Natural Deduction
Testing whether a proposition is a tautology by testing every possible truth assignment is expensive—there are exponentially many. We need a deductive system, which will allow us to construct proofs of tautologies in a step-by-step fashion.
The system we will use is known as natural deduction. The system consists of a set of rules of inference for deriving consequences from premises. One builds a proof tree whose root is the proposition to be proved and whose leaves are the initial assumptions or axioms (for proof trees, we usually draw the root at the bottom and the leaves at the top).
For example, one rule of our system is known as modus ponens. Intuitively, this says that if we know P is true, and we know that P implies Q, then we can conclude Q.
P P ⇒ Q
Q
(modus ponens)
The propositions above the line are called premises; the proposition below the line is the conclusion. Both the premises and the conclusion may contain metavariables (in this case, P and Q) representing arbitrary propositions. When an inference rule is used as part of a proof, the metavariables are replaced in a consistent way with the appropriate kind of object (in this case, propositions).
Most rules come in one of two flavors: introduction or elimination rules. Introduction rules introduce the use of a logical operator, and elimination rules eliminate it. Modus ponens is an elimination rule for ⇒. On the right-hand side of a rule, we often write the name of the rule. This is helpful when reading proofs. In this case, we have written (modus ponens). We could also have written (⇒-elim) to indicate that this is the elimination rule for ⇒.
Rules for Implication
In natural deduction, to prove an implication of the form P ⇒ Q, we assume P, then reason under that assumption to try to derive Q. If we are successful, then we can conclude that P ⇒ Q.
In a proof, we are always allowed to introduce a new assumption P, then reason under that assumption. We must give the assumption a name; we have used the name x in the example below. Each distinct assumption must have a different name.
[x : P]
(assum)
Because it has no premises, this rule can also start a proof. It can be used as if the proposition P were proved. The name of the assumption is also indicated here.
However, you do not get to make assumptions for free! To get a complete proof, all assumptions must be eventually discharged. This is done in the implication introduction rule. This rule introduces an implication P ⇒ Q by discharging a prior assumption [x : P]. Intuitively, if Q can be proved under the assumption P, then the implication P ⇒ Q holds without any assumptions. We write x in the rule name to show which assumption is discharged. This rule and modus ponens are the introduction and elimination rules for implications.
[x : P]
⋮
Q
P ⇒ Q
(⇒-intro/x)
P P ⇒ Q
Q
(⇒-elim, modus ponens)
A proof is valid only if every assumption is eventually discharged. This must happen in the proof tree below the assumption. The same assumption can be used more than once.
RULES OF REPLACEMENT in Natural Deduction
Forms of logical equivalences may also be used as rules of inference. Every substitution instance of a form of logical equivalence is a logical equivalence; moreover, either one of a pair of logically equivalent expressions may be substituted for the other in a proof without loss of validity. Five forms of logical equivalence are given in this section:
~(p · q) is logically equivalent to (~p ∨ ~q).
~(p ∨ q) is logically equivalent to (~p · ~q). These two statements are known as DeMorgan’s Rule (DM).
(p ∨ q) is logically equivalent to (q ∨ p).
(p · q) is logically equivalent to (q · p). These two rules are known as commutativity (Com).
[p ∨ (q ∨ r)] is logically equivalent to [(p ∨ q) ∨ r].
[p · (q ·r)] is logically equivalent to [(p · q) ·r]. These two rules are known as associativity (Assoc). 4. [p · (q ∨ r)] is logically equivalent to [(p · q) ∨ (p ·r)].
5. [p ∨ (q ·r)] is logically equivalent to [(p ∨ q) · (p ∨ r)]. These two rules are known as distribution (Dist).
p is logically equivalent to ~~p. This is known as double negation (DN). These and the remaining rules of replacement may be applied not only to whole lines in proofs—as the rules of implication must be—but to parts of lines as well. Thus, going from (p · q) ⊃ r to (q · p) ⊃ r by the rule of commutativity is quite proper
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