Compound Claims

 Suppose your neighbour says:  “I’ll return your lawn mower or I’ll buy you a new one.”  Has he promised to return your lawn mower? No. Has he promised to buy you a new lawn mower? No.  He has promised to do one or the other

Compound Claims  In this case, we have one claim not two.  A compound claim is one composed of other claims, but which has to be viewed as just one claim.

Examples of Compound Claims  Either some birds don’t fly or penguins aren’t birds.  Columbus landed in South Carolina or on some island near there.  Alternatives are the claims that are the parts of an “or” claim.

The Contradictory of a claim  The contradictory of a claim is one that has the opposite truth-value in all possible circumstances. Sometimes a contradictory is called the negation of a claim.

The Contradictory of a claim  Claim -Spot is barking. -Dick isn’t a student. -Suzy will go to the movies or she will stay home.  Contradictory -Spot isn’t barking. -Dick is a student. -Suzy won’t go to the movies and she won’t stay home.

Contradictory of an “or” claim  A or B has contradictory not A and not B  We sometimes use neither A nor B.  “A” and “B” stand for any claims

Contradictory of an “or” claim  Either Lee will pick up Manuel, or Manuel won’t come home for dinner. Contradictory:  Lee won’t pick up Manuel, and Manuel will come home for dinner.

Contradictory of an “and” claim  Using “and” to join two claims creates a compound, but it’s simpler to consider each claim independently.  Pigs can catch colds, and they can pass colds to humans.  Pigs can catch colds, but dogs can’t.

Contradictory of an “and” claim  A and B has contradictory not A or not B  “But” works the same as “and” in an argument

Reasoning with “or” claims  Determine whether an argument is valid or weak by looking at the role a compound claim plays in it. Example:  Either there is a wheelchair ramp at the school dance, or Manuel stayed home.  But there isn’t a wheelchair ramp at the school dance.  Therefore, Manuel stayed home.

Reasoning with “or” claims  Excluding Possibilities:  A or B  Not A  So B Valid A or B + not A B

Reasoning with “or” claims  Somebody’s cat killed the bird that always sang outside.1  Either it was Sarah’s cat or the neighbour’s cat or some stray.2  Sarah says it wasn’t her cat,3 because hers was in all day.4  My neighbour says her cat never leaves the house.5  So it must have been a stray.6

Reasoning with “or” claims  Here we get from 3 and 4:  Sarah’s cat didn’t kill the bird. a  My neighbour’s cat didn’t kill the bird. b A or B or C + not A + not B C

False dilemmas (bad arguments)  Excluding possibilities is a valid form of argument.  But valid arguments need not be good.  We get a bad argument when the “or” claim doesn’t list all the possibilities.

False dilemmas (bad arguments)  “You’re either going to have to stop smoking those nasty expensive cigars or we’ll have to get rid of Spot (pet dog).”  False Dilemma – A use of reasoning by Excluding Possibilities, but the “or” claim isn’t plausible. Sometimes the false “or” claim itself is called the false dilemma.

Conditionals – and their contradictories  Suppose your instructor says to you: “ If you do well on the final exam, then I’ll give you an A in the course.” - There is no promise, only a conditional promise

Conditional claim  A claim is called a conditional if it can be rewritten as an “if …then …” claim that must have the same truth value

Antecedent and Consequent  In a conditional rewritten as “If A, then B”, the claim A is called the antecedent, and the claim B is called the consequent.

Antecedent and Consequent  “If Dick loves Zoe, he will give her an engagement ring.” “Dick loves Zoe, “ “He will give her an engagement ring.” Antecedent Consequent

Contradictory of a Conditional  If A, then B has contradictory  A, but not B

Valid and weak forms of arguments using conditionals  If Spot barks, then Dick will wake up.  Spot barked.  So Dick woke up. If Spot barks, then Dick will wake up. AB Spot barked. A So Dick woke up. B

The Direct Way ofreasoning with conditionals  If A, then B  A  So B Valid If A, then B + A B This way of reasoning is sometimes called modus ponens

The Indirect Way of reasoning with conditionals  If A, then B  Not B  So not A Valid If A, then B + not B Not A This way of reasoning is sometimes called modus tollens. “not A” and “not B” are shorthand for “the contradictory of A” and “the contradictory of B”.

Affirming the Consequent  If A, then B  B  So A Usually Weak If A, then B + B A

Affirming the Consequent  If it’s the day for the garbageman, then Dick will wake up.  It’s not the day for the garbageman.  So Dick didn’t wake up.

Denying the Antecedent  If A, then B  Not A  So not B Usually Weak If A, then B + not A not B

Denying the Antecedent  If Maria doesn’t call Manuel, then Manuel will miss his class.  Maria did call Manuel.  So Manuel didn’t miss his class.

Contrapositive  The contrapositive of “If A, then B” is “If not B, then not A.” The contrapositive is true exactly when the original conditional is true.  “If you get a speeding ticket, then a policeman stopped you.”  “If a policeman didn’t stop you, then you didn’t get a ticket.”

Contrapositive Valid Usually Weak If A, then B + A B If A, then B + not B Not A If A, then B + B A If A, then B + not A Not B

Necessary and sufficient conditions Example: Dr. E: You’ll pass this exam only if you show up for every class. Suzy: O.K. Suzy: (later) What happened, Dr. E? I attended every single class, and I got an F on the exam. Dr. E: I said you’d pass the exam only if you showed up. It was necessary for you to show up for every class in order to pass, not sufficient. If you didn’t show up, you wouldn’t pass. Of course you needed to get 60% on the exam too.

Necessary and sufficient conditions A only if B means the same as If not B, then not A.

Reasoning from hypotheses Example: Lee: I’m thinking of doing a nursing degree. Maria: That means you’ll have to take summer school. Lee: Why? Maria: Look, you’re in your second year now. Top finish in four years like you told me you need to, you’ll have to take all the upper-division biology courses your last two years. And you can’t take any of those until you’ve finished the three- semester calculus course. So you’ll have to take calculus course. So you’ll have to take calculus over the summer in order to finish in four years.

Reasoning from hypotheses  Maria has proven that if Lee does a nursing degree, then he’ll have to go to summer school. Reasoning from Hypotheses: If you start with an assumption or hypothesis A that you don’t know to be true and make a good argument for B, then what you have established is “If A, then B.”

Reasoning in a chain and the slippery slope Example: Suppose we know that if Dick takes Spot for a walk, then Zoe will cook dinner. And if Zoe cooks dinner, then Dick will do the dishes. Then we can conclude that if Dick takes Spot for a walk, he’ll do the dishes. We can set up a chain of reasoning, a chain of conditionals.

Reasoning in a chain and the slippery slope Reasoning in a Chain with conditionals: If A, then B If B, then C So if A, then C Valid If A, then B + If B then C If A, then C

Reasoning in a chain and the slippery slope  If A, then B  If B, then C  A  So C Valid If A, then B + If B, then C + A C

Slippery Slope Argument A slippery slope argument is one that uses a chain of conditionals some or many of which are dubious