PHIL 200 WEEK TWO NOTES
PHIL 200 WEEK TWO NOTES PHIL 200
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This 5 page Class Notes was uploaded by Alejandra Paz on Sunday January 24, 2016. The Class Notes belongs to PHIL 200 at Ball State University taught by Dr. Thorson in Spring 2016. Since its upload, it has received 49 views. For similar materials see Symbolic Logic in PHIL-Philosophy at Ball State University.
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Date Created: 01/24/16
Wednesday, January 20, 2016 Chapter 6.2: Truth Functions Truth tables gives the truth value of a compound proposition for every possible truth value of its simple components. o Basically, each line of the truth table represents one possible arrangementof truth values. o When creating truth tables, the truth value goes under the connectives Meta variables lower case letters that can stand for any sentence/conjunction whatsoever. o p & q p/q are the conjuncts “p & q” is the conjunction Truth Table Key p q p & q p ∨ q p ⊃ q p ≡ q ~p ~q T T T T T T F F T F F T F F F T F T F T T F T F F F F F T T T T Truth Table Key in Words o In a conjunction (&) It is only true when both variables are true If any part of the conjunction is false, then the whole thing is false. o In a disjunction (∨) It is false only when both variables are false. o In a conditional (⊃) it is false only when the ANTECEDENT is TRUE and the CONSEQUENT is FALSE. o In a biconditional (≡) It is only true when both variables match. i.e. Both variables are true, or both variables are false then it is true. o In a negation (~) The truth value is opposite of what the original says. i.e. p = T, in negation ~p = F Examples (Take for granted that A = T, B = F, C = T) o (A & B) ∨ C T F T F T o (A ≡ B) ⊃ C T F T F T Chapter 6.3: Truth Tables for Propositions Example (1) A C T (A ∨ C) ⊃ T T T T T T T T T T T F T T T F F T F T T T F T T T F F T T F F F F T T F T T T T F T F F T T F F F F T F F F T T F F F F F F T T HOW? o You start by writing down the variables of the propositions on the left o Follow the Truth Table Key and write down the truth variables Tip: For 3 or more variables, since the truth table key only has two variables, begin writing the truth values with the two variables on the right (in the previous example, C and T). • Here is how you add more truth values when there are more than 2 variables: o Copy the previous truth values under (so if you have four truth values, you’ll have 8 Truth Values in total) o For a third variable’s truth values: First four all TRUE Second four all FALSE o Repeat previous 2 steps as more variables are added i.e. If you have four variables, you copy previous truth values (which would be 8) and add them under (so you would have 16 truth values now) o After you have written down the truth values for each variable, you can now find the truth value of the proposition! First, copy the truth values under its respective variable on the proposition side. • Remember that the truth values for propositions go UNDER the connectives Find the truth value (using the truth table key) for each connective by using the truth values of the variables • In the first example, to find the truth value of the disjunction connective, we used the truth values of A and C. • After you have found all of the truth values for the first connective, to find the truth value of the second connective (in the first example, the main connective), you use the truth values of the connective you just found + the truth values of the next variable. o In example one, I used the truth values of A and C to find the truth values of the disjunction connective. o Then I used the truth values of the disjunction connective and the truth values of T to find the truth value of the MAIN connective. Important: Find the truth variables for the MAIN connective LAST. I highlighted the truth values of the main connective. o Tip: Put a line down the truth values you use to find the truth values you’re looking for as you go. More Examples! o Example 2 G T S (G ≡ T) & S T T T T T T T T T T F T T T F F T F T T F F F T T F F T F F F F F T T F F T T T F T F F F T F F F F T F T F T T F F F F T F F F o Example 3 A B ~(A ∨ B) ≡ (~A & ~B) T T F T T T T F F F T F F T T F T F F T F T F F T T T T F F F F T F F F T T T T In example 3, the main connective (highlighted yellow) turns out to have all true values. This is called a tautology. o Example 4 A B A ⊃ B ≡ (A & ~B) T T T T T F T F F T F T F F F T T T F T F T T F F F F F F F T F F F F T In example 4, the main connective (highlighted yellow) turns out to have all false values. This is called a contradiction. **I highlighted the pairs I used to find the truth values the same color (i.e. in example 4, I used A (blue) and B (blue) to get the conditional (⊃) connective, the A (pink) and B (pink) to get the conjunction (&) connective and the conditional (⊃) connective (green) and the conjunction (&) connective (green) to get the main connective (yellow) **For example 3, I used the same coloring scheme BUT note that to get the negation connective I used only green with green, and to find the MAIN connective I used BOLD letters WITH green (so I paired the negation connective with the conjunction connective) Clarifying Statements o Tautologous statement A statement is a tautology if ALL of the truth values under the main connective are TRUE. o Self-contradictory A statement is a contradiction if ALL of the truth values under the main connective are FALSE o Contingent A statement is contingent if at least ONE truth value is TRUE and ONE is FALSE. Friday, January 22, 2016 Chapter 6.4 Meta Variables → lower cases letters that can stand for any sentences whatsoever. o p & q Where p is a conjunct and q is a conjunct • A conjunct is a full sentence o i.e. p = (A & ~B) “p & q” as a whole is a conjunction Invalid → An argument is invalid when it is possible for the premises to be true and the conclusion to be false. o BOTH PREMISES have to be TRUE and the CONCLUSION has to be FALSE. o Truth tables can help us figure out if an argument is valid or invalid Examples (Using Truth Tables!) o Example 1 (6.4 Section 2 #5) K L K ≡ ~L ~ (L & ~K) K ⊃ L T T T F F T T F F T T T T F T T T T F F F T F F F T F T F F T T T F T T F F F F T T F F T F T T o **Highlighted in RED is the CONCLUSION, in GREEN are the PREMISES o Notice, BOTH premises are TRUE but the conclusion is FALSE THEREFORE, the argument is INVALID Example 2 (6.4 Section 2 #3) P N P ≡ ~N N ∨ P T T T F F T T T T F T T T F T T F T F T F T T F F F F F T F F F o **In this example, the conclusion is false AND the premise is false, therefore, the argument is VALID o **REMEMBER: For an argument to be invalid, the CONCLUSION must be FALSE and the PREMISES must be TRUE Example 3 (6.4 Section 2 #6) E Z Z E ⊃ (Z ⊃ E) T T T T T T T T T F F T T F T T F T T F T T F F F F F F T F T F o **If conclusion is a tautology, the argument cannot be invalid. o So in this example, since the conclusion is a tautology, this argument is VALID Different Ways of Writing Premises and Conclusion P ≡ ~N Premise A ∨ B B ⊃ C A ⊃ C N ∨ P Conclusion Premise Premise Conclusion
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