MLS 570Critical ThinkingReading Notes for Fogelin:

Propositional Logic

Fall Term 2006North Central College

Propositional Logic Propositional Logic is aimed at understanding the

concept of validity. Easy to see that one claim follows from another. But WHY can be difficult to see.

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Some Definitions Propositional Connectives

Conjunction: Disjunction Negation Conditional

Propositional Form Substitution Instance Truth tables

Propositional Connectives: Conjunction Propositional connectives permit us to build new

propositions from old ones. What truth conditions govern the word “and”?

That is, under what conditions are propositions containing this connective true.

You list all the possible “truth conditions”John is tall Harry is short John is tall and Harry is short

T T TT F F

F T FF F F

The only conditions under which a conjunction is true is when both the component propositions are true.

Conjunction continued

The conclusion we can draw from this “truth table” is independent of the particular propositions used.

So we can use a general “propositional form” to express it – p & q

p and q are the propositional variables & [ampersand] stands for the ‘and’

A “substitution instance” is when we substitute propositions for the variables

propositional form substitution instancepropositional form substitution instance p&q The rose is red and the daisy is white

General definition of Conjunction using a truth table Different variables can be replaced with the

same proposition, but different propositions cannot be replaced by the same variable.

p q p & qT T TT F FF T FF F F

Checking VALIDITY for Conjunction What you are checking for is whether there is a line where the

premise is true and the conclusion false. In this case there isn’t so the argument is VALID.

(Variables) Premise Conclusionpp qq p & qp & q ppT T T T o.k.T F F TF T F FF F F F

You have explored ALL the possible truth assignments to the variables and on those assignments there is NO assignment on which the conclusion is false when the premise is true..

Summary so far: Validity depends on the form of an argument,

not its particular content. An argument is valid if it is an instance of a valid

argument form An argument form is valid if and only if it has no

substitution instances in which the premises are true and the conclusion is false.

But not all arguments are valid by virtue of their structure.

Disjunction John will win or Harry will win

The truth of the compound proposition depends on the truth of the component propositions.

Disjunction: Exclusive disjunction rules out both disjuncts

being true Inclusive disjunction doesn’t rule out both

disjuncts being true. Logic normally uses the inclusive sense

Truth table for Inclusive Disjunction The “v” is called the “wedge”

p q p v q T T T T F T F T T F F F

The only truth value assignment on which a dis-junction is false is when both disjuncts are false

Negation Negation is another way of creating

a new proposition The symbol for negation is the ‘tilde’ ~ The truth table for negation

p ~pT FF T

Disjunctive SyllogismShe is either using her modem or talking on the phone.She is not using her modem she is talking on her phone.--------------------------

p v q q v p ~p ~q . q p

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The order of the disjuncts doesn’t matter – either version is a disjunctive syllogism

Truth table for Disjunctive Syllogism

premises conclusion

p q p v q ~p q

T T T F T

T F T F F

F T T T T o.k.

F F F T F

How truth-functional connectives work We use the connectives to form new propositions

from old ones. The truth value of the new proposition is a function

of the truth value of the original propositions. If A and B are true propositions and G and H are

false we can easily determine the truth value of the following.

A & B A & G ~A

~G A v H G v H ~A & G

Testing for Validity For an argument to be valid there can never

be an instance [in its truth table] in which the premises are true and the conclusion false.

In a truth table you lay out all the possible truth value assignments for the variables [such as p, q, r, s] in columns and then, for each line [instance], determine the truth of the premises and the conclusion based on those truth value assignments.

Example of a truth table: Please see the “How to Do Truth Tables” handout on

blackboard for more information.

This is a valid argument as there is no truth value assignment on which the conclusion is false while the premises are true.

How to read a truth table What you are checking is to see if there is a truth value

assignment on which the premises are TRUE and the conclusion FALSE. In the case of the table on the previous slide there is only

one line [line # 6] on which both premises are true and on that line the conclusion is also true, so the argument is VALID.

Premise Premise Conclusion

p q r (p v q) (p v r) ~(p v r) q

F T F TT F T TT ok

Example of an INVALID argument from the handout

NOTE: There are three lines on which the premises are true and the conclusion is also true, but all it takes is ONE truth value assignment on which the conclusion is FALSE when the premises are true to make the argument INVALID, which this argument is.

Conditionals: Getting the Truth Table(Don’t worry about the reasoning, just understand the table)

A traditional argument form

A traditional fallacious argument