Derivations overview

Taking stock We’ve come a long way with natural deduction. Before going further, we should take stock, and get some more rigorous definitions in place.

Definitions A Rule of Inference tells when you are allowed, or licensed, to draw a conclusion from one or more sentences or from a whole argument (as represented by a subderivation). Conclusions may only be drawn in accordance with the rules of inference.

Definitions A Derivation is a list of which each member is either a sentence or another derivation. If a first derivation has a second derivation as one of the first derivation's parts, the second derivation is called a Subderivation of the first and the first is called the Outer Derivation, of the second.

Definitions Each sentence in a derivation is a premise or assumption, or a reiteration of a previous sentence from the same derivation or an outer derivation, or a sentence which follows by one of the rules of inference from previous sentences or subderivations of the derivation. Nothing else counts as a legitimate line of a derivation.

Premises and Assumptions Remember the difference between premises and assumptions. Though the rules of inference treat them in just the same way, premises are the unargued sentences assumed at the beginning of the outermost derivation only, whereas assumptions are the unargued sentences assumed at the beginning of subderivations. We always terminate subderivations before the end of the outermost derivation, so assumptions always get discharged, and do not show up as unargued sentences in the outermost derivation.

Scope lines Scope lines help us keep track of our subderivations. More precisely: A Scope Line tells us what sentences and subderivations hang together as a single derivation. Given a vertical scope line, the derivation it marks begins where the line begins and ends where the line ends. The derivation marked by a scope line includes all and only the sentences and subderivations immediately to the right of the scope line.

The trouble with scope lines… A major source of trouble for those starting out with natural deduction is getting scope lines mixed. The conclusions derived within a particular scope line only hold given the assumptions made at the beginning of that scope line. They do not necessarily hold in the outermost derivation. So one cannot reiterate sentences from subderivations in outer derivation. Teller calls this mistake ‘hopping scope lines’. Don’t do it!

For example… In this example, one could not reiterate line 5, ‘~D’, in the outer derivation, say as line 9. This is because ‘~D’ only follows given ‘~G’, which is an assumption of the subderivation, but may well not be true in the outer derivation.

Scope lines So you can’t reiterate sentences from subderivations in their outer derivations. But how about the other way round? Is it permissible to reiterate sentences from outer derivations in subderivations? (And sub-subderivations, and so on?)

Scope lines Yes, absolutely! Subderivations take their special assumptions in conjunction with all the premises and sentences that have been derived from them so far in the proof, and prove things from there. So reiteration is fine from outer derivations to subderivations; and not fine from subderivation to outer derivations.

Examples As with all our topics in logic, practice makes perfect. So here are some practice examples to do: