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Unformatted text preview: of this extension is obvious from Theorem 1.
We set v (a) = v0 (a) for all a of degree 0. Then v is certainly a valuation on
the set of degree 0 propositions.
Suppose that v is a valuation for all propositions of degree less than n extending v0 . If a is a proposition of degree n, then it has a unique formation tree.
The immediate successors of a in the formation tree are labeled by subformulas
of a of degree less than n; hence, these successors have a valuation assigned.
6
Therefore, v has a unique extension to a using the consistency rules V1–V6.
We can conclude that v is a valuation on the set of all proposition of degree n
extending v0 . Therefore, the claim follows by induction.
The key reason that the previous argument by induction works is that the
formation tree is unique. If there would exist several diﬀerent trees for a single
formula, then such a recursive deﬁnition of a valuation would be ambiguous,
27 Theorem Proving, John Wiley & Sons, 1987].
In this section, we have been a little bit pedantic by distinguishing the purely
syntactical form of a proposition such as (a → b) from its meaning M→ (a, b).
Of course, it is a good idea to clearly distinguish between syntax and semantics
until the semantics of the connectives is clearly understood. From now on, we
will abuse notation and freely interpret (a → b) as the function M→ (a, b). Summary Summary. Informally, we can summarize the meaning of the connectives as
follows:
1) The and connective (a ∧ b) is true if and only if both a and b are true.
2) The or connective (a ∨ b) is true if and only if at least one of a, b is true.
3) The exclusive or (a ⊕ b) is true if and only if precisely one of a, b is true.
4) The implication (a → b) is false if and only if the premise a is true and the
conclusion b is false.
5) The biconditional connective (a ↔ b) is true if and only if the truth values
of a and b are the same.
An interpretation of a subset S of Prop is an assignment of truth values to all
variables that occur in the propositions contained in S . We showed that there
exist a unique valuation extending an interpretation of all propositions. 5 Tautologies and Satisﬁability In the previous two sections, we have introduced the language of propositional
logic and gave the propositions a meaning using valuations. In this section, we
28 Equivalences and Applications 29 Remarks
In our formal introduction of propositional logic,
we used a strict syntax with full parenthesizing
(except negations).
From now on, we will be more relaxed about the
syntax and allow to drop enclosing parentheses.
This can introduce ambiguity, which is resolved by
introducing operator precedence rules (from
highest to lowest): 1) negation, 2) and, 3) or & xor,
4) conditional, 5) biconditional
30 Tautologies
A proposition p is called a tautology if and only if
v[[p]] = t holds for all valuations v on Prop.
In other words, p is a tautology if and only if in a truth table it
always evaluates to true regardless of the assignment of truth
values to its variables. Example:
p
F
T ¬p
T
F
31 p⋁¬p
T
T Example of a Tautology p
F
F
T
T q
F
T
F
T ¬p
T
T
F
F ¬p ⋁q
T
T
F
T p→q
T
T
F
T (¬p ⋁q) ≡ (p→q) T
T
T
T Roughly speaking, this means that ¬p ⋁q has the same meaning as
p→q. 32 Logical Equivalence
Two propositions p and q are called logically equivalent if
and only if v[[p]] = v[[q]] holds for all valuations v on Prop. In other words, two propositions p and q are logically
equivalent if and only if p q is a tautology.
We write p ≡ q if and only if p and q are logically equivalent.
We have shown that (¬p⋁q) ≡ (p→q). In general, we can
use truth tables to establish logical equivalences. 33 De Morgan’s Laws
Theorem: ¬ (p⋀q) ≡ ¬p ⋁ ¬q.
Proof: We can use a truth table:
p q (p⋀q) ¬ (p⋀q) ¬p ¬q ¬p ⋁ ¬q F F F T T T T F T F T T F T T F F T F T T T T T F F F F 34 De Morgan’s Laws II Theorem: ¬(p⋁q) ≡¬p ⋀ ¬q
Proof: Use truth table, as before. See Example 2
on page 26 of our textbook. 35 Commutative, Associative, and Distributive Laws
Commutative laws: p∨q ≡q∨p p∧q ≡q∧p
Associative laws: (p ∨ q ) ∨ r ≡ p ∨ (q ∨ r )
(p ∧ q ) ∧ r ≡ p ∧ (q ∧ r )
Distributive laws: p ∨ (q ∧ r ) ≡ (p ∨ q ) ∧ (p ∨ r ) p ∧ (q ∨ r ) ≡ (p ∧ q ) ∨ (p ∧ r )
36 Double Negation Law
We have ¬(¬p)≡p p ¬p ¬(¬p) F T F T F T 37 Logical Equivalences You find many more logical equivalences listed in
Table 6 on page 27.
You should very carefully study these laws. 38 Logical Equivalences in Action
Let us show that ¬(p → q ) ≡ p ∧ ¬q are logically
equivalent without using truth tables.
¬ ( p → q ) ≡ ¬( ¬ p ∨ q ) ≡ ¬ ( ¬ p) ∧ ¬ q
≡ p ∧ ¬q (previous result)
(de Morgan) (double negation law) [Arguments using laws of logic are more desirable than truth tables
unless the number of propositional variables is tiny.]
39...
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