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Natural Deduction
1. Introduction
We now introduce a proof method called natural deduction. ... In a natural deduction proof, we start with the set of premises of the argument we are proving, and we reason in natural logical steps until we reach the argument’s conclusion. ...
Why bother learning natural deduction when we already have the truth table method? ...
· Truth tables (unlike natural deduction) can tell us when an argument is invalid. ... The method is also basically limited to the logic you’ve learnt so far – truth tables can’t be extended
· Natural deduction proofs are similar to the way humans actually reason (in everyday life, maths, the formal sciences, etc). ... In addition, since natural deduction isn’t mechanical, practicing natural deduction techniques strengthens general reasoning ability.
The methods have other relative strengths and weaknesses; as you become more familiar with natural deduction you’ll no doubt notice some of them. ... The First Five Rules
In the natural deduction system we will develop, there are ten basic rules of inference (and a few extra short cut rules). ... To see how it works, look at the following simple example of its use to prove the sequent A ® B, A |- B:
1 (1) A ® B A
2 (2) A A
1,2 (3) B 1,2 MPP
Remember that we start a natural deduction proof by assuming the premises are true and reasoning on from there. ... For each sequent you regarded as valid in exercise 1 above, see if you can produce a natural deduction proof that the sequent is valid. ... For each valid argument, produce a natural deduction proof of the validity of the corresponding sequent. ... Can you think of any conditionals in ordinary English for which this kind of deduction would seem to go wrong? ... 4 Disjunctive Syllogism (DS)
This rule captures the most common way disjunctions are used in natural reasoning. ... This is a famous result, known as the Deduction Theorem. ... So we’ll add a definitional rule to our natural deduction rules of PC – the rule ‘Df «’ states that, for any propositions A and B, we can deduce A « B from the line (A ® B) Ù (B ® A), and vice versa. ... They are extremely useful derived rules of natural deduction. ... Translate each of the following arguments, test for validity by the table method, and provide natural deduction proofs for those arguments that are valid. ... Can you think of any ways in which the natural deduction rules could be plausibly changed so that neither EFQ nor the paradoxes of material implication can be proved? ...
Proof Strategies
As you should be aware by now, natural deduction proofs require creativity, rather than the fairly mechanical approach that tables involve. ...
· It’s part of one of the premises
If the conclusion is a wff that’s hiding in one of the premises, you have to work out an extraction strategy – how can you use the natural deduction rules on that premise to extract the conclusion from it? ... edu/) is a good proof checker for this natural deduction system.
Approximate Word count = 6499
Approximate Pages = 26
(250 words per page double spaced)
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