Intro to Logic: Deduction and Induction

by Luke Muehlhauser on July 11, 2009 in Intro to Logic

Welcome to my course Intro to Logic (index). Here, we learn the basic skills of good thinking and their benefits in real life.

Last time, we learned how to recognize arguments when they appear. Now we are ready to start separating good arguments and bad arguments.

But first, we must distinguish two types of argument.

A deductive argument claims that its premises make its conclusion certain. In contrast, an inductive argument claims that its premises merely make its conclusion probable.

Deductive arguments

Let us consider a deductive argument. If the premises when true succeed in making its conclusion certain, the argument is valid. If the premises when true fail to make its conclusion certain, the argument is invalid. Let’s look at an example:

  1. All mammals have lungs.
  2. All whales are mammals.
  3. Therefore all whales have lungs.

This deductive argument is valid because the conclusion follows with certainty if the premises are true. There is no possible way for the premises to be true and yet the conclusion false. But consider:

  1. All eight-legged creatures have wings.
  2. A spider is an eight-legged creature.
  3. Therefore spiders have wings.

This argument is also valid, because if the premises are true then the conclusion must be true. The problem here is not that the argument is invalid, but that one of the premises is false.

Now consider an argument with true premises and a true conclusion that is invalid:

  1. If I owned all the gold in Fort Knox, I would be wealthy.
  2. I do not own all the gold in Fort Knox.
  3. Therefore I am not wealthy.

Both premises are true, but the conclusion does not follow with certainty. There are many ways to be wealthy without owning all the gold in Fort Knox.

A valid deductive argument with true premises is a sound argument. A sound argument is often called a “proof,” but this term can be misleading. If the premises themselves are absolutely certain, then a sound argument does indeed offer proof, as in the below example:

  1. All bachelors are unmarried.
  2. All bachelors are male.
  3. Therefore all bachelors are unmarried males.

The premises are certain here because they are true by definition, and the argument is sound, so the conclusion is proven. However, consider:

  1. All bachelors are unmarried.
  2. Luke is a bachelor.
  3. Therefore Luke is unmarried.

This is also a sound argument (a valid argument with true premises), but the conclusion is not “proven” in the same way as in the argument above. Why? Because the second premise is an empirical claim about existence, not merely a statement about the meaning of terms. As such, it is always possible it is false. For example, maybe during a drunk night in Vegas I married a stripper but don’t remember the event. Or maybe everything I’ve ever experienced is a fabrication of The Matrix and in the “real world” I am married to a girl named Susan.

So, even a deductive argument cannot offer 100% conclusive proof if one of the premises makes a claim about existence.

Philosophers understand the rules of propositional logic so well that it is rare for one of them to publish an invalid argument. So, in philosophy, nearly all disagreement concerns whether or not the premises of a deductive argument are true or false, probable or improbable – not whether the argument is valid or invalid.

But most of us are not professional philosophers, and we advance invalid arguments all the time. So we’re going to spend some time studying the rules of logic so that we, too, can stop advancing invalid arguments.

Inductive arguments

Inductive arguments do not try to establish their conclusions with certainty. Instead, an inductive argument claims that its premises make the conclusion probable. Inductive arguments cannot be valid or invalid. Instead, they are weak or strong, better or worse. And even when the premises are true and provide very strong support for the conclusion, the conclusion cannot be certain. The strongest inductive argument is not as conclusive as a sound deductive argument.

Here is a simple example:

  1. Most corporation lawyers are conservatives.
  2. Betty Morse is a corporation lawyer.
  3. Therefore Betty Morse is a conservative.

This is a pretty good inductive argument, because (let us say) both premises are true. Thus, the conclusion is more likely true than false.

But now, let’s say we learn something new:

  • Betty Morse is an officer of the American Civil Liberties Union.

And we already know that:

  • Most officers of the American Civil Liberties Union are not conservatives.

Now our inductive argument has been greatly weakened, and it no longer seems probable that Betty Morse is a conservative.

Validity and Truth

Invalid arguments can have any combination of true or false premises and true or false conclusions. A valid argument can also have all these combinations, except that a valid argument cannot have true premises and a false conclusion. Indeed, that is the definition of a valid argument.

Propositions (premises and conclusions) can be true or false. Arguments cannot.

Deductive arguments can be valid or invalid. Inductive arguments and propositions cannot.

It is the task of science and philosophy to determine whether the premises of arguments are true or false, and the purpose of logic is to determine whether deductive arguments are valid or invalid, and whether inductive arguments are strong or weak.

(Also see the post index to this Intro to Logic series.)

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{ 12 comments… read them below or add one }

Lorkas July 11, 2009 at 7:34 am

All bachelors are unmarried.
Luke is a bachelor.
Therefore, Luke is unmarried.
Because the second premise is an empirical claim about existence, not merely a statement about the meaning of terms. As such, it is always possible it is false. For example, maybe during a drunk night in Vegas I married a stripper but don’t remember the event. [bold added]

Could you say a bit more about why you think the statement in bold is true? It seems to me like the scenario you give below makes the conclusion false because it makes premise 2 false, not because it’s a claim about existence.

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lukeprog July 11, 2009 at 7:41 am

Premise 2 is not false, and neither is the conclusion. However, we cannot be 100% certain about either one because premise 2 is an empirical claim. Empirical claims are always probabilistic. Any claim about existence is less certain than the claims of maths.

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Reginald Selkirk July 11, 2009 at 11:21 am

lukeprog: Empirical claims are always probabilistic. Any claim about existence is less certain than the claims of maths.

(Insert old joke about city slickers who can’t tell a horse from a cow.)
 

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Naug July 11, 2009 at 12:18 pm

Lorkas: Could you say a bit more about why you think the statement in bold is true? It seems to me like the scenario you give below makes the conclusion false because it makes premise 2 false, not because it’s a claim about existence.

One should also beware of valid arguments that aren’t sound. A syllogism can be valid, such as the example…
 
Premise 1) All bachelors are unmarried.
Premise 2) Luke is a bachelor.
Conclusion) Therefore, Luke is unmarried.
 
But validity of an argument goes only to its form, not to its truthvalue. For an argument to be sound, the premises must also be in reality. In your example, if Luke was married without his knowledge he would no longer be a bachelor so the argument would no longe be sound, but it would off course still retain its validity, as the form of the argument is unchanged.

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Naug July 11, 2009 at 12:20 pm

Curse my happy triggerfinger. The correct sentance is off course…
“For an argument to be sound, the premises must also true be in reality”

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Kip July 11, 2009 at 12:44 pm

This:

Most corporation lawyers are conservatives.
Betty Morse is a corporation lawyer.
Therefore, Betty Morse is probably a conservative.

seems like a deductive argument to me.  No different than:

If someone is a corporation lawyer, he or she is probably a conservative.
Betty Morse is a corporation lawyer.
Therefore, Betty Morse is probably a conservative.

That’s clearly deductive, and I think it’s identical to the argument you posted.
I think an inductive argument would be more along the lines of:

Every corporation lawyer we’ve ever witnessed is a conservative.
Therefore, all corporation lawyers are conservatives.

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lukeprog July 11, 2009 at 1:28 pm

Kip,

Oops, you’re right – I accidentally put the “probably” into the conclusion. Fixed.

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Taranu September 13, 2009 at 3:35 am

Luke, ontological arguments must be deductive or can they be inductive as well?

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lukeprog September 13, 2009 at 6:03 am

Ontological arguments are, by definition, deductive.

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jun May 23, 2010 at 11:25 pm

sometimes logicians (not necessarily philosophers) tend to treat truth and validity as separate spheres of concern, specially when one deals with symbolic logic.

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wissam October 21, 2010 at 5:21 am

All bachelors are unmarried.
Luke is a bachelor.
Therefore, Luke is unmarried.

This is logically valid! In predicate calculus, this would be:

1. (x) (Bx–>Ux)

2. Bl

3. l=Ul

B: bachelor
U: unmarried

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wissam October 21, 2010 at 5:36 am

Maybe some probability logic is necessary at this stage (esp. since we’re dealing with inductive arguments).

Bayes’ theorem (conditional probability):

Pr(AlB)= [Pr(BlA) Pr(A)]/Pr(B) Pr(B)=/=0.

Other equations can be derived (Note: the symbol I use for “Intersection” is ^; I don’t have the mathematical symbol on my keyboard).

Pr(AlB^C)= [Pr(AlC) Pr(BlA^C)]/P(BlC)

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