Critical Thinking

Chapter 10A Little Propositional Logic

Lecture Notes 2008 McGraw Hill Higher Education

Symbolization When you symbolize an argument, you represent its simple statements with single letters, and then represent the relationship between them (that the argument suggests) with symbols. Example: If Bush won were all going to die. Bush won. Therefore we are all going to die. Where: b = Bush wond = Were all going to dieThe argument gets symbolized like this: b db d

means Therefore

Lecture Notes 2008 McGraw Hill Higher Education

ConjunctionWhen two simple statements are conjoined with an and, we call it a conjunction. We represent each statement as a simple letter, and represent the and with an &. Example: Tina is tall and Sarah is tall.gets symbolized like this: p & qGet it? p = Tina is tallq = Sarah is tall

Lecture Notes 2008 McGraw Hill Higher Education

Other words for andIf you see any of the following words, treat them like and and symbolize the statement with an &but, yet, while, whereas, although, though, however. Be warned: not every statement with and in it is compound. Example: The Knicks and Bulls are playing each other. This is not expressing the facts that each team is playingMistake: The Knicks are playing and the Bulls are playing (i.e., k & b). it is expressing the single fact that the two teams are playing each other. So it would just get a single letter, for instance, e.

Note: What letter you give it really doesnt matter, just as long as you are consistent (use the same letter for that statement every time) and dont use the same letter for two different statements in the same argument.

Lecture Notes 2008 McGraw Hill Higher Education

Truth TablesWhen evaluating validity, you dont worry about the truth value of the statements you are symbolizing. But each statement is either true or false (you just dont know which).Truth Tables allow you to evaluate statements and arguments without knowing truth values by representing all possible truth value combinations.

Lecture Notes 2008 McGraw Hill Higher Education

Truth TablesRecall, we symbolized Tina is tall and Sarah is tall as p&q. We dont know if they are or not, but we can represent all possibilities this way:

Notice: what p&q means is both p and q are true. This means that, unless both p and q are true, p&q will not be true. That is why, above, p&q has a T only on the row on which both p and q both have a T as well.

Lecture Notes 2008 McGraw Hill Higher Education

Using Truth Tables to Examine ValiditySince.an argument is invalid only when it is possible for its premises to be true and the conclusion falseand since truth tables show us all possible truth valueswe can use truth tables to evaluate validity. We use them to determine all the possible truth values, and then look for a row where all the premise are true but the conclusion is false. If we find one, the argument is invalidIf there is no such row, the argument is valid.

Lecture Notes 2008 McGraw Hill Higher Education

Example(1) Tina is tall.(2) Sarah is tall.(3) Therefore, Tina and Sarah are tall. Symbolized: p, q. q. First, represent all the statement letters and their truth values.

Notice:Second row: one T, one F (repeat)First row: two Ts, two Fs,

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ExampleThen, add the premises and the conclusion.

Lecture Notes 2008 McGraw Hill Higher Education

ExampleThen, add the truth values.

Lecture Notes 2008 McGraw Hill Higher Education

ExampleThen look for rows where the premises are all true, and see if the conclusion is false on those rows. If there is such a row, then the argument is invalid. In this case, the only row with all true premises is one in which the conclusion is also true. Thus, the argument is valid.

Lecture Notes 2008 McGraw Hill Higher Education

More ExamplesGrass is green.Therefore, grass is green and the sky is blue.Symbolized: g g & sSince, on the second row, the premise is true but the conclusion is false, the argument is invalid. Shows it to be invalid

Lecture Notes 2008 McGraw Hill Higher Education

NegationWe can easily represent negated statements with a ~.If Sarah is tall is pSarah is not tall is ~p On a truth table:anywhere p has a T ~p will have an F

Lecture Notes 2008 McGraw Hill Higher Education

Example:(1) Tina is not Tall, but Sarah is tall. So, Tina is not tall. Symbolized: ~p, q ~pThe argument is valid. The only row on which both premises are true is a row on which the conclusion is also true.

Lecture Notes 2008 McGraw Hill Higher Education

Another exampleArgument: Frank does not drive a truck. Therefore, Frank doesnt drive a truck and Vinny doesnt drive a minivan. Symbolized: ~f ~f & ~vArgument is invalid. Row 3 is an example of a row with true premises but a false conclusion. To help me keep track of when ~f and ~v are true or false, I made little ts and fsfFf invalid

Lecture Notes 2008 McGraw Hill Higher Education

Whole statement negationsNot only can individual statements be negated~pcompound statements can be too:~(p&q)Since p&q means Both p and q are true ~(p&q) means It is false that both p and q are true. But dont distribute (like in math)~(p&q) is not the same as (~p & ~q)Why? ~(p&q) means they are not both true (at least one is false)(~p & ~q) means they are both false

Lecture Notes 2008 McGraw Hill Higher Education

Proof: ~(p&q) (~p & ~q)

Lecture Notes 2008 McGraw Hill Higher Education

Truth tables with 3 variables(set up)Notice: Third column, one T, one F (repeat)Second column, two Ts, Two Fs (repeat)First column, four Ts, four Fs

Lecture Notes 2008 McGraw Hill Higher Education

Argument: p, ~q & r, p & r Valid because the only row with true premises also has a true conclusion.

Lecture Notes 2008 McGraw Hill Higher Education

Argument: ~(p&q), (~q&r) ~p Shows it is invalid Even though

Lecture Notes 2008 McGraw Hill Higher Education

Disjunctions (or statements)Frank is angry or Hank is tired. gets symbolized: a v t(To make things easier, dont ever use the letter v to symbolize a simple statement.) Exclusive and inclusive or:Exclusive or: a or b means a or b, but not both.Inclusive or: a or b means at least a or b, but maybe both. The convention is to use the inclusive sense. So.

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Disjunction truth tables

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Argument: pvq p Invalid

Lecture Notes 2008 McGraw Hill Higher Education

Be careful with negations of disjunctionsFrank is not angry or Hank is tired.~a v tFrank is not angry or Hank is not tired.~a v ~tIts not the case that Frank is angry or Hank is tired. ~(a v t)Neither is Frank angry nor is Hank tired. ~(a v t)These last two are the same as Frank is not angry and Hank is not tired. (~a & ~t)

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Proof: ~(pvq) ~p&~q look they match

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Conditional (if, then) statementsIf it rained then the ground is wet.Where: r = it rainedw = the ground is wetr wr is the antecedentw is the consequent

Lecture Notes 2008 McGraw Hill Higher Education

Conditional truth tablesTruth table is tricky. pq means every time p is true, q is true. Or when p is true, q is true.

So, only when the antecedent is true and the consequent false, is it the case that the conditional is false.If this confuses you, dont worry It is confusing. Just remember it.

Lecture Notes 2008 McGraw Hill Higher Education

Be careful with negation and conditionalsIf it did not rain, then the game was played.~rpIf it did not rain, then the game was not played.~r ~pIt is not the case that, if it rained then the game was played.~(rp) These all have different meanings.

Lecture Notes 2008 McGraw Hill Higher Education

Be careful with negation and conditionals

Lecture Notes 2008 McGraw Hill Higher Education

Argument: ~(pq), (q v r) (qp) Valid because the only row with true premises also has a true conclusion

Lecture Notes 2008 McGraw Hill Higher Education