Rules of Inference for Natural Deduction
Rules for natural deduction are about introduction and elimination of logical connectives: negation, conjunction, disjunction, implication and bijunction.
For each connective, there are rules to introduce it and eliminate it
Negation
negation introduction
negation elimination
double negation introduction
double negation elimination
Conjunction
and-introduction
and-elimination (conjunction elimination, simplification)
Disjunction
disjunction introduction
disjunction elimination
Implication
implication introduction
implication elimination (modus ponens)
conditional proof (conditional introduction)
modus tollens
Biconditional
biconditional introduction
biconditional elimination
Conjunction
Conjunction introduction (1 rule)
Above the line are the two premises of the rule, and below is the conclusion. We have introduced a conjunction (in the conclusion) where there was none before (in the premises).
Conjunction elimination (2 rules)
Applying these rules is a kind of pattern matching:
Example
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Negation
These rules involve the notion of contradiction.
Double negation
Implication
There is one rule to introduce implication and one to eliminate it.
The rules for implication elimination
There are 2 rules to eliminate implication: modus ponens and modus tollens.
Modus ponens
Modus tollens
modus tollens (MT), modus tolendo tollens
Examples
Note that the order of applying double negation rules and MT is different in these examples; this order is driven by the structure of the particular sequent whose validity one is trying to show.
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The rule for implication introduction
Disjunction
The rules for disjunction introduction
The rules for disjunction elimination
The Kreisel–Putnam rule
The Kreisel–Putnam rule or Independence of Premise
is admissible in the Intuitionistic Propositional Calculus (IPC). In fact, it is admissible in every superintuitionistic logic.
is not an intuitionistic tautology, hence KPR is not derivable in IPC. In particular, IPC is not structurally complete.
Defeasibility
it refers to the possibility of a particular principle, rule or understanding being overridden in appropriate circumstances.
In modern argumentation theories, arguments are regarded as defeasible passages from premises to a conclusion. Defeasibility means that when additional information (new evidence or contrary arguments) is provided, the premises may be no longer lead to the conclusion (non-monotonic reasoning). This type of reasoning is referred to as defeasible reasoning. For instance we consider the famous Tweedy example:
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