(Week 4)
In the previous lecture we defined a valid argument as an argument with the feature that if the premises were true, the conclusion would also have to be true. An understanding of necessary and sufficient
conditions, and the conditional propositions in which they occur, will now allow us to determine the validity or invalidity of an important kind of argument - conditional arguments.
4.1 Valid Conditional Argument Forms
There are two main forms of valid conditional argument. These are called "affirming the sufficient
condition", and "denying the necessary condition". (This classification of conditional argument forms is from Thinking Clearly , by Jill LeBlanc)
4.1.1 Affirming the sufficient condition
Consider the following standardised argument:
(6a)
1: If you have a blood alcohol level of more than 0.05, you are not legally allowed to drive.
2: You have a blood alcohol level of more than 0.05.
C: You are not legally allowed to drive.
The first premise of this argument is a conditional proposition. Because it is in "if... then..." form, we know that the sufficient condition is "you have a blood alcohol level of more than 0.05", and the necessary condition is "you are not legally allowed to drive".
The second premise affirms the sufficient condition: "You have a blood alcohol level of more than 0.05".
The conclusion states the necessary condition of the conditional premise: " You are not legally allowed to drive".
As we established in the last lecture, if a conditional statement is true, and its sufficient condition holds, then the necessary condition also holds. What that tells us in this case is that if the first premise (the
conditional statement) is true, and the second premise (the sufficient condition) is true, then the conclusion (the necessary condition) must also be true.
This argument, then, is valid, since if its premises are true, its conclusion must also be true.
This will be the case for any argument which has this structure:
(6b)
1 [Conditional premise]
2 [sufficient condition of conditional premise]
C [ necessary condition of conditional premise]
Some examples are:
(6c)
1 If we are only aware of our own mental states, we cannot know whether the external world exists.
2 We are only aware of our own mental states.
C We cannot know whether the external world exists.
(6d)
1 All cats hate water 2 Fluffy is a cat C Fluffy hates water
(6e)
1 We couldn't live on Mars if there is no oxygen on Mars 2 There is no oxygen on Mars
C We couldn't live on Mars
This argument form is referred to as affirming the sufficient condition. As with all the conditional argument forms we will be looking at, the name of the form describes the role of the non-conditional premise, relative to the conditional premise.
4.1.2 Denying the necessary condition
The other basic valid conditional argument form is that of an argument such as:
(6f)
1 If we are going to get there on time, we will have to leave here by 7 o'clock 2 We won't be leaving here by 7 o'clock
C We won't get there on time.
The conditional premise of this argument states that leaving by 7 o'clock is a necessary condition for getting there on time. The second premise says that we won't be leaving by 7. So if leaving by 7 is necessary for getting there on time, but we won't be leaving by 7, it follows that we won't get there on time. So this argument is also valid. If both the premises were true, the conclusion would also have to be true.
This form is called denying the necessary condition, since the non-conditional premise denies the necessary condition of the conditional premise.
(6g)
1 [Conditional proposition]
2 [denial of the necessary condition of the conditional]
C [denial of the sufficient condition of the conditional]
By "denying" here, we really mean "contradicts" or "says the opposite of". So the following would also be an example of denying the necessary condition:
(6h)
1 If the grass is artificial, it will not grow.
2 The grass does grow C The grass isn't artificial
Since here, also, the non-conditional premise says the opposite of the necessary condition, and the conclusion says the opposite of the sufficient. Other examples of this form are:
(6i)
1 Only those who have talent should be allowed to do karaoke 2 Bert has no talent
C Bert should not be allowed to do karaoke
(6j)
1 They would have contacted us if we'd won the lottery.
2 They haven't contacted us C: We didn't win the lottery
(6k)
1 If you're nice to the dog, he doesn't bite.
2 The dog bit you
C: You weren't nice to him
Confirm for each of these arguments that the second premise denies the necessary condition of the first.
4.2 Invalid Conditional Argument Forms
There are also two common invalid forms of conditional argument. Superficially they are similar in form to the valid arguments - again, we have a conditional premise, and a premise which affirms or denies one of its conditions. In the case of the two invalid forms, however, the truth of the premises does not guarantee the truth of the conclusion.
4.2.1 Affirming the necessary condition
Suppose we know it is true that:
If Bill is a flautist, then Bill is a musician.
The sufficient condition here is "Bill is a flautist", and the necessary condition is "Bill is a musician".
Suppose that we also know that the necessary condition holds - that Bill is a musician. Would this entitle us to conclude that he is therefore a flautist? It would not - the fact that the necessary condition holds tells us nothing about whether the sufficient does or not. He could play the trombone, or sing, for example.
For that reason, the argument:
(7a)
1 If Bill is a flautist, then Bill is a musician.
2 Bill is a musician C Bill is a flautist.
is invalid. The premises could be true without the conclusion also being true. Following the convention that the name given to the form refers to how the non-conditional premise relates to the conditional premise, this form will be called affirming the necessary condition.
Other examples of this form are:
(7b)
1 If someone is a known criminal, they won't be accepted into the police force 2 Al was not accepted into the police force
C Al is a known criminal
(7c)
1 All supermodels are over 6 foot 2 Roger is over 6 foot
C Roger is a supermodel
(7d)
1 Only adults can vote in Australian elections.
2 President Bush is an adult.
C President Bush can vote in Australian elections.
In each case, it would be possible for the premises to both be true, but the conclusion false. These arguments, then, are invalid.
4.2.2 Denying the sufficient condition
The other most common invalid conditional argument form is denying the sufficient condition . (7e)
1 If Bill is a flautist, then Bill is not a musician
2 Bill is not a flautist.
C Bill is not a musician
In this case, the second premise denies the sufficient condition of the first. Again, this is an invalid form, since the fact that Bill's being a flautist would be sufficient for his being a musician, does not make it necessary. He could be a musician even though he is not a flautist, so the failure of the sufficient condition tells us nothing about whether or not the necessary condition holds.
Other examples of arguments of this form are:
(7f)
1 The rich have a responsibility to help the poor.
2 I am not rich.
C I do not have a responsibility to help the poor.
(7g)
1 If something contains a lot of salt, it is bad for you 2 This chocolate fudge cake does not contain a lot of salt C This chocolate fudge cake is not bad for you
(7h)
1 You are not allowed to drive if your blood alcohol level is over 0.05 2 Your blood alcohol level is not over 0.05
C You are allowed to drive
4.3 Examples
We will now evaluate some conditional arguments. They may occur within larger arguments, either as the main argument or a subargument, but when this is the case remember that it is only the part of the argument which is a conditional argument we can show to be valid or invalid by these methods.
(8a) UN intervention in Northern Ireland could succeed only if there was a solution to which both parties would agree. There is, however, no such solution, since both sides are convinced of the rightness of their respective causes. It follows that UN intervention in Northern Ireland cannot succeed.
First, work out what the basic components of the argument are, and identify any premise or conclusion indicators:
[UN intervention in Northern Ireland could succeed only if there was a solution to which both parties would agree]. [There is, however, no such solution] , since [both sides are convinced of the rightness of their respective causes]. It follows that [UN intervention in Northern Ireland cannot succeed].
The conclusion here is contained in the last sentence "UN intervention in Northern Ireland cannot succeed".
The main premises in support of this conclusion are "UN intervention in Northern Ireland could succeed only if there was a solution to which both parties would agree" and "There is no such solution".
The main argument could be standardised as:
1: UN intervention in Northern Ireland could succeed only if there was a solution to which both parties would agree
2: There is no solution to which both parties would agree
C: UN intervention invention in Northern Ireland cannot succeed.
The occurrence of the premise indicator "since" tells us that the remaining premise, "Both sides are convinced of the rightness of their respective causes" is a premise in support of 2.
The whole argument, then, is standardised as follows:
1: UN intervention in Northern Ireland could succeed only if there was a solution to which both parties would agree
2.1: Both sides are convinced of the rightness of their respective causes.
2: There is no solution to which both parties would agree
C: UN intervention invention in Northern Ireland cannot succeed.
The main argument here is a conditional argument. Premise 1 is the conditional premise, so the first thing to do is work out which condition is the necessary condition and which is sufficient. Sentences involving "only"
tend to emphasise what is necessary, so a good way to determine the conditions in a sentence involving
"only if" or "only" is just to ask yourself what it is saying is necessary - which condition is it saying must hold for the other condition to hold. This sentence asserts that there being a solution to which both parties would agree is necessary for UN intervention to succeed. So we have:
Sufficient condition: UN intervention could succeed
Necessary condition: There is a solution to which both parties would agree.
The name of the form comes from how the non-conditional premise relates to the conditional premise. The non-conditional premise in this conditional argument is "There is no solution to which both parties would agree".
The second premise therefore denies the necessary condition of the first. This is one of the valid forms of conditional argument. So, this conditional argument is valid, since it denies the necessary condition.
One more long example:
(8b) [It would be reasonable for Australia to criticise Japan for its continued whaling if Australia did not also have industries which involve killing animals.] But [many animals are killed for the sake of Australian
industry], since [meat production is an important Australian industry], and [killing animals is necessary for the production of meat]. So [Australia cannot reasonably criticise Japan for whaling].
Here, we have two conditional arguments, where one is a subargument of the other. The argument can be standardised as follows:
1: It would be reasonable for Australia to criticise Japan for its continued whaling if Australia did not also have industries which involve killing animals
2.1 Killing animals is necessary for the production of meat
2.2 Meat production is an important Australian industry 2: Many animals are killed for the sake of Australian industry C: Australia cannot reasonably criticise Japan for whaling.
The main argument and the subargument can each be evaluated by our methods above.
First, the main argument. The conditional premise of the main argument is premise 1, which is of the form A, if B. It says that if Australian industries did not involve the killing of animals, that would be sufficient for its being reasonable for Australia to criticise Japan for whaling.
The second premise is "Many animals are killed for the sake of Australian industry". This premise denies the sufficient condition of premise 1. (Remember that "denying" here just means "contradicts" or "says the opposite of", so it doesn't matter which premise contains the "not").
The main argument, therefore, is invalid, since it denies the sufficient condition.
The subargument is also a conditional argument .The conditional premise of the subargument is premise 2.1
"Killing animals is necessary for the production of meat". Here, the statement of the necessary condition is quite explicit. The non-conditional premise of the subargument, "Meat production is an important
Australian industry", can be understood to affirm the sufficient condition.
(If you're unsure of this, think of the argument in the following form: " Killing animals is necessary for the production of meat, Australian industry involves the production of meat, therefore Australian industry involves killing animals.")
It follows that the subargument affirms the sufficient condition, and is therefore valid. A conditional
argument may occur as a main argument or a subargument. Conditional arguments contain one conditional premise, and one premise which affirms [denies] one of the conditions. The conclusion will affirm [deny] the other condition.
The valid forms are affirming the sufficient condition, and denying the necessary condition. The invalid forms are affirming the necessary condition, and denying the sufficient condition
It doesn't matter which order the premises appear in the argument. There should be one conditional premise, and one premise which either affirms or denies one of the conditions of the conditional, but their order is unimportant.
4.4 Using Diagrams To Check Your Answers
If you found diagrams helpful in identifying necessary and sufficient conditions, you may also find them useful as a way to check the validity or invalidity of conditional arguments.
In the last lecture, a diagram was used to represent the necessary and sufficient conditions in the sentence:
Only students are eligible for the discount.
Now consider the following conditional arguments, based on that conditional statement:
(9a)
1 Only students are eligible for the discount 2 Ed is not a student
C Ed is not eligible for the discount and
(9b)
1 Only students are eligible for the discount 2 Ed is a student
C Ed is eligible for the discount.
Recall that the diagram represents the information given in the conditional statement.
Using this diagram as an aid, we can see that the first argument is valid. The second premise tells us that Ed is not a student, so if we were to place Ed on the diagram, he would have to be outside the circumference of the larger circle. But then if he were outside the larger circle, he would have to be outside the inner circle as well, so he could not be eligible for the discount.
The second argument, however, is invalid. We know from premise 2 that Ed is a student, but all that tells us is that Ed is somewhere inside the larger circle. It does not guarantee that he is also inside the inner circle.
So it would be possible for the premise to be true, and the conclusion false - there is space on our diagram for someone who is a student, but ineligible for the discount.
4.5 Validity (Deductive Validity)
In the last lecture, we established a way to decide whether a given conditional argument was valid or invalid, by looking at its form. The important feature of a valid argument was that the truth of its premises would guarantee the truth of its conclusion - it would be impossible for the premises to all be true, but the conclusion false.
An argument is valid if it would be impossible for all the premises to be true but the conclusion false. If an argument is invalid, however, it would be possible for the premises to be true and the conclusion false. If an argument is invalid, then even if all the premises are true, this does not establish the truth of the conclusion.
The analysis of validity and invalidity need not, of course, be restricted to conditional arguments. Consider, for example, the following argument:
(1a)
1: Either whales are fish, or whales are mammals.
2: Whales are not fish.
C: Whales are mammals
This argument is clearly valid, since it would be impossible for the premises to be true and the conclusion false. If we know that whales are either fish or mammals, and they're not fish, it follows that they must be mammals.
Similarly, these are valid arguments: if their premises are true, their conclusions must also be true:
(1b)
1Nothing is both a fish and a mammal 2 A whale is a mammal
C A whale is not a fish.
(1c)
1 Whales live underwater 2 Whales are mammals
C Whales are mammals which live underwater.
There are many other forms of argument whose validity we could assess by similar methods, and formal logic is concerned with exploring in detail ways of proving arguments valid or invalid.
Our concern in this course, however, is to examine ways in which arguments may be good or bad which go beyond an analysis of their validity or invalidity.
Consider, for example, the following argument:
(1d)
Every flame I have ever put my hand in has burnt me. Therefore, if I put my hand in this new flame, it will burn me.
Is this argument valid? Does the truth of its premise guarantee the truth of its conclusion?
It does not guarantee its conclusion even if its premise is true, because it is possible that putting my hand in this new flame might not burn me. Perhaps this flame is entirely different from every other flame I have experienced, and would have some entirely different effect. It would be possible for the argument to have its premise true and conclusion false, so the argument is not valid.
But is this a reason to doubt the conclusion? Should you really think that despite all the experience of the past, you should be unsure about the effect of this future flame, and about whether or not to thrust your hand into it?
Even if we do not always think of our inferences in terms of explicit arguments, the argument above clearly involves a kind of reasoning we engage in all the time. We do feel confident about the conclusions drawn through such reasoning, although the truth of those conclusions may not follow, with absolute certainty, from the premises.
Because arguments can be good, convincing arguments without being valid in the sense we have defined, it will be necessary to distinguish arguments which can be evaluated as valid or invalid from those where such an evaluation is inappropriate. Whether an argument is valid or invalid was said to be a matter of whether the conclusion of the argument would be guaranteed by the truth of the premises. So it will only be
appropriate to assess an argument as valid or invalid if the arguer intends the premises and conclusion to be related in this way.
4.6 Deductive And Inductive Arguments
The purpose of any argument, we have seen, is to convince an audience to accept a conclusion, by providing
The purpose of any argument, we have seen, is to convince an audience to accept a conclusion, by providing