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propositional logic - Propositional Logic In categorical...

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1 Propositional Logic In categorical syllogism, we deal with terms that are related by quantifiers and copula in four different manners. In propositional logic, we deal with propositions as the basic units of meaning. All complicated propositions or arguments are constructed from propositions that are basic or simple.
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2 Simple/basic propositions are those that are the simplest and contain within themselves no other simple propositions. Compound propositions are composed of a number of simple propositions. Imagine that when you think, write or speak, you are actually constructing a model with the basic units.
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3 Simple/basic propositions John is a man. Mary is beautiful. Paul stole the watch.
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4 Compound propositions It is not the case that John is clever . If you give me the money , then I will be rich. Either Mary is a liar or John misunderstands Mary .
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5 Symbols We use capitalized letters to represent propositions. So if the proposition “John is clever” is A and the proposition “Mary is happy” is B, then the compound proposition “John is clever and Mary is happy” is A&B.
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6 Logical operators We have five logical operators: 1) ~ negation (not, it is not the case that) 2) • conjunction (and, but, also) 3) v disjunction (or, unless) 4) implication (if …then, only if) 5) ≡ equivalence (if and only if)
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7 In some books, ‘&’ or ‘ ’ is used for and; ¬ ’ is used for not; ‘ ’ is used for imply; ’ is used for equivalent.
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8 Logical operators are logical connectives that join the simple propositions into a logically meaningful compound proposition. Example: It is not the case that E ~E B and C B • C Either C or D C v D If A then B A B H if and only D H ≡ D
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9 Note that A B is a conditional proposition expressing the relation of material implication. Generally, it may be read as our normal “If … then”. Exceptional cases will be encountered later. A is the antecedent and B is the consequent . ≡ is called the material equivalence.
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10 Only ~ is placed in front of an expression. Other operators are placed between expressions.
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11 Main Operator The main operator is that operator in a compound proposition that governs the largest component(s) in the proposition. It is also the last operator to be dealt with in finding out the truth value (T or F) of a compound proposition. Compare to our use of +, - , × , ÷ . In 4 × (3 + 2), “ × ” is the main mathematical operator.
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12 The main operator of the following propositions is “~” ~B ~(A B) ~((A≡ F) • (D ≡ E))
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13 Main operator is “•”. K • ~L (E v F) • (G v H) ((R T) v (S U)) • ((W ≡ X) v (Y ≡ Z))
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14 The main operator is “v”.
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