Logic

what is an argument? People argue all the time― that is, they

have arguments. It is not often, however, that in the course

of having an argument people actually give an argument.

Indeed, few of us have ever actually stopped to consider what it means to give an argument or what an argument is in the first place.

Yet, there is a whole discipline devoted to just this: logic.

general definition An argument is a set of statements that

includes: at least one premise which is intended to

support (i.e. give reason to believe) a conclusion.

So according to this definition is the following set of statements an argument? Ms. Mayberry left to go to work this morning.

Whenever she does this, it rains. Therefore, the moon is made of blue cheese.

Yes. We know this is an argument because of the word “therefore”:

• Ms. Mayberry left to go to work this morning. Whenever she does this, it rains. Therefore, the moon is made of blue cheese.

This typically indicates that the final sentence is intended to follow from (that is, be supported by) the preceding sentences.

Of course, it is easy to see that this is not a good argument. The premises of the argument seem to be irrelevant

to (that is, they do not support) the conclusion.

other examples

Does God exist? P1) If there is unnecessary evil in the world,

then God does not exist. P2) There is unnecessary evil in the world. C) Therefore, God does not exist.

Is ethics absolute or relative? P1) If there were absolute truth about

morality, then cultures would not disagree about morality.

P2) Cultures do disagree about morality. C) Therefore, there is no absolute truth about

morality.

The premises in these arguments may be true or false.

Either way, in each of these examples the premises support the conclusion. That is, if they are true, then they give us

reason to believe the conclusion. But this raises an important question:

What does it mean to say that premises of an argument support the conclusion?

The move from the premises to the conclusion is called an inference.

The premises support the conclusion only if the inference is good. Our question, then, concerns what it

means for an inference to be good.

validity

An argument is valid when it contains a good deductive inference.

A deductive inference is good just in case, given the truth of the premises, the conclusion must also be true – that is, if there is no way for the premises to be true and the conclusion to be false.

In short, the conclusion must be true, assuming the truth of the premises. It is extremely important to note that the above

definition does not say that the premises of the argument are true. Rather we assume that the premises are true and try to determine whether, given this assumption, the conclusion must be true as well.

validity

P1) If the animal in the barn is a pig, then it is a mammal.

P2) The animal in the barn is a pig. Therefore, the animal in the barn is a

mammal.

The truth of the premises guarantees the truth of the conclusion.

There is simply no rational way to accept the truth of both premises and still deny the conclusion. That is why it is valid.

validity

P1) If the animal in the barn is a pig, then it is a mammal.

P2) The animal in the barn has feathers and lays eggs.

Therefore, the animal in the barn is a mammal.

invalid• The premises do not guarantee the conclusion.

soundness

Of course, valid arguments with obviously false or highly controversial premises are of little real value.

What we must strive for are arguments with obviously true or relatively uncontroversial premises. That is, what we want are valid arguments with

true premises: i.e., sound arguments. A deductive argument is sound just in case

the argument is valid and, in addition, all of its premises are true.

P1) If the animal in the barn is a pig, then it is a mammal.

P2) The animal in the barn is a pig. C) Therefore, the animal in the barn is a mammal. valid sound

P1) If the animal in the barn is a pig, then it is purple.

P2) The animal in the barn is a pig. C) Therefore, the animal in the barn is purple. valid NOT sound

deductive logical structure

There is an important difference the form and the content of an argument. The form is its logical structure. The content is its subject matter, or what it’s

about. Deductive arguments are valid because

they involve the right sort of logical structure.

Content is irrelevant for validity But not for soundness (why?).

if-then conditionals The if-then conditional plays an important

role in many deductive arguments: P1) If Joe is a father, then he is a male. P2) Joe is a father. C) Therefore, he is a male.

(P1 is a conditional) – conditionals have two parts

If Joe is a father (antecedent), then he is a male (consequent).

If Joe is a father, then he is a male.

There are two valid things you can do with a conditional:

you can affirm the antecedent, or you can deny the consequent.

modus ponens (affirming the antecedent)

P1) If the moon is made of blue cheese, then pigs fly.

P2) The moon is made of blue cheese. C) Therefore, pigs fly.

P1) If it’s raining, then the streets are wet. P2) It’s raining. C) Therefore, the streets are wet.

These arguments bear an obvious similarity to one another. This is because they both have the same form. Roughly: P1) If this, then that. P1) p q P2) This. P2) p C) Therefore, that. C) Therefore, q.

modus tollens (denying the consequent)

P1) If the moon is made out of blue cheese, then pigs fly.

P2) Pigs don’t fly. C) Therefore, the moon is not made out of blue

cheese. P1) If my car can get us to Denver, then it is working

properly. P2) My car is not working properly. C) Therefore, my car cannot get us to Denver.

Once again, these arguments bear an obvious similarity to one another, which is the form: P1) If this, then that. P1) p q P2) Not that. P2) ¬ q C) Therefore, not this. C) Therefore, ¬ p

If Joe is a father, then he is a male.

There are two invalid things you can do with a conditional:

you can deny the antecedent, or you can affirm the consequent.

affirming the consequent P1) If the moon is made out of blue cheese, then

pigs fly. P2) Pigs fly. C) Therefore, the moon is made out of blue cheese.

P1) If my car can get us to Denver, then it is working properly.

P2) My car is working properly. C) Therefore, my car can get us to Denver.

Once again, these arguments bear an obvious similarity to one another, which is the form: P1) If this, then that. P1) p q P2) That. P2) q C) Therefore, this. C) Therefore, p.

denying the antecedent P1) If the moon is made of blue cheese, then pigs

fly. P2) The moon is not made of blue cheese. C) Therefore, pigs don’t fly.

P1) If it’s raining, then the streets are wet. P2) It’s not raining. C) Therefore, the streets are not wet.

These arguments bear an obvious similarity to one another. This is because they both have the same form. Roughly: P1) If this, then that. P1) p q P2) Not This. P2) ¬ p C) Therefore, not that. C) Therefore, ¬ q.

four if-then structures

Valid: P1) If p, then q.

P2) p. C) Therefore, q.

Valid: P1) If p, then q.

P2) not q. C) Therefore, not p.

Invalid: P1) If p, then q.

P2) not p. C) Therefore, not q.

Invalid: P1) If p, then q.

P2) q. C) Therefore, p.

a question

So, why is it valid to affirm the antecedent (modus

ponens) deny the consequent (modus

tollens) But not to

affirm the consequent deny the antecedent

Consider the following premise: If Joe is a father, then he is a male.

If Joe is a father, does it follow that he is a male? Yes. This is modus ponens.

If Joe is not a male, does it follow that he is not a father? Yes. This is modus tollens.

Consider the following premise: If Joe is a father, then he is a male.

If you know that Joe is a male, then can you conclude that he must be a father? No. This is affirming the consequent.

If Joe is not a father, does it follow that he is not a male? No. This is denying the antecedent.

formal fallacies It is important to realize that the reason

that these arguments forms are invalid is that their logical structures do not guarantee the truth of their conclusions. Hence, these argument forms are always

invalid.

Because affirming the consequent and denying the antecedent are fallacies that arise simply in virtue of argument form, they are called formal fallacies.

necessary & sufficient conditions

An “if-then” conditional is composed of two propositions, p and q, related by the connective ‘if, then’ (or ‘→’). The first proposition (i.e., the one that

follows the ‘if’) is part of the antecedent of the conditional

The second proposition (i.e., the one that follows the ‘then’) is part of the consequent.

sufficient conditions Suppose I say, ‘If you give birth to a baby, then you are

a mother.’

What I am saying is that the antecedent (i.e., giving birth to a baby) is enough (it is all you need) to make it true that you are a mother.

In general, we will say that the antecedent of a conditional is a sufficient condition for its consequent. Thus, giving birth to a baby is a sufficient condition for being

a mother.

This is why modus ponens is deductively valid. p → q p q

necessary conditions

‘p → q’ means that p is enough (sufficient) for q It also means that q is required (necessary) for p So, if you give birth to a baby, you must be a

mother (being a mother is required). That is, you can’t have given birth to a baby, but not be a mother.

In this way, the consequent of the conditional is a necessary condition for its antecedent. If the antecedent is true, then the consequent must be true as well Being a mother is a necessary condition for giving

birth to a baby. This is why modus tollens is deductively valid.

p → q ¬ q ¬ p

If you give birth to a baby, then you’re a mother.

You giving birth is a sufficient condition for being a mother. So, you’ve given birth to a baby only if you are a

mother.

You being a mother is a necessary condition for giving birth. So, you are a mother if you’ve given birth to a

baby.

another example P1) If Maria is alive, then she’s breathing.

The conditional (1st premise) states that: Maria’s being alive is a sufficient condition for her

breathing. So, on the assumption that we have the following

premise: P2) Maria is alive.

It follows that: C) She is breathing. (modus ponens)

Conversely, Maria’s breathing is a necessary condition for her being alive. P2) Maria’s not breathing. C) Maria is not alive. (modus tollens)

If Maria is alive, then Maria is breathing.

Maria’s being alive is a sufficient condition for her breathing. So, more generally, x is alive only if x is

breathing.

Maria’s breathing is a necessary condition for her being alive. So, more generally, x is breathing if x is alive.

natural language if-thens

If p, then q. Assuming p, q. Whenever p, q. Given p, q. Provided p, q. p only if q. (or Only if q, p.) A necessary condition of p is q.

p if q. This translates as if q, then p. So does: p when q. p since q. p in case q. p so long as q. A sufficient condition of p is q.

counterexamples

An if-then conditional p q is false just in case: p is not sufficient for q, or q is not necessary for p, or p is true while q is false.

So, to show that a conditional is false, you must show that the antecedent (p) is true but the consequent (q) is false.

Such a situation is called a counterexample.

let’s consider some examples:

Being red is a [?] condition for being scarlet. Being a horse is a [?] condition for being a

mammal. Being a female is a [?] condition for being a

sister. Being a father is a [?] condition for being a

male. Being tall is a [?] condition for being a good

BB player.

let’s consider some examples:

Being red is a [necessary] condition for being scarlet.

Being a horse is a [sufficient] condition for being a mammal.

Being a female is a [necessary] condition for being a sister.

Being a father is a [sufficient] condition for being a male.

Being tall is a [NEITHER] condition for being a good BB player.

Being red is a [necessary] condition for being scarlet. If the ball is scarlet, then it is red. If the ball is red, then it is scarlet.

Being a father is a [sufficient] condition for being a male. If he is male, then he is a father. If he is a father, then he is male.

Being a three sided figure is a [BOTH] for being a triangle. If it is a 3-sided figure, then it is a triangle. If it is a triangle, then it is a 3-sided figure.

bi-conditionals

x is a triangle only if x is a three-sided figure (necessity) x is a triangle if x is a three-sided figure (sufficiency) x is a triangle if and only if x is a three-sided figure

The last says that being a three sided figure is both necessary and sufficient for being a triangle.

Since it’s the combination of two conditionals, this is called a biconditional. Conditional 1: x is a triangle x is a three-sided figureConditional 2: x is a three-sided figure x is a triangleBi-conditional: x is a triangle x is a three-sided figure

analysis

In certain cases, to give necessary and sufficient conditions is to give a definition or an analysis.

To give an analysis of x is to state what x is. Analyses or definitions in our sense are not to be

confused with what you find in dictionaries, which often simply list various uses of words without stating what it is to be that to which the words refer.

Since one of the primary aims of philosophy is to understand the nature of things (to state what they are), philosophers are particularly interested in such biconditionals.

Consider a simple example of an analysis: x is a bachelor if and only if

(i) x is an adult,(ii) x is male, and(iii) x is unmarried.

The first thing to notice is that the analysis is stated as a biconditional (if and only if).

The second thing to notice is that we want analyses to hold necessarily.

For example, it turns out that no bachelors are over ten feet tall.

Does this mean that the following is a good analysis? x is a bachelor if and only if

(i) x is an adult,(ii) x is male,(iii) x is unmarried, and(iv) x is under ten feet tall.

This is not a good analysis. Why? Because an adult unmarried male over ten

feet tall would still be a bachelor. In other words, being under ten feet tall is not

essential to being a bachelor – it’s not part of what it is to be a bachelor.

counterexamples, again

An analysis is false just in case: there is a possible situation in which one

side holds while the other does not. To show that an analysis is false you simply

have to find a possible situation in which one side of the biconditional is true while the other side is false – that is, a possible situation in which the truth-values of the two sides differ.

Again, this is called a counterexample.

a philosophical example What is it to know that p (where p is any

proposition)? Consider the view that knowledge is true

belief. x knows that p iff

(i) x believes that p, and(ii) it is true that p.

In order to see if this is a good analysis, we need to evaluate this biconditional.

To do this, we must ask: Is each condition on the right hand side

necessary for knowledge? Are the conditions on the right hand side jointly

sufficient for knowledge?

in-class exercise

Give an analysis of love. x loves y iff … ? …

extra credit

Evaluate the following analysis of parent: x is a (biological) parent of y iff

(i) x is an ancestor of y, and (ii) x is not an ancestor of an ancestor of y.

Find a counterexample to this analysis. (There is at least one.)