By Gunter Lumer
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Additional info for Algebres de fonctions et espaces de Hardy
Note that, in the last example, we could have appealed to the result obtained in the ﬁrst one where we showed that the inclusive disjunction of any proposition and its negation is a tautology. 2 we have a proposition p ∧ q and its negation ( p ∧ q). Hence, by the previous result, the inclusive disjunction ( p ∧ q) ∨ ( p ∧ q) is a tautology. The proposition ( p ∧ q) ∨ ( p ∧ q) is said to be a substitution instance of the proposition p ∨ p. ¯ The former proposition is obtained from the latter simply by substituting p ∧ q for p throughout.
1. Some monkeys eat bananas. All monkeys are primates. Therefore some primates eat bananas. 2. All cars are dangerous weapons. No dangerous weapons should be given to children. Therefore cars should not be given to children. 3. No reasonable man approves of wars. Jack approves of wars. Therefore Jack is not a reasonable man. 4. All gamblers are bound for ruin. No one bound for ruin is happy. Therefore no gamblers are happy. 5. All computer scientists are clever or wealthy. No computer scientist is wealthy.
Two propositions as similar as ‘Bill has green eyes’ and ‘Jeff has green eyes’ would have to be symbolized by p and q. We have as yet no means of expressing the fact that both propositions are about ‘green eyes’. Predicate Logic 29 A predicate describes a property of one or several objects or individuals. Examples of predicates might be: (a) (b) (c) (d) (e) . . is red. . has long teeth. . enjoys standing on his head. . has spiky leaves. . cannot be tolerated under any circumstances. The space in front of these predicates can be ﬁlled in with the names of objects or individuals where appropriate to form a proposition which may be true or false in the usual way.
Algebres de fonctions et espaces de Hardy by Gunter Lumer