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PHIL 210 - Terms (Midterm)

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PHIL 210

Section 2.1 – Valid and Sound Arguments Logical Consequence – when a statement follows validly from given premises, it is also a logical consequence of the premises Section 2.2 – Methods of Proof Identity Elimination – given b=c, anything that holds true of b also holds true of c Identity Introduction – a=a can always be inferred from any set of premises, even no premises Identity Symmetry – given a = b, b = a *Transitivity of Identity – if a = b, and b = c, then a = c Chapter 4 – The Logic of Boolean Connectives Truth-functional – describes connectives, which, when used in a complex sentence, allows one to know the truth value of the complex sentence simply by looking at the truth values of the sentence's immediate constituents. Examples include ^, v, and the negation connective. Section 4.1 – Tautologies and Logical Truth Logical necessity – sentences that cannot be false, no matter the premises. Such a sentence is true in every logically possible circumstance. Ex. a = a Three forms of derivation: 1) TW 2) Truth tables 3) If one can prove a sentence using no premises whatsoever Logical possibility – a sentence that could be (or could have been) true, at least on logical grounds Ex. Going faster than the speed of light. Tautology – a simple kind of logical necessity; it is any sentence whose truth table has only T's in the column under its main connective. All tautologies are logically necessary; their truth is guaranteed by the structure and meaning of the truth-functional connectives, independent of the truth values of its constituent sentences. However, not all logically necessary claims are tautologies. Ex. a = a Section 4.2 – Logical and Tautological Equivalence Logical Equivalence – sentences that have the same truth values in every possible circumstance; having the same truth conditions Tautological Equivalence – sentences that can be seen to be equivalent simply in virtue of the meanings of the truth-functional connectives. All tautologically equivalent sentences are logically equivalent, but the reverse is usually not true. DeMorgan's Laws 1) The negation of a conjunction is logically equivalent to the disjunction of the negations of the original conjuncts. 2) The negation of a disjunction is equivalent to the conjunction of the negations of the original disjuncts Section 4.3 – Logical and Tautological Consequence Logical Consequence – sentence S is a logical consequence of set of premises P1... Pn, if it is impossible for the premises all to be true while the conclusion S is false Logical truths are sentences that are a logical consequence of a set of premises. Logically equivalent sentences are sentences that are a logical consequence of each other. Tautological consequence – a strict form of logical consequence; P is a tautological consequence of Q when all instances in which Q is true are also instances in which P is true. Section 4.5 – Pushing Negation Around Substitution of Equivalents – if P and Q are logically equivalent, then the results of substituting one for the othe in the context of a larger sentence are also logically equivalent. Ex. If P is logically equivalent to Q, then S(P) is logically equivalent to S(Q). Examples of substitution of logical equivalents: 1) Associativity of Conjunctions A sentence P ^ (Q ^ R) is logically equivalent to (P ^ Q) ^ R, which is in turn equivalent to P ^ Q ^ R. 2) Associativity of Disjunctions A sentence P v (Q v R) is logically equivalent to (P v Q) v R, which is in turn equivalent to P v Q v R. 3) Commutativity of Conjunctions A conjunction P ^ Q is logically equivalent to Q ^ P. 4) Commutativity of Disjunctions A disjunction P v Q is logically equivalent to Q v P. 5) Idempotence of Conjunctions A conjunction P ^ P is equivalent to P. Thus, with Commutativity, P ^ Q ^ P is equivalent to P ^ Q. 6) Idempotence of Disjunctions A disjunction P v P is equivalent to P. Thus, with Commutativity, P v Q v P is equivalent to P v Q. Section 4.6 – Conjunctive and Disjunctive Normal Forms Distributive laws: 1) Distribution of ^ over v: P ^ (Q v R) is logically equivalent to (P ^ Q) v (P ^ R) 2) Distribution of v over ^: P v (Q ^ R) is logically equivalent to (P v Q) ^ (P v R) Section 5.1 – Valid Inference Steps 1) Conjunction elimination (simplification) From P ^ Q, we can infer either P or Q. 2) Conjunction introduction From P and Q, we can infer P ^ Q. 3) Disjunction introduction (addition) From P, we can infer P v Q. Section 5.2 – Proof by Cases Disjunction elimination (proof by case): If our goal is S, and we have proven P v Q, then if
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