Today’s Topics

Review Logical Implication & Truth Table Tests for Validity

Truth Value Analysis Short Form Validity Tests Consistency and validity (again) Substitution instances (again)

Logical Implication

One statement logically implies another if, but only if, whenever the first is true, the second is true as well

If a statement, S1, implies S2 then the conditional (S1 S2) will be a tautology

Implication is the validity of the conditional.

Determining whether S1 Logically Implies S2 Construct a truth table with columns for S1

and S2.

If there is no row in which S1 is true and S2 false, then S1 implies S2.

If there is no row in which S2 is true and S1 is false, then S2 implies S1.

NOTE: Logical Equivalence is Mutual Implication

Equivalence is the validity of the bi-conditional

Truth Table Tests for Validity (and Non-validity) Construct a column for each premise in the

argument Construct a column for the conclusion 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 ValidityP Q P Q P ~Q

T T T T

T T T T

T T

T

Verdict: NOT VALID, row 1

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

Testing for Validity

P Q (P Q) P ~P Q PT T T T F T TT F F T F F TF T F T T T FF F F T T F F

 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 ValidityP Q P Q ~Q ~P

T T T

T T

T T T

T T

Verdict: VALID, no invalidating rows

Truth Value Analysis

Sometimes we can know the truth value of a compound statement without knowing the truth values of each component simple statement.

Sometimes we don’t need a full truth table. Since truth tables get very large very quickly (e.g.,

8 variables produces 256 rows) this is good news. Download the Handout on Truth Value Analysis

and read it.

Examples We know that a conditional with a false antecedent

is true, so, if ‘P’ is false, then P (Q v (R S)) is TRUE, no matter what the

truth values of ‘Q,’ ‘R,’ and ‘S’ happen to be! Similarly, since a conjunction with a false

conjunct is false, if any one of ‘P,’ ‘Q,’ ‘R,’ or ‘S’ is false, then

P (Q (R S)) is FALSE no matter what the truth values of the others.

Rules for truth value analysis A conjunction with a false conjunct is false A disjunction with a true disjunct is true A conditional with a false antecedent or a

true consequent is true A biconditional with a true component has

the same truth value as the other component A biconditional with a false component has

a truth value opposite the other component

Try a few exercises

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.

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: We can skip constructing ANY rows in

which the conclusion is true. Assume the conclusion to be false, and

assign truth values to the simple statements in it accordingly.

Using those assignments, try to make all the premises true.

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

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

Download the Handout Truth Value Analysis Validity Tests and read the explanation. Now read it again.

Now work the problems and discuss your answers via the bulletin board

Testing for Consistency A set of statements is consistent if, but only

if, it is possible for all of the members of the set to be true.

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.

If there is NO such row, then the set is inconsistent.

Consistency and Validity (Again)

Consistency is closely related to validity If the premises of a argument are consistent

with the negation of the conclusion, then the argument is non-valid.

If the premises of a argument are inconsistent with the negation of the conclusion, then the argument is valid.

Statement Forms and Substitution Instances A statement form is a mix of sentential

variables and logical operators (which remain constant)

Every WFF’s is a substitution instances of a basic statement form

WFF’s are also substitution instances of other (non-basic) statement forms

Substitution Instance A compound WFFFis a

substitution instance of the statement formYif, but only if, Fcan be obtained by replacing each sentential variable inYwith a WFF, using the same WFF for the same sentential variable throughout.

For example:

~(~A B) is a substitution instance of p, ~p, ~(p q), and ~(~p q) 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 Understanding the difference between

sentences and sentence forms and between variables and constants is crucial to understanding logic

Variables and Constants

In statement forms, the lower case letters are sentential variables, they stand for complete statements but are not themselves statements

The logical operators in statement forms are constants, they do not change in the instances of the form

Every substitution instance of a statement form has the same dominant operator as the form

Argument Forms and Substitution Instances 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

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

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.

Discuss your answers, questions on the bulletin board.

1. 1.           p2. 2.               ~p 3. 3.               p v q4. 4.               p q5. 5.               ~(p q)6. 6.               ~p q7. 7.               ~p (q v r)8. 8.               (p v q) r9. 9.               p q10. 10.            ~(p q)11. 11.             ~p ( q v r)

A. ~[(P Q) R]

B. ~(Q v R) ~(R S)

Key Ideas Logical implication & truth table tests Truth Value Analysis shortcuts constructing

full truth tables by ignoring rows that could not be invalidating rows.

Testing for consistency, using a consistency test to test for validity

Constants and variables in statement forms

Thus endeth the first unit

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.

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).