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.
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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)
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.
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.
NOTE: Logical Equivalence is Mutual Implication n Equivalence is the validity of the bi- conditional
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.
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
Testing for Validity PQ P Q P ~Q TTTT TTTT TT T n Verdict: NOT VALID, row 1
Test the following argument for validity: (P ● Q) P, ~P, Q P
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
Test the following argument for validity: (P Q), ~ Q ~P
Testing for Validity PQ P Q ~Q ~P TTT TTT TTT TT Verdict: VALID, no invalidating rows
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
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.
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
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.
Short Form Validity Tests (Truth Value Analysis of Validity)
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.
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.
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.
If making the conclusion false forces at least one premise to be false, then the argument is valid.
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
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.
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
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.
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.
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
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.
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
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
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
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
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.
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)
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
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).