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Today’s Topics n Review Logical Implication & Truth Table Tests for Validity n Truth Value Analysis n Short Form Validity Tests n Consistency and validity (again) n Substitution instances (again)

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Logical Implication n One statement logically implies another if, but only if, whenever the first is true, the second is true as well n If a statement, S 1, implies S 2 then the conditional (S 1 S 2 ) will be a tautology n Implication is the validity of the conditional.

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Determining whether S 1 Logically Implies S 2 n Construct a truth table with columns for S 1 and S 2. n If there is no row in which S 1 is true and S 2 false, then S 1 implies S 2. n If there is no row in which S 2 is true and S 1 is false, then S 2 implies S 1.

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NOTE: Logical Equivalence is Mutual Implication n Equivalence is the validity of the bi- conditional

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Truth Table Tests for Validity (and Non-validity) n n Construct a column for each premise in the argument n n Construct a column for the conclusion n n Examine each row of the truth table. Is there a row in which all the premises are true and the conclusion is false. If so, the argument is non-valid. If not, then the argument is valid.

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When using a truth table test for validity, one is looking for an Invalidating Row (or a Counter- Example Row). Failure to find an invalidating row shows that the argument is valid. Test the following argument for validity: P ▼Q, P, ~Q

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Testing for Validity PQ P Q P ~Q TTTT TTTT TT T n Verdict: NOT VALID, row 1

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Test the following argument for validity: (P ● Q) P, ~P, Q P

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Testing for Validity PQ (P Q) P ~PQ P P P P TT T T T TFT T TF F T F TFF T FT TT F FF TF F n Verdict: NON VALID! In ROW 3 all the premises are true and the false conclusion

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Test the following argument for validity: (P Q), ~ Q ~P

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Testing for Validity PQ P Q ~Q ~P TTT TTT TTT TT Verdict: VALID, no invalidating rows

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Truth Value Analysis n Sometimes we can know the truth value of a compound statement without knowing the truth values of each component simple statement. n Sometimes we don’t need a full truth table. n Since truth tables get very large very quickly (e.g., 8 variables produces 256 rows) this is good news. n Download the Handout on Truth Value Analysis and read it. Handout

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Examples n We know that a conditional with a false antecedent is true, so, if ‘P’ is false, then n P (Q v (R S)) is TRUE, no matter what the truth values of ‘Q,’ ‘R,’ and ‘S’ happen to be! n Similarly, since a conjunction with a false conjunct is false, if any one of ‘P,’ ‘Q,’ ‘R,’ or ‘S’ is false, then n P (Q (R S)) is FALSE no matter what the truth values of the others.

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Rules for truth value analysis n A conjunction with a false conjunct is false n A disjunction with a true disjunct is true n A conditional with a false antecedent or a true consequent is true n A biconditional with a true component has the same truth value as the other component n A biconditional with a false component has a truth value opposite the other component

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Try a few exercises n Download the Handout Truth Value Analysis Exercises and determine whether each formula is true, false or undecided give the assumptions. I call this a resolution of the truth value of a statement. Handout n Discuss your answers via the bulletin board.

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Short Form Validity Tests (Truth Value Analysis of Validity)

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When using a truth table test for validity, one is looking for an Invalidating Row (or a Counter- Example Row). Failure to find an invalidating row shows that the argument is valid.

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In an invalidating row, the conclusion must be false: n We can skip constructing ANY rows in which the conclusion is true. n Assume the conclusion to be false, and assign truth values to the simple statements in it accordingly. n Using those assignments, try to make all the premises true.

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n If you succeed, if it is possible to make all the premises true while the conclusion is false, the argument is non-valid. n If you fail, if it is impossible to make the premises true after making the conclusion false, the argument is valid.

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If making the conclusion false forces at least one premise to be false, then the argument is valid.

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NOTE: If more than one assignment of truth values makes the conclusion false, you MUST test each assignment. ANY combination of truth values that results in true premises and a false conclusion invalidates the argument

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NOTE: This method is most valuable when the conclusion is falsified by only one or two combinations of truth values. Hence, it is most valuable when the conclusion is either a conditional or a disjunction.

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Try a few on your own n Download the Handout Truth Value Analysis Validity Tests and read the explanation. Now read it again. Handout n Now work the problems and discuss your answers via the bulletin board

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Testing for Consistency n A set of statements is consistent if, but only if, it is possible for all of the members of the set to be true. n If there is ANY row in a truth table for a set of statements in which each of the statements is true, then the set is consistent. n If there is NO such row, then the set is inconsistent.

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Consistency and Validity (Again) n Consistency is closely related to validity n If the premises of a argument are consistent with the negation of the conclusion, then the argument is non-valid. n If the premises of a argument are inconsistent with the negation of the conclusion, then the argument is valid.

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Statement Forms and Substitution Instances n A statement form is a mix of sentential variables and logical operators (which remain constant) n Every WFF’s is a substitution instances of a basic statement form n WFF’s are also substitution instances of other (non-basic) statement forms

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Substitution Instance A compound WFF is a substitution instance of the statement form if, but only if, can be obtained by replacing each sentential variable in with a WFF, using the same WFF for the same sentential variable throughout. A compound WFF is a substitution instance of the statement form if, but only if, can be obtained by replacing each sentential variable in with a WFF, using the same WFF for the same sentential variable throughout.

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For example: n ~(~A B) is a substitution instance of n p, ~p, ~(p q), and ~(~p q) n However, while ‘~~A’ is a substitution instance of ‘~~p,’ ‘A’ is not, even though ‘A’ and ‘~~A’ are logically equivalent

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Logical Form and Logical Equivalence are not the same n Understanding the difference between sentences and sentence forms and between variables and constants is crucial to understanding logic

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Variables and Constants n In statement forms, the lower case letters are sentential variables, they stand for complete statements but are not themselves statements n The logical operators in statement forms are constants, they do not change in the instances of the form n Every substitution instance of a statement form has the same dominant operator as the form

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Argument Forms and Substitution Instances n Each and every legitimate use of a rule of inference or equivalence involves a substitution instance (or instances) of the statement form(s) that occur in the rule n A rule can be applies only to substitution instances of the forms that occur in the rule

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Let’s try to determine which WFFs are instances of which statement forms n For each statement form in the left hand column, determine whether or not each WFF in the right hand column is an instance of it. n Discuss your answers, questions on the bulletin board.

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1. 1. p 2. 2. ~p 3. 3. p v q 4. 4. p q 5. 5. ~(p q) 6. 6. ~p q 7. 7. ~p (q v r) 8. 8. (p v q) r 9. 9. p q 10. 10. ~(p q) 11. 11. ~p ( q v r) A. ~[(P Q) R] B. ~(Q v R) ~(R S)

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Key Ideas n Logical implication & truth table tests n Truth Value Analysis shortcuts constructing full truth tables by ignoring rows that could not be invalidating rows. n Testing for consistency, using a consistency test to test for validity n Constants and variables in statement forms

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Thus endeth the first unit n Download the Sample Exam for Sample Exam # 1. Take the exam, give yourself 50 minutes. Early Wednesday I will post a key to the sample exam. We can have a review for the exam via the bulletin board. Sample Exam Sample Exam n Honor system, no collaborating on the exam (and, since the person you cheat off of might be more clueless than you, it REALLY isn’t a good idea in logic).

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