Deductive Arguments: Validity and Truth

(Baronett, Section 1 F-G)

 

When we evaluate arguments, we begin by evaluating the inferential claim, then move to the factual claim.

Validity and Soundness apply to deductive arguments.

A valid deductive argument is an argument such that it is impossible for the premises to be true and the conclusion false. The conclusion follows with strict necessity from the premises.

On the other hand, an invalid deductive argument is a deductive argument such that it is possible for the premises to be true and the conclusion false. The conclusion does not follow with strict necessity from the premises.

Note that there is no middle ground--an argument cannot be "almost" valid.

Examples:

 

All television networks are media companies.

NBC is a television network.

Therefore, NBC is a media company.

Imagine that the premises are true (and they are), and then ask: is it possible for the conclusion to be false? Not in this case. 

 

All automakers are computer manufacturers.

United Airlines is an automaker.

Therefore, United Airlines is a computer manufacturer.

Again, imagine that the premises are true (they are not), and then ask: is it possible for the conclusion to be false? Again, not in this case, even though the premises themselves are false.

Note: The truth or falsity of the individual premises and conclusion has nothing to do with the validity of the argument! It is valid or invalid regardless of the truth value of its premises. In a valid argument, we say that if the premises are true, then the conclusion must be true. Logical analysis comes first, then truth value analysis.

Another example:

 

All banks are financial institutions.

Smith-Barney is a financial institution.

Therefore, Smith-Barney is a bank.

True premises, false conclusion. This form is invalid. The relationship between the premises and conclusion determines validity. Note that any (EVERY!) deductive argument with actually true premises and an actually false conclusion is, by definition, invalid.

A sound argument is a valid argument with true premises. Notice that, by definition, a sound argument will have a true conclusion as well. An unsound argument is a deductive argument that is invalid, has one or more false premises, or both. Review table page 28.

The validity of a deductive argument is determined by its form. Consider two examples:

 

A. All dogs are cats. All cats are snakes. Therefore, all dogs are snakes.

 

B. No mammals are beagles. No mammals are dogs. Therefore, no beagles are dogs.

 

Okay, all of the premises and conclusions are false, but that doesn’t matter for our analysis of logical form. To isolate the form, we’ll substitute letters for terms. Thus, the first argument has this form:

All D are C

All C are S

All D are S

 

The second argument has this form:

 

No M are B

No M are D

No B are D

 

We can provide substitution instances of the argument form, so long as the logical form remains intact. The question is, can the form yield a substitution instance which yields true premises and a false conclusion? Consider the table on page 29. The first form is valid; the second is invalid. Why?

 

Counterexamples:

 

Logical analysis shows a deductive argument to be valid or invalid. When we add the results of truth analysis, deductive arguments are either sound or unsound.

 

A substitution instance having true premises and a false conclusion is called a counterexample. We can use this method in a systematic fashion to demonstrate and argument’s invalidity. To use the method, we first isolate the form of the argument, and then construct a substitution instance having true premises and a false conclusion, which, as we recall, is the very definition of invalidity.

 

For categorical syllogisms. Consider:

 

All adlers are bobkins.

All crockers are bobkins.

Therefore, all adlers are crockers.

Which has this form:

 

All A are B

All C are B

All A are C

Why is this form invalid?

Note the substitution instance:

 

All women are featherless bipeds

All men are featherless bipeds

All women are men

True

True

False

 

Thus, this form is invalid, and any uniform substitution instance of that form results in an invalid argument. Here is another counterexample:

 

All cats (A) are animals (B).

All dogs (C) are animals (B).

All cats (A) are dogs (C).

This argument has true premises and a false conclusion. Thus, this is proven invalid.

 

Note: For categorical syllogisms, use dogs, cats, mammals, fish, and animals to substitute for the terms. Also remember that in logic, “some” means “at least one.” So, “some dogs are mammals” is a true statement. We cannot infer from “some dogs are mammals” that some of them are not mammals.

 

To facilitate the process, begin by making the conclusion false, then fill in the premises.

 

But, the categorical syllogism is but one form of deductive arguments. Consider:

 

If the government imposes import restrictions, the price of automobiles will rise. Therefore, since the government will not impose import restrictions, it follows that the price of automobiles will not rise.

To represent this form, we need a different approach. We use letters to represent entire statements rather than terms. In this case, we get:

 

If G, then P.

Not G.

Therefore, not P.

If we give the following substitution instance:

G = Abraham Lincoln committed suicide.

P = Abraham Lincoln is dead.

We get:

 

If Abraham Lincoln committed suicide, then Abraham Lincoln is dead.

Abraham Lincoln did not commit suicide.

Therefore, Abraham Lincoln is not dead.

We have true premises and a false conclusion. Yes, that first premise is true.

 

Identifying basic argument forms requires that we be familiar with basic deductive forms. First, write the argument with premises first and conclusion last. Then identity the form by noting the "form words." Leave the form words as they are and substitute the non-form words with letters. For categorical syllogisms, form words are "all," "no", "are", and "not." For hypothetical syllogisms, "if," "then," and "not" are form words. For other arguments, we have "either", "or", "both", and "and."

 

Another example:

 

All movie stars are actors who are famous, because all movie stars who are famous are actors.

 

If we replace “movies stars,” “actors,” and “famous” with the letters M, A, and F, we get this argument form:

 

        All M who are F are A.

        All M who are A are F.

Counterexample:

 

        All humans who are fathers are men.

        Therefore, all humans who are men are fathers.

 

Remember: The counterexample method cannot prove validity, only invalidity.

 

 

Inductive Arguments: Strength and Truth

Strength and Cogency apply to inductive arguments.

A strong inductive argument is an inductive argument such that it is improbable that the premises be true and the conclusion false. A weak inductive argument, on the other hand, is one in which the conclusion probably does not follow from the premises.

Again, assume that the premises are true, and then we determine whether, based on that assumption, the conclusion is probably true.

Examples:

 

All crows we have seen to date are black. Therefore, probably the next crow we see will be black.

 

All meteorites found to this day have contained sugar. Therefore, probably the next meteorite found will contain sugar.

Both are strong. The first argument, though, is cogent while the second is uncogent. Cogent: A strong inductive argument with all true premises.

 

During the past fifty years, inflation has consistently reduced the value of the American Dollar. Therefore, industrial productivity will probably increase in the years ahead.

Notice: Both of these may well be true, but the premise provides no evidence for the conclusion. It is weak. Again, the strength of an argument is simply the degree to which the premises, when assumed true, support the conclusion.

Review the table on page 39.

Note: For both inductive and deductive arguments, the only arrangement of truth and falsity relevant to validity or soundness is when we have true premises and a false conclusion. Difference: the strength of an inductive argument is a matter of degree. Validity is either/or.