Tautology, Contradiction and Contingency in Mathematics, Schemes and Mind Maps of Mathematical logic

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Module 1. f Mathematic Lesson 3. Tautology, Contradiction and Contingency As mentioned at the start of our discussions about logical operators, compound propositions can be constructed using one or more logical operators. With this, if more than one operator is involved, you need to be guided by the rules of precedence of these logical operators as shown in Table 7, in order to solve for the correct truth value of the constructed compound proposition. Table 7. Precedence of Logical Operators Logical Operator Precedence ~ 1 z 3 4 5 Tij<> Table 7 specifically shows that negation (~) always precedes any other logical operators. For example, the proposition p A ~q > ~r means p A (~q) > (~r). Also, it shows that conjunction (A) and disjunction (Vv) are coequal, and so are conditional (>) and biconditional (<) operators. To indicate precedence between each pair, the use of parentheses is necessary. However, if parentheses are not used, operations should be done from left to right. For example, the proposition p V q Ar means (p Vq) Ar and the proposition pAq Vr means (pAgq)Vr. On the other hand, the proposition p e@q-r means (p © gq) 9 and the proposition p > g er means (p> g) er. Lastly, conjunction and disjunction take precedence over conditional and biconditional operators. For example, p > qVr @ s Atmeansp > (qV1r) @ (S At). In general, we can construct compound propositions by combining one or more propositional variables using one or more logical operators. We can systematically solve for the possible truth values of such compound proposition using truth tables. These compound propositions can be categorized into three types according to their possible truth values - tautologies, contradictions and contingencies. Definition 7. Tautology, Contradiction and Contingency A compound proposition that is always true, no matter what the assumed truth values of the propositional variables, is called a tautology. Acompound proposition that is always false is called a contradiction. A compound proposition that is neither a tautology nor a contradiction is called a contingency. LOGIC AND SET THEORY CYRENE A. CASPE / LEYTE NORMAL UNIVERSITY 16