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By David Patrick, Naoki Sato
This e-book comprises the answer within the textual content Intermediate Counting & likelihood.
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Extra info for Intermediate Counting and Probability (Solution Manual)
Example text
B ) P & Q , PI- -Q. ( c ) P Q , -Q t- -P. (4 P t P V Q . (e) P-Q, Q - Rt-P-R. 3 FORMALIZATION The language consisting of the symbolic notation introduced in the previous section is called the language o f propositional logic. , how these sentences may be formalized), and then by stating explicit grammatical rules (formation rules) for the language itself. The process of formalization converts an English sentence or argument into a sentence form or argument form, a structure composed of sentence letters and logical operators.
2) Either Rembrandt painted the Mona Lisa or Michelangelo did. Rembrandt didn't d o it. :. Michelangelo did. (3) Either he's at least 18 or he's a juvenile. He's not at least 18. :. He's a juvenile. It is easy to see that these three arguments are all deductively valid. Their common form is known by logicians as disjunctive syllogism, and can be represented as follows: Either P or Q. It is not the case that P :. Q. The letters 'P' and 'Q' function here as placeholders for declarative' sentences.
A logically necessary statement is a statement whose very conception or meaning requires its truth; its falsehood, in other words, is logically impossible. Here are some examples: Either something exists, or nothing at all exists. 2+2=4. No smoker is a nonsmoker. If it is raining, then it is raining. Everything is identical with itself. ) Logically necessary statements have the peculiar property that if one occurs as the conclusion of an argument, then the argument is automatically deductive, regardless of the nature of the premises.