Today’s Topics

Review Logical Implication & Truth Table Review Logical Implication & Truth Table Tests for ValidityTests for Validity

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

Logical Implication

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

If a statement, SIf a statement, S11, implies S, implies S22 then the then the

conditional (Sconditional (S11 S S22) will be a tautology) will be a tautology

Implication is the validity of the Implication is the validity of the conditionalconditional..

Determining whether S1 Logically Implies S2

Construct a truth table with columns for SConstruct a truth table with columns for S11

and Sand S2.2.

If there is no row in which SIf there is no row in which S11 is true and S is true and S22

false, then Sfalse, then S11 implies S implies S22..

If there is no row in which SIf there is no row in which S22 is true and S is true and S11

is false, then Sis false, then S22 implies S implies S11. .

NOTE: Logical Equivalence is Mutual Implication

Equivalence is the validity of the bi-Equivalence is the validity of the bi-conditionalconditional

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 ValidityPP QQ P P Q Q PP ~Q~Q

TT TT TT TT

TT TT TT TT

TT TT

TT

Verdict: NOT VALID, row 1Verdict: NOT VALID, row 1

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

Testing for Validity

PP QQ (P (P Q) Q) P P ~P~P QQ PP

TT TT T TT T FF TT TT

TT FF F TF T FF FF TT

FF TT F TF T TT TT FF

FF FF F TF T TT FF FF

  Verdict: NON VALID! In ROW 3 all the Verdict: NON VALID! In ROW 3 all the premises are true and the false conclusionpremises are true and the false conclusion

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

Testing for ValidityPP QQ P P Q Q ~Q~Q ~P~P

TT TT TT

TT TT TT

TT TT TT

TT TT

Verdict: VALID, no invalidating rowsVerdict: VALID, no invalidating rows

Truth Value Analysis

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

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

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

and read it.and read it.

Examples

We know that a conditional with a false We know that a conditional with a false antecedent is true, so, if ‘P’ is false, thenantecedent is true, so, if ‘P’ is false, then

P P (Q v (R (Q v (R S)) is TRUE, no matter what the S)) is TRUE, no matter what the truth values of ‘Q,’ ‘R,’ and ‘S’ happen to be!truth values of ‘Q,’ ‘R,’ and ‘S’ happen to be!

Similarly, since a conjunction with a false Similarly, since a conjunction with a false conjunct is false, if any one of ‘P,’ ‘Q,’ ‘R,’ or ‘S’ conjunct is false, if any one of ‘P,’ ‘Q,’ ‘R,’ or ‘S’ is false, thenis false, then

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

Rules for truth value analysis

A conjunction with a false conjunct is falseA conjunction with a false conjunct is false A disjunction with a true disjunct is trueA disjunction with a true disjunct is true A conditional with a false antecedent or a A conditional with a false antecedent or a

true consequent is truetrue consequent is true A biconditional with a true component has A biconditional with a true component has

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

a truth value opposite the other componenta truth value opposite the other component

Try a few exercises

Download the Download the HandoutHandout Truth Value Truth Value Analysis Exercises and determine whether Analysis Exercises and determine whether each formula is true, false or undecided each formula is true, false or undecided give the assumptions. I call this a give the assumptions. I call this a resolution of the truth value of a statement.resolution of the truth value of a statement.

Discuss your answers via the bulletin board.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 We can skip constructing ANY rows in

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

assign truth values to the simple statements assign truth values to the simple statements in it accordingly.in it accordingly.

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

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

If you If you failfail, if it is impossible to , if it is impossible to make the premises true after make the premises true after making the conclusion false, the making the conclusion false, the argument is argument is valid.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 Download the HandoutHandout Truth Value Truth Value Analysis Validity Tests and read the Analysis Validity Tests and read the explanation. Now read it again.explanation. Now read it again.

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

Testing for Consistency

A set of statements is A set of statements is consistentconsistent if, but only if, but only if, it is possible for all of the members of if, it is possible for all of the members of the set to be true.the set to be true.

If there is ANY row in a truth table for a set If there is ANY row in a truth table for a set of statements in which each of the of statements in which each of the statements is true, then the set is consistent.statements is true, then the set is consistent.

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

Consistency and Validity (Again)

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

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

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

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

variablesvariables and and logical operatorslogical operators (which (which remain constant)remain constant)

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

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

Substitution Instance

A compound WFFA compound WFFis a is a substitution instance of the statement substitution instance of the statement formformif, but only if, if, but only if, can be can be obtained by replacing each sentential obtained by replacing each sentential variable invariable inwith a WFF, using the with a WFF, using the same WFF for the same sentential same WFF for the same sentential variable throughout.variable throughout.

For example:

~(~A ~(~A B) is a substitution instance of B) is a substitution instance of p, ~p, ~(p p, ~p, ~(p q), and ~(~p q), and ~(~p q) q) However, while ‘~~A’ is a substitution However, while ‘~~A’ is a substitution

instance of ‘~~p,’ ‘A’ is not, even instance of ‘~~p,’ ‘A’ is not, even though ‘A’ and ‘~~A’ are logically though ‘A’ and ‘~~A’ are logically equivalentequivalent

Logical Form and Logical Equivalence are not the same Understanding the difference between Understanding the difference between

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

Variables and Constants

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

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

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

Argument Forms and Substitution Instances Each and every legitimate use of a rule of Each and every legitimate use of a rule of

inference or equivalence involves a inference or equivalence involves a substitution instance (or instances) of the substitution instance (or instances) of the statement form(s) that occur in the rulestatement form(s) that occur in the rule

A rule can be applies only to substitution A rule can be applies only to substitution instances of the forms that occur in the ruleinstances 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 For each statement form in the left hand column, determine whether or not each column, determine whether or not each WFF in the right hand column is an WFF in the right hand column is an instance of it.instance of it.

Discuss your answers, questions on the Discuss your answers, questions on the bulletin board.bulletin board.

1.1. 1.           p1.           p2.2. 2.               ~p 2.               ~p 3.3. 3.               p v q3.               p v q

4.4. 4.               p 4.               p q q

5.5. 5.               ~(p 5.               ~(p q) q)

6.6. 6.               ~p 6.               ~p q q

7.7. 7.               ~p 7.               ~p (q v r)(q v r)

8.8. 8.               (p v q) 8.               (p v q) r r

9.9. 9.               p 9.               p q q10.10. 10.            ~(p 10.            ~(p q) q)11.11. 11.             ~p (11.             ~p ( q v r) q v r)

A.A. ~[(P ~[(P Q) Q) R] R]

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

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

full truth tables by full truth tables by ignoringignoring rows that could rows that could not be invalidatingnot be invalidating rows. rows.

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

Constants and variables in statement formsConstants and variables in statement forms

Thus endeth the first unit

Download the Download the Sample Exam Sample Exam for Sample Exam # for Sample Exam # 1. Take the exam, give yourself 50 minutes. 1. Take the exam, give yourself 50 minutes. Early Wednesday I will post a key to the sample Early Wednesday I will post a key to the sample exam. We can have a review for the exam via the exam. We can have a review for the exam via the bulletin board. bulletin board.

Honor system, no collaborating on the exam (and, Honor system, no collaborating on the exam (and, since the person you cheat off of might be more since the person you cheat off of might be more clueless than you, it REALLY isn’t a good idea in clueless than you, it REALLY isn’t a good idea in logic).logic).