Valid Arguments Brief Preview Decision making whether in science or elsewhere involves reasoning based on evidence Question when does some piece of information count as good evidence for or against a conclusion When does some piece of information evidence serve to support confirm or count against disconfirm a hypothesis or theory To answer these questions we need to discuss arguments Brief Review Statements are sentences that have a truth value value are either true or false Arguments are sets of statements some of which serve as premises for others which are conclusions Valid arguments are arguments in which if the premises are true the conclusion must also be true Sound arguments are valid arguments with true premises 1 Review continued True or False A valid argument cannot have a false conclusion A sound argument cannot have a false conclusion An argument with a true conclusion is sound The conclusion of a valid argument with false premises is false Conditional Statements Conditional statements consist of two component statements linked by the logical connective IF THEN If and then are not indicator words words they are not marking premises and conclusions of an argument If it rains today there will be no picnic is not an argument It simply asserts a conditional relationship between two statements Compare On account of the fact that it is raining today there will be no picnic Conditional Statements 2 IF THEN is a truth functional connective the truth of a compound statement depends only on the truth values of the component statements If A then B is false when the antecedent is true and the consequent is false Otherwise it is true If you trespass then you will be arrested is is is is false if you trespass and are not arrested true if you trespass and are arrested true if you do not trespass and are not arrested true if you do not trespass and are arrested The last case may seem surprising but of course there are other reasons you might be arrested 2 Conditional Statements 3 IF A THEN B is NOT equivalent to IF B THEN A IF A THEN B is false when A is true and B is false IF B THEN A is false when B is true and A is false IF A THEN B is equivalent to IF not B THEN not A If you trespass then you will be arrested is equivalent to If you are not arrested then you did not trespass Conditional Statements 4 IF THEN versus ONLY IF Compare If you trespass then you will be arrested False if you trespass and are not arrested Only if you trespass will you be arrested False if you don t trespass and are arrested B ONLY IF A is equivalent to If B then A If you were arrested then you trespassed THERE IS NO IF IN ONLY IF Conditional statements 5 UNLESS can also be used to assert conditional relations Unless you complete the assignment you will not get promoted says the same thing as If you do not complete the assignment you will not get promoted or If you get promoted then you completed the assignment 3 Sufficient Conditions When a conditional statement uses general terms e g dog mammal it expresses relations between categories of things that satisfy those terms If something is a dog then it is a mammal Presents a relation between being a dog and being a mammal It asserts that meeting the first condition being a dog suffices for meeting the second condition being a mammal If then suffices for Necessary Conditions Since a true conditional statement cannot have a true antecedent and a false consequent the consequent of a conditional expresses something that is necessary if the antecedent is true If something is a dog then it is a mammal Asserts that meeting the second condition being a mammal is necessary for meeting the first condition being a dog If then is necessary for If versus Only if again What follows the if of a conditional is a sufficient condition What follows only if is a necessary condition You can vote only if you are at least 18 years old Being 18 is a necessary condition for voting If you are able to vote then you are at least 18 years old Being able to vote is sufficient evidence that you are at least 18 years old 4 Practice with conditionals Assume Sales are increasing T We need to build a new plant F Our sales force is less effective F We have excess production capacity T Whenever sales are increasing we need to build a new plant T F F T F F T F T T T T T F T F F T T F T If we do not need to build a new plant then our sales are not increasing Only if sales are increasing do we need to build a new plant We do not need to build a new plant only if we have excess production capacity Unless we have excess production capacity we need to build a new plant Only if our sales force is less effective are our sales not increasing Unless sales are increasing we need to build a new plant Using conditionals in inference There are two ways to use a conditional statement in a valid inference one obvious one less so The obvious way From IF A THEN B affirm A From this it follows that B Why If B weren t true and A is true If A then B would be rendered false So the following form is VALID If A then B A B Modus ponens Using conditionals in inference 2 The less obvious way From IF A THEN B what happens if B is denied If B is false and A is true then what is the truth value of IF A THEN B It is false Thus A cannot be true when the whole conditional is true Thus If A then B Not B Not A is VALID Modus tollens 5 Uses of conditional arguments in scientific reasoning Modus ponens is most commonly invoked to make predictions from a hypothesis If malaria is transmitted by mosquitoes and we eliminate the mosquitoes malaria will decline Malaria is transmitted by mosquitoes and we are eliminating the mosquitoes Malaria will decline Modus tollens is most commonly invoked to confirm or falsify a hypothesis based on the truth of falsity of a prediction Invalid conditional arguments Not all arguments that start with conditional statements are valid What can you conclude validly from X If A then B Not A Denying the Antecedent INVALID Remember to be valid it must be that if the premises were true the conclusion would also have to be true What conclusion has to be true in this case Both B and not B are compatible with the premises There is no valid argument here Invalid conditional arguments 2 What about if we start with X If A then B B Affirming the consequent INVALID What conclusion has to be true in this case Both A and Not A are …