Logic 2: Induction, Deduction, Validity & Consistency

An argument consists of a series of complete sentences that can be true or false.  The statement, “It is raining” can be true or false, but the question “Is it raining?” is neither true nor false, and neither is the exclamation, “Stop raining!”.  In formal logic, if a statement is true, it is assigned the truth value ‘T’, and if a statement is false, it is assigned the truth value ‘F’.  In binary code, used by binary computers, the number 1 stands for true, and 0 stands for false.

In formal logic, statements are exclusively and absolutely true or false.  They cannot be both true and false or somewhat true and somewhat false.  This is called the Principle of Non-contradiction.  Also, statements cannot be neither true nor false.  This is called the Principle of Bivalence.  Formal logic is only concerned with absolute truth, not relative truth.  If Suzette is the third tallest person in a group of ten, we would say she is relatively tall but not absolutely the tallest.  Similarly, if Suzette tells someone ten things, and one of them turns out to be false, we would say that Suzette relatively told the truth, but in formal logic, if any part of a statement or set of statements is false, then the truth value of the statement or sum of statements is false, as each must be assigned an absolute truth value.

In an argument, there are two types of statements, premises and conclusions.  Premises are used to support and prove conclusions.  Often, premises are indicated by use of the words ‘because’, ‘since’ or ‘given’, and conclusions are indicated by the words ‘therefore’, ‘so’, ‘then’, ‘thus’ and ‘hence’.  In a formal logical argument, the conclusion always comes last.  In a casual, everyday argument, the conclusion can sometimes come first, or in the middle.  Consider the following:

John is a fool.  He put a fork in the microwave and it caught fire.

Anyone who would do that is an idiot.

Notice that the conclusion, or point of the argument, is that John is a fool and this comes first, and then the premises, which support the conclusion, come after.  The point of the argument is not that John put a fork in the microwave, or that anyone who would do that is a fool.  If we were to put the conclusion last, as we do in a formal argument, it would read:

John put a fork in the microwave and it caught fire.

If someone puts a fork in the microwave such that it catches fire, that person is a fool.

Therefore, John is a fool.

There are two types of arguments, inductive and deductive.  Inductive arguments generalize what is likely true based on a gathering of evidence.  Consider the following:

I read many books as a freshman.

I read many books as a sophomore.

I read many books as a junior.

Therefore, I will likely read many books as a senior.

Notice that the conclusion is probably true, assuming that one continues on to be a senior and that senior year is much like previous years.  While this inductive argument does not prove its conclusion with absolute certainty, it does present us with evidence that generally leads us to its conclusion.  Because formal logic requires that we assign an absolute truth value rather than a relative truth value to each statement, formal logic has no use for inductive arguments.

For deductive arguments, if the premises are absolutely certain then the conclusion should also be absolutely certain.  Formal logic is concerned only with deductive arguments.  Consider the following deductive argument, in which the first two sentences are premises which lead to the third, the conclusion:

All people can use logic.

Whoever can use logic can think rationally.

Therefore, all people can think rationally.

A deductive argument can be wrong in two different ways.  First, the argument can be invalid.  An argument is valid if the conclusion follows from the premises, such that it is impossible for the premises to be true and the conclusion false.  If an argument’s premises are true and the conclusion is false, it is invalid.  It is important to distinguish between truth and validity.  The conclusion of an argument can be true and yet the argument is invalid, as the conclusion does not follow from the premises.  Consider the following argument:

Rainforests get much rainfall.

Honduras has rainforests.

Therefore, Tegucigalpa is the capital of Honduras.

All three of these statements are true, but the conclusion does not follow from the premises.  We can imagine that the capital of Honduras could be changed to another city, such that the premises are true but the conclusion is false.

The second way that a deductive argument can be wrong is that the premises can be false.  An argument can be valid, such that the conclusion follows from the premisses, but false if its premises are false.  Consider the following three arguments:

All puppies are green.

Everything that is green is evil.

Therefore, puppies are evil.

___________________________________________________

Elevators play jazz music.

Anything that plays jazz music is sent by the Devil.

Therefore, elevators are sent by the Devil.

___________________________________________________

The enemy is a threat.

If something is a threat, it should be completely destroyed.

The enemy should be completely destroyed.

These are all perfectly valid arguments.  If the premises were true, the conclusion would also be true.  Unfortunately, they are also false arguments, because at least one of each argument’s premises is false.  It is not true that puppies are green, nor that all green things are evil, nor that jazz music is sent by the Devil, nor that that which threatens us must be completely destroyed.  Often, when people call an argument illogical or irrational, it is not that the structure of argument is flawed or invalid, but that the assumptions on which the argument is built are false or misleading.  While it is important to think rationally and give valid arguments for one’s thinking, this does not guarantee that one’s conclusions are correct or one’s argument will be successful, as our assumptions may be wrong.

Most statements we make in arguments, whether or not they are premises or conclusions, are contingent, could be either true or false depending on the situation.  If I say, “It is raining”, this is true or false depending on the weather at the time.  When it is raining, my statement is true, and when it is not raining, my statement is false.  There are some statements, however, that are not contingent, that are necessarily true or necessarily false no matter what the circumstances.  If we assume that it must be either raining or not raining, with no middle ground, and I say, “It is raining or it is not raining”, my statement is necessarily true no matter what the weather is.  If a statement is necessarily true, it is a tautology, and is not contingent.  If I say, “It is both raining and not raining”, and we assume it must either be raining or not raining, with no middle ground, my statement is necessarily false no matter what the weather is.  If a statement is necessarily false, it is a contradiction, and is not contingent.

Not only can a statement contradict itself, such as “It is raining and not raining”, but a set of statements can contradict itself if two or more of the statements in the set contradict each other.  If a set of statements does not contain a contradiction, if it is possible for all of the statements to be true at the same time, we say that the set is consistent.  If a set of statements contain statements that contradict each other, we say that the set is inconsistent.  If a witness in a trial says things that contradict each other, we similarly say that their testimony is inconsistent.  Consider the following argument:

It is raining.

It is not raining.

Therefore, it is either raining or it is not raining.

The conclusion of this argument is necessarily true, a tautology, however the two premises, both contingent, contradict each other, and thus the argument, a set of statements, is inconsistent.  Interestingly, even though the argument is inconsistent, it is valid.  In accord with the definition of validity, it is impossible for the premisses to be true and the conclusion false, as the premises cannot both be true, and it is impossible for the conclusion to be false.  This is yet another example of a valid argument which is also a bad argument.

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