todayÕs lecture 2/23/10jdblair/documents/truth_tables_3.pdf · todayÕs lecture 2/23/10 chapter...

Today’s Lecture2/23/10

Chapter 7.3

Using truth tables to evaluate arguments

Announcements

HW: Ex 7.3 pg. 320 Part B (1-19 odd)

Quiz on Thurs (the 25th): be ready to construct one or twotruth tables.

Exams back at the end of class (key).

Flyer on the California Pre-Doctoral Program

Truth Tables for Arguments (the goal)

The goal of constructing a truth table for any argumentform is to find out whether it’s possible to come up with acounter-example. This can be done by finding out all thepossible truth-values of the premises and conclusion.Because much of the time the premises and conclusion willbe compound statements, and because compoundstatements are truth functional, we can determine thepossible truth-values of the premises and conclusion byfinding out the possible truth-values of the atomicstatements that comprise the premises and conclusion inparticular and the argument in general.

Truth Tables for Arguments (some review)

Valid argument: one in which it’s impossible for itspremises to be true while its conclusion is false.

Invalid argument: one in which it's possible for itspremises to be true while its conclusion is false.

Note that if an argument has premises that are actually trueand a conclusion that is actually false, then it's an invalidargument. Actuality entails possibility. If something isactually the case, then it's not impossible; which is to sayit's possible. So if it's actually the case that an argument'spremises are true while its conclusion is false, that entailsthat it's possible for the argument's premises to be truewhile its conclusion is false.

A valid argument form is one in which every substitutioninstance is a valid argument.

An invalid argument form is one in which somesubstitution instances will be invalid arguments.

[An argument form is simply a pattern of inference thatconsists in nothing but variables that are connected bylogical operators or logical words e.g. "if-then". Asubstitution instance is a set of English statements thatuniformly replace the variables (and operators if need be)].

A counter-example to an argument form is a substitutioninstance that has premises that are (actually) true and aconclusion that is (actually) false.

A counter-example shows that the argument form inquestion is an invalid one. How? If we can come up witha counter-example (which is an invalid argument), we'vethereby shown that, of the form in question, there areindeed some substitution instances that are invalidarguments.

A truth table for an invalid argument form shows us thatit's possible to come up with at least one counter-example,

thereby showing it to be an invalid form. It does this by

the row(s) that indicates all the premises are true while theconclusion is false.

Consider…

P Q P ! Q Q " P

T T T T T

T F T F T

F T T T F invalid

F F F F F

See each row as representing a class of possible substitution instances.Take row 1 for example, it's possible to come up with any number ofsubstitution instances that have true premises and a true conclusion.

Now notice row 3. This indicates that it's possible to come up withany number of counter-examples—substitution instances that have alltrue premises and a false conclusion.

Using row 3 of the preceding truth table as our guide, wecan think of an actual counter example:

Either Biden is President of the U.S. (F) or Obama is President of theU.S. (T). -- The entire disjunction is true

Obama is President of the U.S (T).

So, Biden is President of the U.S (F).

A valid argument form cannot have a counter-examplebecause the form is such that it's impossible to come upwith a substitution instance that is an invalid argument.Since this is the case, it's impossible to come up with asubstitution instance that has true premises and a falseconclusion.

If a truth table shows that it's impossible come up with acounter-example, then it shows the argument form inquestion to be valid.

Steps for Constructing a Truth Table

A # B, ~B, " ~A

Step 1: isolate the distinct statement letters (the statementletters stand in for atomic statements) and list them on theleft side of the vertical line in the order they appear in thesymbolic argument.

A # B, ~B, " ~A

Step 2: assign all the possible truth-values for thestatement letters. [Since statement letters stand forstatements, they have a truth-value (i.e. they can be eithertrue or false)].

Step 2 continued:

The possible truth-values for the statement letters willappear in rows. Since there are 2 possible truth values(truth and falsehood), a truth table should have 2n rows,where n stands for the number of distinct statement lettersin the symbolic argument. So a symbolic argument with 2distinct statement letters will have four rows. A symbolicargument with 3 distinct statement letters will have 8 rows.

A # B, ~B, " ~A

A B A # B ~B " ~A

Step 3: List the premises and conclusion of the symbolic argument onthe right side of the vertical line.

A # B, ~B, " ~A

A B A # B ~B " ~A

T T T F F

Step 4: Proceeding row by row, fill in the possible truth-values of the premises and conclusion according to theassignment of possible truth-values of the statement letters,as well as our 'primitive' truth tables.

A # B, ~B, " ~A

A B A # B ~B " ~A

T T T F F

T F F T F

A # B, ~B, " ~A

A B A # B ~B " ~A

T T T F F

T F F T F

F T T F T

A # B, ~B, " ~A

A B A # B ~B " ~A

T T T F F

T F F T F

F T T F T

F F T T T

A # B, ~B, " ~A

A B A # B ~B " ~A

T T T F F

T F F T F

F T T F T

F F T T T

Valid.

Step 5: Indicate whether the argument form is valid or invalid. Anargument form is valid if there is no row in the truth table thatindicates all the premises are true while the conclusion is false. Anargument form is invalid if there is at least one such row. Importantly,if the argument is invalid, indicate which row(s) of the truth table isthe 'invalidating row'.

Practice

Construct a truth table for the following:

~P # ~R " ~( P # R)

Practice

P R ~P # ~R " ~( P # R)

T T T F

Practice

P R ~P # ~R " ~( P # R)

T T T F

T F T T

Practice

P R ~P # ~R " ~( P # R)

T T T F

T F T T

F T F F

Practice

P R ~P # ~R " ~( P # R)

T T T F

T F T T

F T F F

F F T F

Practice

P R ~P # ~R " ~( P # R)

T T T F

T F T T

F T F F

F F T F

invalid by rows 1 and 4.

A $ B " A • B

Practice

A B A $ B " A • B

T T T T

Practice

A B A $ B " A • B

T T T T

T F F F

Practice

A B A $ B " A • B

T T T T

T F F F

F T F F

Practice

A B A $ B " A • B

T T T T

T F F F

F T F F

F F T F

Practice

A B A $ B " A • B

T T T T

T F F F

F T F F

F F T F

invalid by row 4

Practice

(H • B) # S " B # S

H B S (H • B) # S " B # S

T T T T T

H B S (H • B) # S " B # S

T T T T T

T T F F F

H B S (H • B) # S " B # S

T T T T T

T T F F F

T F T T T

H B S (H • B) # S " B # S

T T T T T

T T F F F

T F T T T

T F F T T

H B S (H • B) # S " B # S

T T T T T

T T F F F

T F T T T

T F F T T

F T T T T

H B S (H • B) # S " B # S

T T T T T

T T F F F

T F T T T

T F F T T

F T T T T

F T F T F

H B S (H • B) # S " B # S

T T T T T

T T F F F

T F T T T

T F F T T

F T T T T

F T F T F

F F T T T

H B S (H • B) # S " B # S

T T T T T

T T F F F

T F T T T

T F F T T

F T T T T

F T F T F

F F T T T

F F F T T

H B S (H • B) # S " B # S

T T T T T

T T F F F

T F T T T

T F F T T

F T T T T

F T F T F

F F T T T

F F F T T

invalid by row 6

Ex 7.3 pg. 320 Part B (1-19 odd)

A B A • ~B " ~(A#B)

T T F F

T F T T

F T F F

F F F F

E G ~E # ~G " G # E

T T T T

T F T T

F T F F

F F T T

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