed sub derivation as a sort of box, as you are under the
scope of an assumption the box is open and you can see the formulas under its
different definitions for auxiliary assumptions.
SD is an auxiliary
assumption, together with the formulas that fall under its scope is a sub
FA is a formula is
accessible at a given stage when it is obtained under assumptions all of which
continue to apply. SA is a sub derivation that is accessible at a given stage
when it is obtained under assuptions all of which continue to apply.
All of this
definitions become more concrete as we turn now to the rules of our official
system natural deduction.
with the presentation of our official natural deduction system with rules whose
application is just to sentential forms that can form involving.
contradiction appears easily at the level of atomics and negated atomics.
of our full system of natural deduction includes all the rules for the
sentential part of natural deduction.
We remain that the
strategy are modified only to accommodate the parallels between son methods.