Arguments

Learning Outcomes

  • Discern between an inductive argument and a deductive argument
  • Evaluate deductive arguments
  • Analyze arguments with Euler diagrams (a form of Venn diagram) or truth tables
  • Use logical inference to infer whether a statement is true

A logical argument is a claim that a set of premises support a conclusion. There are two general types of arguments: inductive and deductive arguments.

strategy for success

Practice the examples and TRY IT problems in this section as much as you can. The ideas of logic are powerful and help to make us more powerful out in the world.

As always, practice and repetition will help you to gain familiarity. Practicing as many different problems as you can will give you broader exposure to logical situations and will help you obtain a deeper understanding overall.

Argument types

An inductive argument uses a collection of specific examples as its premises and uses them to propose a general conclusion.

A deductive argument uses a collection of general statements as its premises and uses them to propose a specific situation as the conclusion.

To help you understand the difference between inductive and deductive, here are definitions found on the Dictionary website.

What does inductive mean?

Inductive is used to describe reasoning that involves using specific observations, such as observed patterns, to make a general conclusion. This method is sometimes called induction. Induction starts with a set of premises, based mainly on experience or experimental evidence. It uses those premises to generalize a conclusion.

For example, let’s say you go to a café every day for a month, and every day, the same person comes at exactly 11 am and orders a cappuccino. The specific observation is that this person has come to the cafe at the same time and ordered the same thing every day during the period observed. A general conclusion drawn from these premises could be that this person always comes to the cafe at the same time and orders the same thing.

While inductive reasoning can be useful, it’s prone to being flawed. That’s because conclusions drawn using induction go beyond the information contained in the premises. An inductive argument may be highly probable, but even if all the observations are accurate, it can lead to incorrect conclusions.

In our basic example, there are a number of reasons why it may not be true that the person always comes at the same time and orders the same thing.

Additional observations of the same event happening in the same way increase the probability that the event will happen again in the same way, but you can never be completely certain that it will always continue to happen in the same way.

That’s why a theory reached via inductive reasoning should always be tested to see if it is correct or makes sense.

What does deductive mean?

Deductive reasoning (also called deduction) involves starting from a set of general premises and then drawing a specific conclusion that contains no more information than the premises themselves. Deductive reasoning is sometimes called deduction (note that deduction has other meanings in the contexts of mathematics and accounting).

Here’s an example of deductive reasoning: chickens are birds; all birds lay eggs; therefore, chickens lay eggs. Another way to think of it: if something is true of a general class (birds), then it is true of the members of the class (chickens).

Deductive reasoning can go wrong, of course, when you start with incorrect premises. For example, look where this first incorrect statement leads us: all animals that lay eggs are birds; snakes lay eggs; therefore, snakes are birds.

The scientific method can be described as deductive. You first formulate a hypothesis—an educated guess based on general premises (sometimes formed by inductive methods). Then you test the hypothesis with an experiment. Based on the results of the experiment, you can make a specific conclusion as to the accuracy of your hypothesis.

What is the difference between inductive vs. deductive reasoning?

Inductive reasoning involves starting from specific premises and forming a general conclusion, while deductive reasoning involves using general premises to form a specific conclusion.

Conclusions reached via deductive reasoning cannot be incorrect if the premises are true. That’s because the conclusion doesn’t contain information that’s not in the premises. Unlike deductive reasoning, though, a conclusion reached via inductive reasoning goes beyond the information contained within the premises—it’s a generalization, and generalizations aren’t always accurate.

Try It

Example

The argument “when I went to the store last week I forgot my purse, and when I went today I forgot my purse. I always forget my purse when I go the store” is an inductive argument.

The premises are:

I forgot my purse last week
I forgot my purse today

The conclusion is:

I always forget my purse

Notice that the premises are specific situations, while the conclusion is a general statement. In this case, this is a fairly weak argument, since it is based on only two instances.

Example

The argument “every day for the past year, a plane flies over my house at 2pm. A plane will fly over my house every day at 2pm” is a stronger inductive argument, since it is based on a larger set of evidence.

Evaluating inductive arguments

An inductive argument is never able to prove the conclusion true, but it can provide either weak or strong evidence to suggest it may be true.

Many scientific theories, such as the big bang theory, can never be proven. Instead, they are inductive arguments supported by a wide variety of evidence. Usually in science, an idea is considered a hypothesis until it has been well tested, at which point it graduates to being considered a theory. The commonly known scientific theories, like Newton’s theory of gravity, have all stood up to years of testing and evidence, though sometimes they need to be adjusted based on new evidence. For gravity, this happened when Einstein proposed the theory of general relativity.

A deductive argument is more clearly valid or not, which makes them easier to evaluate.

Evaluating deductive arguments

A deductive argument is considered valid if all the premises are true, and the conclusion follows logically from those premises. In other words, the premises are true, and the conclusion follows necessarily from those premises.

When the argument contains quantifiers such as “all”, “some”, or “none (no)”, we can use a form of a Venn diagram referred to as an Euler (pronounced Oiler) diagram to evaluate it.  See the example below.

Example

The argument “All cats are mammals and a tiger is a cat, so a tiger is a mammal” is a valid deductive argument.

The premises are:

All cats are mammals
A tiger is a cat

The conclusion is:

A tiger is a mammal

Both the premises are true. To see that the premises must logically lead to the conclusion, one approach would be to use an Euler diagram. From the first premise, we can conclude that the set of cats is a subset of the set of mammals. From the second premise, we are told that a tiger lies within the set of cats. From that, we can see in the Euler diagram that the tiger also lies inside the set of mammals, so the conclusion is valid.

A Venn diagram with a large circle labeled Mammals and a smaller circle labeled Cats contained within the Mammals circle. A red X labeled Tiger is in the circle labeled Cats.

Try It

No similar questions are available for this example.

Analyzing Arguments with Venn/Euler diagrams

To analyze an argument with a Venn/Euler diagram

  1. Draw a Venn/ Euler diagram based on the premises of the argument.
  2. If the premises are insufficient to determine the location of an element, indicate that.
  3. The argument is valid if it is clear that the conclusion must be true.  In other words, the diagram cannot be drawn contrary to the conclusion.

Example

Premise: All firefighters know CPR
Premise: Jill knows CPR
Conclusion: Jill is a firefighter

From the first premise, we know that firefighters all lie inside the set of those who know CPR. From the second premise, we know that Jill is a member of that larger set, but we do not have enough information to know if she also is a member of the smaller subset that is firefighters.

Fig4_2_2

Since the conclusion does not necessarily follow from the premises, this is an invalid argument, regardless of whether Jill actually is a firefighter.

It is important to note that whether or not Jill is actually a firefighter is not important in evaluating the validity of the argument; we are only concerned with whether the premises are enough to prove the conclusion.

In addition to these categorical style premises of the form “all ___,” “some ____,” and “no ____,” it is also common to see premises that are implications.

Example

Premise: If you live in Seattle, you live in Washington.
Premise: Marcus does not live in Seattle.
Conclusion: Marcus does not live in Washington.

From the first premise, we know that the set of people who live in Seattle is inside the set of those who live in Washington. From the second premise, we know that Marcus does not lie in the Seattle set, but we have insufficient information to know whether or not Marcus lives in Washington or not. This is an invalid argument.

Fig4_2_3

Example

Consider the argument “You are a married man, so you must have a wife.”

Some arguments are better analyzed using truth tables.

Example

Consider the argument:

Premise: If you bought bread, then you went to the store
Premise: You bought bread
Conclusion: You went to the store

While this example is hopefully fairly obviously a valid argument, we can analyze it using a truth table by representing each of the premises symbolically. We can then look at the implication that the premises together imply the conclusion. If the truth table is a tautology (always true), then the argument is valid.

We’ll let B represent “you bought bread” and S represent “you went to the store”. Then the argument becomes:

Premise: [latex]B{\rightarrow}S[/latex]
Premise: [latex]B[/latex]
Conclusion: [latex]S[/latex]

Next, we will take our premises and combine them using a conjunction (AND).  Then, along with the conclusion, we form an implication.  We want the form to be [ (Premise 1) Λ (Premise 2) ] → Conclusion.

To test the validity, we look at whether the combination of both premises implies the conclusion; is it true that [latex]\left[\left(B{\rightarrow}S\right){\wedge}B\right]{\rightarrow}S[/latex]?

[latex]B[/latex] [latex]S[/latex] [latex]B{\rightarrow}S[/latex] [latex]\left(B{\rightarrow}S\right){\wedge}B[/latex] [latex]\left[\left(B{\rightarrow}S\right){\wedge}B\right]{\rightarrow}S[/latex]
T T T T T
T F F F T
F T T F T
F F T F T

Since the truth table for [latex]\left[\left(B{\rightarrow}S\right){\wedge}B\right]{\rightarrow}S[/latex] is a tautology (always true), this is a valid argument.

 

Analyzing arguments using truth tables

To analyze an argument with a truth table:

  1. Represent each of the premises symbolically
  2. Create a conditional statement (an implication), joining all the premises with AND (Λ) to form the antecedent, and using the conclusion as the consequent.  We want the form [ (Premise 1) Λ (Premise 2) ] → Conclusion.
  3. Create a truth table for that statement. If it is always true, then the argument is valid.  When the last column of the truth table has all T’s, we call that a tautology which means the argument is valid.

Example

Premise: If I go to the mall, then I’ll buy new jeans.
Premise: If I buy new jeans, I’ll buy a shirt to go with it.
Conclusion: If I go to the mall, I’ll buy a shirt.

Let M = I go to the mall, J = I buy jeans, and S = I buy a shirt.

The premises and conclusion can be stated as:

Premise: [latex]M{\rightarrow}J[/latex]
Premise: [latex]J{\rightarrow}S[/latex]
Conclusion: [latex]M{\rightarrow}S[/latex]

We can construct a truth table for [latex]\left[\left(M{\rightarrow}J\right)\wedge\left(J{\rightarrow}S\right)\right]{\rightarrow}\left(M{\rightarrow}S\right)[/latex]

[latex]M[/latex] [latex]J[/latex] [latex]S[/latex] [latex]M{\rightarrow}J[/latex] [latex]J{\rightarrow}S[/latex] [latex]\left(M{\rightarrow}J\right)\wedge\left(J{\rightarrow}S\right)[/latex] [latex]M{\rightarrow}S[/latex] [latex]\left[\left(M{\rightarrow}J\right)\wedge\left(J{\rightarrow}S\right)\right]{\rightarrow}\left(M{\rightarrow}S\right)[/latex]
T T T T T T T T
T T F T F F F T
T F T F T F T T
T F F F T F F T
F T T T T T T T
F T F T F F T T
F F T T T T T T
F F F T T T T T

From the truth table, we can see this is a valid argument.

The previous problem is an example of a syllogism.

Syllogism

A syllogism is an implication derived from two others, where the consequence of one is the antecedent to the other. The general form of a syllogism is:

Premise: [latex]p{\rightarrow}q[/latex]
Premise: [latex]q{\rightarrow}r[/latex]
Conclusion: [latex]p{\rightarrow}r[/latex]

This is sometime called the transitive property for implication.

more on the transitive property

The transitive property appears regularly in the various branches of mathematical study. For example, the transitive property of equality states

if [latex]a = b[/latex] and [latex]b = c[/latex] then [latex]a = c[/latex].

Example

Premise: If I work hard, I’ll get a raise.
Premise: If I get a raise, I’ll buy a boat.
Conclusion: If I don’t buy a boat, I must not have worked hard.

Try It

Is this argument valid?

Lewis Carroll, author of Alice in Wonderland, was a math and logic teacher, and wrote two books on logic. In them, he would propose premises as a puzzle, to be connected using syllogisms.

Example

Solve the puzzle. In other words, find a logical conclusion from these premises.

All babies are illogical.

Nobody who can manage a crocodile is despised.

Illogical persons are despised.

Let B = is a baby, D = is despised, I = is illogical, and M = can manage a crocodile.

Then we can write the premises as:

[latex]B{\rightarrow}I\\M{\rightarrow}{\sim}D\\I{\rightarrow}D[/latex]

From the first and third premises, we can conclude that [latex]B{\rightarrow}D[/latex]; that babies are despised.

Using the contrapositive of the second premised, [latex]D{\rightarrow}{\sim}M[/latex], we can conclude that [latex]B\rightarrow\sim{M}[/latex]; that babies cannot manage crocodiles.

While silly, this is a logical conclusion from the given premises.

Logical Inference

Suppose we know that a statement of form [latex]P{\rightarrow}Q[/latex] is true. This tells us that whenever P is true, Q will also be true. By itself, [latex]P{\rightarrow}Q[/latex] being true does not tell us that either P or Q is true (they could both be false, or P could be false and Q true). However if in addition we happen to know that P is true then it must be that Q is true. This is called a logical inference: Given two true statements we can infer that a third statement is true. In this instance true statements [latex]P{\rightarrow}Q[/latex] and P are “added together” to get Q. This is described below with [latex]P{\rightarrow}Q[/latex] stacked one atop the other with a line separating them from Q. The intended meaning is that [latex]P{\rightarrow}Q[/latex] combined with P produces Q.

[latex]P{\rightarrow}Q\\\underline{P\,\,\,\,\,\,\,\,\,\,\,\,}\\Q[/latex] [latex]\,\,P{\rightarrow}Q\\\underline{{\sim}Q\,\,\,\,\,\,\,\,\,\,\,\,}\\{\sim}P[/latex] [latex]\,\,P{\vee}Q\\\underline{{\sim}P\,\,\,\,\,\,\,\,\,\,\,\,}\\Q[/latex]

Two other logical inferences are listed above. In each case you should convince yourself (based on your knowledge of the relevant truth tables) that the truth of the statements above the line forces the statement below the line to be true.

Following are some additional useful logical inferences. The first expresses the obvious fact that if P and Q are both true then the statement [latex]P{\wedge}Q[/latex] will be true. On the other hand, [latex]P{\wedge}Q[/latex] being true forces P (also Q) to be true. Finally, if P is true, then [latex]P{\vee}Q[/latex] must be true, no matter what statement Q is.

[latex]\,\,P\\\underline{\,\,Q\,\,\,\,\,}\\P{\wedge}Q[/latex] [latex]\underline{P{\wedge}Q}\\P[/latex] [latex]\underline{\,P\,\,\,\,\,\,\,\,\,}\\\,P{\vee}Q[/latex]
The first two statements in each case are called “premises” and the final statement is the “conclusion.” We combine premises with [latex]{\wedge}[/latex] (“and”). The premises together imply the conclusion. Thus, the first argument would have [latex]\left(\left(P{\rightarrow}Q\right){\wedge}P\right){\rightarrow}Q[/latex]

An Important Note

It is important to be aware of the reasons that we study logic. First, the truth tables we studied tell us the exact meanings of the words such as “and,” “or,” “not,” and so on. For instance, whenever we use or read the “If…, then” construction in a mathematical context, logic tells us exactly what is meant. Second, the rules of inference provide a system in which we can produce new information (statements) from known information. Finally, all of us are faced with making sense of arguments posed to us everyday.  Do we believe what one political candidate promises enough to vote for them?  Do we think one type of car is better than another?