PHIL 2100: Introduction to Philosophy
Dr. Robert Lane
Lecture Notes: Wednesday June 9, 2010

 

[2.] Logic.

 

As we have already seen, philosophers do not simply announce their theories to the world without any evidence to back them up. Philosophy depends on reasoning and evidence. Philosophers test existing theories, and develop new ones, primarily by reasoning. They give arguments to support their claims:

 

argument (df.): a set of statements some of which (the argument’s premises) are intended to serve as evidence or reasons for thinking that another statement (the argument’s conclusion) is true.

 

Some simple examples of arguments:

 

 

 

 

As you can see, an argument in logic is not a fight, quarrel or disagreement. It is a statement put forward along with reasons for thinking that it is true. To give an argument is to make a claim and attempt to support that claim with evidence.

 

As Sober points out, the use of arguments to support claims in not unique to philosophy; “mathematicians do this, as do economists, physicists, and people in everyday life.” (p.7) What sets philosophy apart from other areas of inquiry is not its use of arguments but the kinds of questions that it attempts to answer.

 

Still, because philosophy does rely so heavily on argumentation, before we can really do philosophy seriously, we need to know at least a little about how to tell whether a given argument is good or bad.

 

So we need to learn a little about

 

logic (df.): the area of philosophy that studies reasoning and arguments; it is particularly concerned with what distinguishes good arguments from bad ones.

 

 

[2.1.] What Makes an Argument Good or Bad?

 

This is a very hard question to answer completely. In a way, the entire Critical Thinking (PHIL 2020) course taught here at UWG is a semester-long answer to this question.

 

But a short answer will do for present purposes: an argument is good when it is rationally persuasive, i.e., when it would be rational, or reasonable, to be convinced by the argument and thus accept its conclusion.

 

So far, so good… but exactly what is it about an argument that makes it rationally persuasive?

 

An argument is good (i.e., rationally persuasive) when it meets two conditions:

 

1. it has true premises;

2. there is a logical connection between premises and conclusion.

 

The first condition is easy to understand. It would be irrational to accept a conclusion that is supported by premises that you know to be false.

 

Understanding the second condition is a bit trickier…

 

Logically good arguments can be sorted into two broad categories, the second of which can be divided further...  (see chart on p.8)

 

logically good arguments

Deductively Valid

Not Deductively Valid

·         inductively strong

·         abductively strong

 

 

We’ll talk about good arguments that are not deductively valid arguments tomorrow. For now we’ll focus on deductively valid arguments...

 

 

[2.2.] Deductively Valid Arguments.

 

Consider the following simple argument:

 

All mammals are animals. (premise)

All dogs are mammals. (premise)

Therefore, all dogs are animals. (conclusion)

 

This argument is displayed in a way that makes it very explicit which statements are the premises and which is the conclusion. But most of the time, when you come across an argument in your reading (even in your philosophy reading), it will not be indented from the margin, with each statement on a separate line. Normally, it will be written in prose form.  Further, the conclusion may come before the premises. For example, this argument could be written: “Of course dogs are animals; they’re mammals, and all mammals are animals.”

 

The word “therefore” indicates that the conclusion is supposed to be supported by the premises. (For this reason, “therefore” is sometimes referred to as a conclusion indicator. Other conclusion indicators are “so” and “hence.)

 

Notice that in the mammals argument, there is a

·         very tight connection between premises and conclusion; more specifically:

·         the conclusion follows from the premises; this means that

·         if the premises were true, then the conclusion would have to be true as well.

 

This “follows from” property is known as deductive validity:

 

deductive validity, a.k.a. validity (df.): A valid argument is one in which

1.      if the premises were true, then the conclusion would have to be true as well.

2.      the truth of the premises would guarantee the truth of the conclusion;

3.      it is impossible for the premises to be all true and the conclusion to be false at the same time.

·         These are three equivalent ways of defining validity.

·         IMPORTANT: In logic and philosophy, the word “valid” does not mean exactly the same thing that it does in ordinary English. For the purposes of this class, “valid” means only what it is defined to mean above: if the premises were true, then the conclusion would have to be true, as well.

 

 

[2.2.1.] Examples of Deductively Valid Arguments.

 

To illustrate validity, I will use slightly different examples than Sober (see his on p.9):

 

 

All humans are mortal.

All politicians are human.

Therefore, all politicians are mortal.

 

 

All dogs are mammals.

All pit bulls are dogs.

Therefore, all pit bulls are mammals.

 

These two arguments have the same logical form: the blueprint according to which the argument is constructed. This is the logical form shared by these two arguments

 

 

Form #1

 

All Bs are Cs.

All As are Bs.

All As are Cs.

 

 

We can arrive at those first two arguments by filling in the values for “B,” “C” and “A” as follows:

 

B: humans

C: mortal

A: politicians

B: dogs

C: mammals

A: pit bulls

 

There are loads of other possible arguments with this logical form:

 

  B: spiders

        C: arachnids

        A: black widows

 

        B: dogs

        C: fluent in German

        A: pit bulls

 

Any argument that has this logical form will be deductively valid, no matter what A, B and C stand for.

 

 

Now you’re ready for two important facts about validity:

 

IMPORTANT FACT #1

 

Whether or not an argument is valid has nothing to do with the actual truth of its premises and conclusion. Validity does not require true premises, or a true conclusion. The only impossible combination in a valid argument is: true premises and a false conclusion. (See the table on p.12.)

 

So this argument…

 

All dogs are fluent in German.

All pitbulls are dogs.

Therefore, all pitbulls are fluent in German.

 

is deductively valid: IF the premises were true, then the conclusion would have to be true as well.

 

This is why it is important to define validity this way:

1. if the premises were true, then the conclusion would have to be true as well.

2. the truth of the premises would guarantee the truth of the conclusion

 

 …instead of this way:

(*)  if the premises are true, then the conclusion must be true as well;

the truth of the premises does guarantee the truth of the conclusion.

 

(*) are not good definitions of validity, because they imply that the premises must actually be true in order for the argument to be valid. But that’s wrong: the premises do not actually have to be true in order for the argument to be valid.

 

Why call an argument with false premises valid? Remember: “valid” is a technical term. It does not mean the same in logic as it does in its normal, every day usage.

 

 

IMPORTANT FACT #2

 

Whether or not an argument is valid usually depends on nothing but its logical form; it never depends on whether its premises are actually true or false.[2]

 

Here’s another form:

 

Form #2

No Bs are Cs.

All As are Bs.

Therefore, no As are Cs.

 

 

Filling in either set of values…

 

 

        B: dogs

        C: fluent in German

        A: pit bulls

 

 

  B: spiders

        C: arachnids

        A: black widows

 

gives a valid argument. The first yields an argument with all true premises; the second yields an argument with one true premise and one false premise. But even if the premises or conclusion were all false, the argument would still be valid.
 

 

So far we’ve only looked at logical forms with blank spaces that can be filled in with names of categories, or sets, or kinds of thing. Other logical forms have blank spaces that can be filled in with entire statements.

 

Form #3:

If p, then q

p________

Therefore, q

 

For example:

 

If Usher lives in Atlanta, then Usher lives in Georgia.

Usher lives in Atlanta.

Therefore, Usher lives in Georgia.

p: Usher lives in Atlanta.

q: Usher lives in Georgia.

 

If it is raining, then the streets are wet.

It is raining.

Therefore, the streets are wet.

p: It is raining.

q: The streets are wet.

 

 

[2.2.2.] Soundness.

 

soundness (df.): A sound argument is an argument that (1) is deductively valid and (2) has all true premises.

 

Some of the arguments we’ve examined so far have been sound…

 

 

All humans are mortal.

All politicians are human.

Therefore, all politicians are mortal.

 

 

All dogs are mammals.

All pit bulls are dogs.

Therefore, all pit bulls are mammals.

 

But others have not been sound:

 

All dogs are fluent in German.

All pitbulls are dogs.

Therefore, all pitbulls are fluent in German.

 

 

[2.2.3.] Invalidity.

 

 

From the definition of validity, we can derive a definition of invalidity:

 

deductively invalidity, a.k.a. invalidity (df.): an invalid argument is one in which the truth of the premises would not guarantee the truth of the conclusion. I.e. (in other words) it is possible for the premises to be true and the conclusion false at the same time.

 

Here are two examples of invalid arguments with true premises and false conclusions:

 

Form #4:

All Bs are Cs.

No As are Bs.

Therefore, no As are Cs.

 

All fish are swimmers. (T)

No mammals are fish. (T)

Therefore, no mammals are swimmers. (F)

 

Form #5:

If p, then q.

q________

Therefore, p

 

If Jimmy Carter is from Texas, then he is from the USA. (T)

Jimmy Carter is from the USA. (T)

Therefore, Jimmy Carter is from Texas. (F)

 

But remember that it is possible for an invalid argument to have true premises and a true conclusion:

 

Form #4:

All Bs are Cs.

No As are Bs.

Therefore, no As are Cs.

 

All emeralds are green (T)

No crows are emeralds. (T)

Therefore, no crows are green. (T)

 

Form #5:

If p, then q.

q________

Therefore, p

 

If Lane is from Selma, then Lane is from Alabama. (T)

Lane is from Alabama. (T)

Therefore, Lane is from Selma. (T)

 

 

 

IMPORTANT FACT #3

 

It’s easy to turn an invalid argument into a valid one… but it’s not easy to turn an invalid argument into a good valid argument. For example, this is an invalid argument:

 

All pit bulls are dogs. (T)

Therefore, all poodles are dogs. (T)

 

We can easily make it valid by adding the premise “All poodles are pit bulls”:

 

All pit bulls are dogs. (T)

All poodles are pit bulls. (F)

Therefore, all poodles are dogs. (T)

 

But this does not transform this into a good valid argument, since the premise that we have added is false. The logical aspect of the argument is now good, but since it has a false premise, overall it is not be a good argument.

 

Remember, just being valid does not make an argument good – it must also have premises all of which are true. Arguments purport to give reasons or evidence for thinking that a conclusion is true. But if those reasons are themselves false, then the argument is doing a pretty lousy job, no matter how good the logical relationship is between premises and conclusion.

 

 

[2.2.4.] Common Valid Argument Forms.

 

Some of the argument forms we’ve looked at have names…

 

Form #3:

If p, then q.

p________

Therefore, q.

 

If it’s raining, then the streets are wet.

It’s raining.

Therefore, the streets are wet.

Name:

modus ponens

 

VALID

 

Form #5:

If p, then q.

q________

Therefore, p.

 

If it’s raining, then the streets are wet.

The streets are wet.

Therefore, it’s raining.

Name:

affirming the consequent

 

INVALID

 

 

A relevant term that you need to know…

 

conditional (df.): an if-then statement. Conditionals need not be expressed in the form “if p, then q” – “if p, q” is a conditional, as is “q, if p”. The “if” part of a conditional is called its antecedent (in “if p, then q,” “p” is the antecedent). The “then” part of a condition is called its consequent (in “if p, then q,” “q” is the consequent).

 

Two other common argument forms involving conditionals are:

 

Form #6:

If p, then q.

not-q________

not-p

 

If it’s raining, then the streets are wet.

The streets are not wet.

It’s not raining.

Name:

modus tollens

 

VALID

 

Form #7:

If p, then q.

not-p________

not-q

 

If it’s raining, then the streets are wet.

It’s not raining.

The streets are not wet.*

Name:

denying the antecedent

 

INVALID

 

*This conclusion does not follow from the two premises, i.e., it is possible for the premises to be true and yet this conclusion false, for example, if there is not a cloud in the sky but there is a broken water main that is spilling water out into the streets.

 

IMPORTANT FACT #4

 

Whether or not an argument is valid does not depend on the order of the premises.  In any of the examples we’ve examined, you can change the order of the premises without changing the logical connection between premises and conclusion.

 

IMPORTANT FACT #5

 

Arguments are not limited to having two premises; they can have any number of premises whatsoever. I have stuck to relatively short, two-premise arguments so far just to make it easier for you to understand the logical points we’re considering. But later on we will examine arguments that have more than two premises.

 

[2.4.] Circular Reasoning / Begging the Question.

 

We have looked at two important characteristics of arguments:

 

1.      The truth values of the premises.

 

2.      The logical connection between premises and conclusion – so far we’ve only looked at arguments with the tightest possible connections (deductively valid arguments) and those with not-very-tight-at-all connections. (Soon we’ll look at arguments which are not valid but which still have some degree of tightness between premises and conclusion.)

 

We’ve said that an argument that’s valid is not necessarily good – if a valid argument has false premises, it’s a bad argument. So are all valid arguments that have true premises (i.e., sound arguments) good? No…

 

Some such arguments commit the mistake known as

 

begging the question (df.): an argument makes the mistake of begging the question when it assumes in its premises the very claim that it is supposed to be proving.

 

explicit:

 

The current President is a Democrat.

Therefore, the current President is a Democrat.

 

 

This is obviously bad—an argument is supposed to persuade you rationally that its conclusion is true. Just saying the conclusion over again doesn’t do this.

 

implicit:

 

If the Bible says that God exists, then God exists.

The Bible says that God exists.

Therefore, God exists.

 

 

This is not a good argument, because the first premise presupposed the truth of the conclusion. No one would accept that first premise unless he or she already believed the conclusion.

 

 

[2.5.] Good Arguments That Aren’t Deductively Valid.

 

[2.5.1.] Security and Informativeness.

 

We have seen that valid arguments have maximum security, i.e., they have the tightest possible logical connection between premises and conclusion.

 

But valid arguments are not very informative. In other words, the information in the conclusion does not go far beyond the information contained in the premises. Valid arguments have minimum informativeness.

 

As logical security drops, informativeness increases, and vice versa.

 

Consider this argument, which is not valid but which has higher informativeness than any we looked at yesterday:

 

In a recent survey, 70% of students in this class responded that they were pro-choice.

Therefore, 70% of the student body is pro-choice.

 

(Compare this to Sober’s example of a phone survey of voters in a specific county, asking about their political party affiliations; p.20.)

 

Because it is invalid, the truth of the premise would not guarantee the truth of the conclusion (the members of the class might not be representative of the student body as a whole, or they might be lying.).

 

But it can provide something weaker than a guarantee: the truth of the premise(s) can make the conclusion more probable.

 

Probability, or likelihood, is a matter of degree… an argument can have premises the truth of which makes its conclusion extremely probable, very probable, somewhat probable, etc. And this makes perfect sense, since security is a matter of degree.

 

100%

99%

 

 

 

50%

 

 

 

0%

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valid

less than valid, but still extremely secure

 

Any degree of security below the maxim (i.e., below an absolute guarantee of the truth of the premises by the truth of the conclusion) is referred to as strength:

 

logical strength, a.k.a. strength (df.): an argument is logically strong to the degree that the truth of the premises makes the conclusion probable, or likely, or provides good reason for thinking that the conclusion is true.

 

 

Sober’s gambling analogy (p.21) illustrates how informativeness and security vary:

·         A deductive argument is like a bet that is absolutely safe (1:1 odds), but the payoff is $0. The extreme conservative strategy is only to place absolutely safe bets.

·         A non-deductive argument is like a bet with some payoff (> $0) but also with some risk (< 1:1 odds). A thoughtful risk taker might make bets like this.

 

In our reasoning, why shouldn’t be just be extreme conservatives? Why engage in reasoning that is less than secure than deductive reasoning?

·         The answer lies in the fact that logical security and informativeness are inversely proportional: as one goes up, the other goes down.

·         So unless we engage in thoughtful risk taking with our reasoning, we will never reach conclusions that tell us anything new.

This would have especially bad consequences for our ability to justify two types of belief:

 

1.      Beliefs About the Future (“predictive beliefs”). Many of our ordinary everyday beliefs are based on reasoning that is not deductively valid…

 

Every day up to now, the sun has risen in the morning.

Therefore, the sun will rise tomorrow morning.

 

I have before never grown a foot in height overnight.

Therefore, I will not wake up tomorrow and be one foot taller than I am now.

 

Neither one of these arguments is deductively valid, since in each one it is possible for the premise to be true and the conclusion false at the same time.

 

2.      Universal Generalizations. Science is concerned to establish the truth of universal generalizations, statements which are true for all places and times (e.g. “e = mc2”, “all crows are black,” “pure water boils at 212º F at standard pressure”). The only premises from which science can begin have to do with particular observed cases, and universal conclusions cannot validly follow from those.

 

Crow a is black, crow b is black, crow c is black…

Therefore, all crows are black.

 

Reasoning that results in beliefs about the future or in universal generalizations are never deductively valid, but they are still capable of being logically strong. There are two types of non-deductive but logically strong reasoning:

·         induction

·         abduction

 

 

Stopping point for Wednesday June 9. For next time, read Sober ch.3. Tomorrow you will have a pop quiz over today’s lecture notes AND over this reading.

 



[1] The second and third examples are from Kahane & Cavander, Logic and Contemporary Rhetoric.

[2] To say that it always depends on nothing but its logical form is an overstatement. There are some valid arguments the validity of which does not depend on their logical form, at least not obviously, for example: “his object is red all over; therefore, this object is not green all over.” So learn this important fact, but also remember that there are a few exceptions.




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