PHIL 4160: Symbolic Logic
Dr. Robert Lane
Lecture Notes: Thursday January 9, 2014

 

[1.5.] “Deduction and Induction.”

 

[1.5.1.] Three Ways to Evaluate an Argument.

 

There are three aspects of an argument, and it is possible to evaluate each of these aspects separately (i.e., to judge how good or bad each aspect is, independent of the others):[1]

 

·         logical: the logical connection between premises and conclusion, i.e., the degree to which the premises provide evidential support to the conclusion (if the premises are all true, then what is the likelihood that the conclusion is true?).

 

·         factual: the truth or falsity of the premises.

 

·         rhetorical: the persuasiveness of the argument to a given audience; if the argument succeeds in persuading the audience to accept its conclusion, then it is rhetorically effective, even if its logic is bad and it includes false premises.

 

In this course, we will usually focus only on the logical evaluation of arguments.

 

 

[1.5.2.] Validity.

 

As we saw last time, validity is a desirable logical characteristic of arguments. Here is a more complete definition:

 

validity (df.): A valid argument is one in which

1.      the truth of the premises would guarantee the truth of the conclusion;

2.      it is impossible for the premises to be all true and the conclusion to be false at the same time;

3.      if the premises were true, then the conclusion would have to be true as well.

(These are three equivalent definitions of “validity”; they all mean the same thing.)

 

An argument that is not valid is invalid:

 

invalidity (df.): an invalid argument is one in which

1.      the truth of the premises would not guarantee the truth of the conclusion;

2.      it is possible for the premises to be all true and the conclusion to be false at the same time;

(These are two equivalent definitions of “invalidity”; they mean the same thing)

 

 

EXAMPLES (p.5):

 

1.      All wars are started by miscalculation.

2.      The Iraq conflict was a war.

\ 3. The Iraq conflict was started by miscalculation.

 

1.      If Bonny has had her appendix taken out, then she doesn’t have to worry about getting appendicitis.

2.      She has had her appendix taken out.

\  3. She doesn’t have to worry about getting appendicitis.

 

These arguments are valid, whether or not they have true premises. As we saw last time, an argument can have false premises and still be valid. Remember that validity involves only the logical connection between premises and conclusion (i.e., the logical aspect of the argument); it does not involve the actual truth or falsity of the premises (i.e., the factual aspect of the argument). So the following argument is valid, even though it has two false premises:

 

1.      All cars are fueled by peanut butter.

2.      Giraffes are cars.

\  3. Giraffes are fueled by peanut butter.

 

IMPORTANT: In logic and philosophy, the word “valid” does not mean exactly the same thing that it does in ordinary English. For the purposes of this class, “valid” means only what it is defined to mean above: if the premises were true, then the conclusion would have to be true, as well.

 

A note about terminology: Validity is frequently called “deductive validity.” In fact, the names “validity” and “deductive validity” are interchangeable. Similarly, the names “invalidity” and “deductive invalidity” are interchangeable.

 

 

[1.5.3.] Inductive Strength.

 

Your text describes validity in terms of information “contained” in the premises and conclusion. (p.5) This seems to be a correct description of many valid arguments, including the two given above. But it doesn’t seem correct about all valid arguments, e.g.,

 

1.      Barack Obama is President of the United States.

\  2. Either Barack Obama is President of the United States or the moon is made of cheese.

 

This is a valid argument, but it does not seem correct to say that all the information contained in the conclusion is contained – even implicitly – in the premise.

 

The textbook describes an “inductive argument” as an argument that has a conclusion that goes beyond what is contained in its premises (p.6). But as we just saw, the conclusions of some deductively valid arguments also go beyond what is contained in their premises.

 

It is better not to think of arguments as divided into two classes, deductive and inductive.[2] It is more accurate to think of two standards which we can apply in evaluating the logical aspect of any argument.[3]

 

We have already seen one standard: we can ask whether the truth of the premises would guarantee the truth of the conclusion. If so, the argument is valid; if not, it is invalid.

 

Here is the second standard: we can ask whether the truth of the premises would (not guarantee the truth of the conclusion, but) make the truth of the conclusion likely, or probable. If so, then the argument is inductively strong:

 

inductive strength (df.): an inductively strong argument is one in which the truth of all the premises would make it probable/likely that the conclusion is true, but would not guarantee that the conclusion is true.

 

Examples of inductively strong arguments given in the text on p.6:

 

1.      The sun has always risen every morning so far.

\ 2. The sun will rise tomorrow (or every morning).

 

1.      All of the movies produced in recent years by George Lucas have been successful.

\ 2. The latest film produced by Lucas will be successful.

 

Two important points to notice about inductive strength:

A.     it comes in degrees, unlike validity, which is all-or-nothing.

B.     the conclusion of an inductively strong argument can be false, even if all its premises are true (unlike the conclusion of a valid argument, which cannot be false if the argument’s premises are all true)

 

So you should think of arguments as being divisible into three groups:

·         deductively valid

·         inductively strong (to some degree)

·         neither deductively valid nor inductively strong (to any degree)

This is less problematic than the two-way division suggested by the textbook, into deductive and inductive arguments.

 

 

 

 

 

 

 

 

[1.6.] “Argument Forms.”

 

An argument form is the logical structure or “skeleton” of an argument.[4]

 

See the argument about Art on p.7, which uses the following form:

 

1.      ____________ or ........................

2.      It’s not true that ________________

\ 3. ........................

 

The Art argument is valid because of its form. Any argument with this form is valid.

 

 

EXAMPLES (pp.7-8):

·         arguments 1 through 4 are all valid; they have the same form as the Art argument

·         arguments 5 through 8 are all valid; they share this form:

 

  1. If _______________, then .......................
  2. ______________.

\ 3. ..........................

 

Any argument with this form is valid.

 

We will be learning a number of different valid argument forms. (Most of the rules on the inside front cover of your textbook correspond to some valid argument form or other.)

 

 

[1.7.] “Truth and Validity.”

 

The only impossible combination of (a) validity/invalidity, (b) truth/falsehood and (c) premises/conclusion is: a valid argument with all true premises and a false conclusion. Every other combination is possible.

 

See EXAMPLES on pp.9-10.

·         Notice that four arguments (the unnumbered example about reading this book (immediately precedes no.6); 6; 7 and 10) have the same invalid form.

 

So, from the fact that an argument is invalid, you cannot infer anything about the truth or falsity of its premises or conclusions.

 

EXERCISE 1-2 (pp.10-11)

§  complete ALL problems: for the even-numbered questions, come up with different examples than those in the back of the book; we will go through these at the beginning of the next class.

 

 

 

 

[1.8.] “Soundness.”

 

soundness (df.): A sound argument is an argument that (1) is valid and (2) has all true premises. If an argument lacks either of these characteristics, it is unsound.

 

Notice that in evaluating an argument as sound or unsound, you are commenting on both its logical aspect and its factual aspect.

 

EXAMPLES, p.11-12.

 

 

[1.9.] “Consistency.”

 

consistency (df.): A consistent set of statements is one in which it is possible for all statements to be true (it does not matter whether any of them are actually true; what matters is only that it is possible that they all be true).

 

inconsistency (df.): An inconsistent set of statements is one in which it is impossible for all statements to be true; at least one of them must be false.

 

EXAMPLES, p.13. One of these is particularly difficult: “Harry, the barber, shaves all of those, and only those, who do not shave themselves.” This is an inconsistent sentence: it cannot possibly be true. This is revealed when we consider whether Harry shaves himself (and is thus not one of those who do not shave themselves.) If he does shave himself, then he doesn’t (since he only shaves people who don’t shave themselves); and if he doesn’t shave himself, then he does (since he shaves all of those who don’t shave themselves).[5]

 

Although consistency and validity are different, there is an intimate connection between them:

 

An argument is valid if and only if the set consisting of the argument’s premises and the denial of its conclusion is inconsistent. If that set is consistent, then it is possible for all the premises to be true and the conclusion false, and this is exactly what makes an argument invalid.

 

EXERCISE 1-3 (p.15)

§  complete ALL of these; check your answers to the even questions against the back of the book; we will go through the odd problems at the beginning of the next class.

 

 

Stopping point for Thursday January 9. For next time: complete exercises 1-2 and 1-3, then read pp.19-32 (chapter 2 sections 1 through 9)

 

 

 



[1] Haack, Philosophy of Logics (1978, ch.2). Haack also notes that if humans were perfectly rational, there would be no need to attend to the rhetorical aspect of an argument separately from the other two aspects, since only arguments with true premises having an appropriate connection between premises and conclusion would persuade anyone.

 

[2] The textbook authors might be half-conscious of the problems that arise when we think of arguments as being either deductive or else inductive. Although they give a definition of “inductive argument” (in terms of information “contained” in the conclusion), they never offer a definition of “deduction” or “deductive argument”—instead they start by defining deductive validity.

 

[3] Here I follow Haack 1978, p.12. She says she’s following B. Skyrms, e.g. Choice and Chance, 1966, ch.1.

 

[4] The textbook gives a more precise, technical definition in chapter four: “A group of sentence forms, all of whose substitution instances are arguments.” (p.124) We will need to cover more material before absorbing this definition.

[5] This is a variation on a famous problem discovered by Bertrand Russell (1872-1970), now called Russell’s Paradox. For more information, see A. D. Irvine, “Russell’s Paradox,” The Stanford Encyclopedia of Philosophy (Summer 2009 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/sum2009/entries/russell-paradox/>.

Symbolic Logic Homepage | Dr. Lane's Homepage | Phil. Program Homepage

This page last updated 1/9/2014.

Copyright © 2014 Robert Lane. All rights reserved.

UWG Disclaimer