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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 (1 st 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 figure Conditional 2: x is a three-sided figure  x is a triangle Bi-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.)