An Introduction to Deductive Arguments

10 Deduction and Argument Form

Section 1: Introduction

To this point we have focused on learning how to identify and represent arguments, and have introduced the basics of argument evaluation.  However, the practice of putting these evaluative principles to work can vary from one kind of argument to another.  The remainder of this book is largely devoted to learning how to evaluate some of the most common and important kinds of arguments.  As we saw in Chapter 7, one way we can classify arguments is on the basis of their logical strength.  Recall that deductive arguments have maximal logical strength: the premises, if true, guarantee the truth of the conclusion.  In contrast, inductive arguments have less than perfect logical strength.  In Unit #5, we will focus in on inductive arguments.  In this unit, on the other hand, we will take a closer look at deduction.  Although deductive arguments are not particularly common in everyday life, they have a number of distinctive features.  Clarifying these features will not only help us understand deduction, but will give us a deeper understanding of argument evaluation more broadly.  In this chapter, we will start by identifying one distinctive feature of deductive arguments in particular, indefeasibility, and we will discuss its relation to argumentative form.  We will then identify some common deductive argument forms, and end the chapter with a look at some argumentative forms that people commonly confuse with deductive ones.

Section 2: Defeasible and Indefeasible Arguments

Most arguments are defeasible.  To say an argument is defeasible is to say that its logical strength is sensitive to new information.  Consider the following example.

Ex. 1:

Dominic: How are you going to get to the airport tonight?

Zoe: The interstate seems like the best option, since normally traffic is pretty light at that time of the evening.

Dominic: But the baseball game will be finishing right about the time you’ll be on the road, and all those people will be trying to get home.

Zoe: Oh right; I’ll guess I’ll take the back way.

Let us set aside numbers for a moment and represent Zoe’s initial inference like this:

Premise 1: Typically, traffic on the interstate is pretty light at that time of evening.

Conclusion: So, taking the interstate is the best option.

Though we don’t know all the relevant details, Zoe’s initial inference seems like a reasonably good argument.  How does adding the new information about the baseball game affect Zoe’s reasoning?

The new information Dominic adds into the mix does not show that the premise in Zoe’s argument is false, and so doesn’t challenge the factual correctness of the argument.  Rather the new information undercuts the logical strength of Zoe’s initial inference. The fact that the baseball game will be finishing at that time means traffic will probably be pretty heavy (though normally it is light at that time).  As a consequence, taking the interstate is not the best option.  Zoe’s final argument looks like this:

Premise 1: Typically, traffic on the interstate is pretty light at that time of evening.

Premise 2: The baseball game will likely be finishing about the time you’ll be on the road, and all those people will be trying to get home.

Conclusion: So, taking the interstate is not the best option (and the back way is).

Zoe’s initial inference was defeasible: although considered by itself P1 was good evidence for C1, this inference was sensitive to new information.  Indeed, in this case the introduction of P2 undermined the evidential support P1 gave to C1, so that when taken together P1 and P2 do not give us good reason to believe C1 (as Zoe’s reasoning shows).

Aerial shot of Dodger Stadium
“Dodger Stadium-Downtown L.A.” by kla4067 CC BY 2.0

Here is another example.  Suppose a detective is investigating a murder.  In the course of the investigation she discovers that the murder weapon was owned by the victim’s sister, the sister’s fingerprints were found on the gun, and the sister had a strong motive.  These facts strongly suggest that the sister is the murderer.  But again, this argument is sensitive to new information.  Say the detective subsequently discovers, for example, security camera footage showing she was 25 miles away at the time of the murder.  This piece of information completely undermines the previous argument.  The addition of this new evidence doesn’t undermine the argument by showing that one of the premises is false; rather the premises no longer support the conclusion given this addition.

We are constantly updating our beliefs in light of new information, and anytime we do so we are thinking in terms of defeasible arguments.  What does all of this have to do with deductive arguments?  Introducing the concept of defeasibility is useful for highlighting a distinctive feature of deductive arguments, namely that they are not defeasible.  That is, deductive arguments are indefeasible.  An argument is indefeasible when (and only when) its logical strength is not sensitive to new information.  Add as much relevant information as you want to a deductive argument; the information will never change the logical support the original premises lent to the conclusion.  To be clear, new information can challenge or contradict the truth of a premise in an indefeasible argument; that is, we can certainly discover that a premise we initially thought was true, is actually false.  Thus, while new information can lead us to change our mind about the factual correctness of an indefeasible argument, it can never lead us to change our mind about its logical strength.  This should strike you as odd.  How can an argument be insensitive in this way?

The answer falls right out of the definition of a deductive argument.  If the truth of the premises genuinely guarantees the truth of the conclusion, then nothing anyone might learn can change the logical support the premises give to the conclusion.  After all, that is what ‘guarantee’ means in this context: in a deductive argument it is impossible for the premises to be true and the conclusion false.

Section 3: Deduction and Argument Form

As we just saw, deductive arguments are indefeasible on account of the fact that in a deductive argument the truth of the premises guarantees the truth of the conclusion.  This raises a question: namely, how do they do this?  That is, how do deductive arguments guarantee the truth of the conclusion (if the premises are true)?  The answer is that in a deductive argument the conclusion is already contained within the premises.  By granting the premises you thereby automatically grant the conclusion.  Another way of putting this is that deductive arguments are indefeasible because of the way information is arranged within them—that is, through their form.  Let’s look at an example of a deductive argument.

  1. If Alpha Centauri is more than a light-year away from Earth, then it is further away from Earth than Jupiter.
  2. Alpha Centauri is more than a light-year away from Earth.
  3. Alpha Centauri is further away from Earth than Jupiter.

To see this argument’s form, we need to simplify its content.  To this end, we will replace each independent clause with a variable (we will use capital letters, A, B, C, etc.).  The first premise is a conditional claim that contains two independent clauses.  The first is in the antecedent and is prefaced by the word ‘if’: Alpha Centauri is more than a light-year away from Earth.  Call this A.  The second independent clause is in the consequent of the conditional, and is prefaced with the word ‘then’: it [Alpha Centauri] is further away from Earth than Jupiter.  Call this B.  By substituting the variables into the argument, we can see its form:

  1. If A, then B.
  2. A.
  3. B.

It is because the argument exhibits this basic structure that it is deductive.  It may be surprising, but no matter what information you substitute for the variables ‘A’ and ‘B’, the resulting argument will be deductive if the information is presented in this pattern.  This particular argumentative form is very common and is called Modus Ponens.

Before looking at some other deductive forms, it is important to make two notes.  First, for deductive forms the order of the premises doesn’t matter.  A standardized argument like the one above claims only that, when taken together, 1 and 2 imply 3.  It does not say that a person has to discover or think 1 first and 2 second (or vice versa).  Thus, the following is also an instance of the form Modus Ponens:

  1. A.
  2. If A, then B.
  3. So, B.

Second, many common deductive forms have conditional premises.  This raises a question about what kind of conditional is at work here: strict or strong?  Recall that in Chapter 8, we said that strict conditionals claim that the truth of the antecedent is always accompanied by the truth of consequent, no exceptions! (e.g. if a shape is a square, then it is a rectangle).  Whereas strong conditionals claim only that the truth of the antecedent is usually or mostly accompanied by the truth of the consequent (e.g. if you jump out a three-story window, then you will get hurt).  Conditionals in deductive arguments are always strict.  They have to be: think about the case of Modus Ponens above.  If the conditional were only strong and at least sometimes A is true without B, then knowing A is true wouldn’t guarantee that B is true as Modus Ponens asserts.

Section 4: Common Deductive Forms

While there are many deductive argument forms, we will isolate here only the most common ones.  We have already seen the form known as Modus Ponens (MP).  A similar deductive form is called Modus Tollens (MT).  Arguments of this kind have the following form (note: the symbol ‘~’ is shorthand for ‘not’).

  1. If A, then B.
  2. ~B.
  3. ~A.

Arguments of this form are deductive because the strict conditional tells us that if the antecedent of the conditional is true, so is the consequent.  As a result, if the consequent is not true, then neither is the antecedent.  To see this, consider the following example:

  1. If Kendall has been convicted of murder, then Kendall is a felon.
  2. It is not true that Kendall is a felon.
  3. So, it is not true that Kendall has been convicted of murder.

A similar argumentative form makes use of two strict conditionals.  These arguments are called Hypothetical Syllogisms (HS) and have the following form:

  1. If A, then B.
  2. If B, then C.
  3. So, if A, then C.

The term ‘hypothetical’ is referring to the fact that both premises are conditionals and tell you will happen in a hypothetical situation. Here is a hypothetical syllogism with content:

  1. If Kendall has been convicted of murder, then Kendall is a felon.
  2. If Kendall is a felon, then she cannot vote.
  3. So, if Kendall has been convicted of murder, then Kendall cannot vote.

One last form is called a Disjunctive Syllogism (DS).  A disjunctive syllogism is perhaps the most obviously deductive of the argument forms discussed here.  These arguments have the following form:

  • 1) Either A or B.
  • 2) ~A.
  • 3) So, B.

‘Disjunction’ is the grammatical term of an ‘or’ statement (and the terms of disjunction are called “disjuncts”).  We have seen examples like this in previous homework.  Consider the following:

  1. Either the butler committed the murder or the judge did it.
  2. It was not the butler.
  3. So, the judge committed the murder.

The first premise is a disjunction, and its disjuncts are ‘the butler committed the murder’ and ‘the judge committed the murder’.  It is important to be clear that there are many deductive argument forms besides these.  Moreover, these argument forms can be strung together into more complicated deductive arguments. For example, imagine the conclusion of a Hypothetical Syllogism is If C, then D, and the conclusion of a Disjunctive Conclusion is C.  Bringing these two pieces of information together you could deduce, by Modus Ponens, that D is true.

Section 5: Illusory Forms

There are a number of argumentative patterns that resemble genuinely deductive forms and are commonly (but mistakenly) seen to be deductive forms themselves.  The first of these is called Denying the Antecedent (DA).  Arguments of this kind have the following form:

  1. If A, then B.
  2. ~A.
  3. ~B.

Instances of denying the antecedent are not deductive arguments because the premises do not guarantee the truth of the conclusion.  Strict conditionals say that if the antecedent is true, then so is the consequent.  It does not tell us what is true when the antecedent is false.  For example:

  1. If Will is enrolled in Biology 101, then Will is enrolled in a science course.
  2. Will is not enrolled in Biology 101.
  3. So, Will is not enrolled in a science course.

If the premises are true, they do not guarantee the conclusion because there are many different science courses Will might be enrolled in besides BIO 101 (e.g. Chemistry 101, Astronomy 101).

2 dimensional house that looks 3 dimensional
“Illusory House II” by Frankenstein CC BY-NC 2.0

The second illusory argument form we will discuss is called Affirming the Consequent (AC).   Arguments of this kind have the following form:

  1. If A, then B.
  2. B.
  3. So, A.

Instances of affirming the consequent are not deductive forms because the strict conditional only tells us that if A is true, then B is too.  It doesn’t tell us what is the case when B is true.  Let us turn to Will again:

  1. If Will is enrolled in Biology 101, then Will is enrolled in a science course.
  2. Will is enrolled in a science course.
  3. So, Will is enrolled in Biology 101.

Again, the premises, if true, do not guarantee the truth of the conclusion for the same reason.  There are many different science courses Will might be enrolled in, and the fact that he is enrolled in a science course doesn’t tell us that it must be Biology 101.  Importantly, there are many other non-deductive argument forms besides these two.  We have focused on Denying the Antecedent and Affirming the Consequent because they are particularly deceptive.  That is, many people treat arguments with these forms as if they were deductive, even though they are not.

Exercises

Exercise Set 10A:

Directions: Determine whether each of the following is true or false, and explain your answer.

#1:

A deductive argument can have a false premise.

#2:

A deductive argument can have a false conclusion.

#3:

A deductive argument can have all false premises and a false conclusion.

#4:

A deductive argument can have all false premises and a true conclusion.

#5:

A deductive argument can have all true premises and a false conclusion.

Exercise Set 10B:

Directions: For each of the following, first, determine whether the argument is an instance of one of the forms identified above, and second whether the argument is deductive or not.  Assume that the conditionals are all strict. Note: some arguments may not be of any of the kinds we have discussed so far.

#1:

If Zala is a senior, then she can register for this class, but since she is not a senior, I guess she cannot register for the class.

#2:

If an infectious disease has an R0 value less than 10, then it is less contagious than chickenpox.  This virus has an R0 value less than 10, and so the virus is less contagious than chickenpox.

#3:

If the proposed explanation is correct, then the specimen will dilate.  Therefore, since the specimen did not dilate, we know the proposed explanation is incorrect.

#4:

Your problem has got to be a bad fuel pump, since it is either a bad fuel pump or bad filter and I know it isn’t the filter.

#5:

If the state adopts water restriction policies, then you won’t be allowed to water your lawn, and if you aren’t allowed to water your lawn, then it will die.  So, if the state adopts water restriction policies, then your lawn will die.

#6:

I strongly suspect the head of the accounting office has been cheating the company somehow.  I know how much she gets paid, and she has been living beyond her means for years.

#7:

If proposed explanation is correct, then the specimen will dilate.  Since the specimen did dilate, we can conclude that the proposed explanation is correct.

#8:

If your business is certified by the SBA, then it is eligible for tax credits on the installation of new solar panels.  Since your business is not certified by the SBA, then it is not eligible for these tax credits.

#9:

If public high school education in this state were truly equitable, then all high-school students would have equal access to AP courses in History, English Language and Composition, and Calculus.  But less than ½ the high schools in the state offer students AP courses in these three subjects.   It follows that public high school education in this state is not equitable.

Exercise Set 10C:

#1:

Given what we’ve learned in this chapter, what do you make of the following inference?

If Japan makes its airports more convenient for the Japanese public, Japanese tourism in Europe will increase. If Japanese tourism in Europe increases, then either European facilities in key centers will be enlarged or crowding in key centers will occur.  Japan is making its airports more convenient for the Japanese public, but European centers are not expanding their tourist facilities.  So we can expect crowding in main European tourist centers.

#2:

Suppose that your team is going to a tournament, and you need to find somebody to drive the college van.  The driver has to be at least 22 and have an unrestricted driver’s license.  A friend of a friend, suggests Sam.  You talk to her briefly and she claims she meets both conditions, and you reason as follows:

  1. Sam has an unrestricted driver’s license and is at least 22 years old.
  2. If Sam has an unrestricted driver’s license and is at least 22 years old, then she is eligible to drive the college van.
  3. So, Sam is eligible to drive the college van.

First, what form is this deductive argument?  Suppose that you subsequently learn from Sam’s roommate that she actually doesn’t have an unrestricted license (despite what she said).  This will surely lead you to revise your earlier thinking, but doesn’t this show that that not all deductive arguments are defeasible?  After all, a new piece of information led you to change your mind.  Is this really an example of a defeasible deductive argument?  Why or why not?

 

 

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Arguments in Context by Thaddeus Robinson is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License, except where otherwise noted.

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