Logic: A Quick Primer

While I am sure that many of my readers will be familiar with logic, I prefer to include this primer to the site for completeness, and to define and clarify some of the terms that I will invariably use in my analyses and criticisms of various philosophies.

Reference:
Thomas, Stephen Naylor. Practical Reasoning in Natural Language, 4th Ed., Prentice-Hall, NJ. 1997.

Reason

A reason is any statement given in support, justification, or explanation of some action, claim, expectation, prediction, or assertion.

Conclusion

A conclusion is any statement that an author or speaker tries to support by reasons in a discourse. Conclusions are claims that claim to be supported in an argument.

Reasoning

A discourse containing reasoning is one in which one or more statements are set forth as making probable, proving, justifying, or explaining other statements in the same discourse. Put differently, a discourse containing reasoning is one in which certain claims or alleged facts are offered as justification or explanation of something.

Using the word “reason” and “conclusion,” we may define “reasoning” as any discourse in which some statement is given as a reason for some conclusion.

Argument

Discourses containing reasoning are called arguments. An argument, then, refers to any discourse having a certain logical structure – a structure containing reasoning.

The discourse,

I am twenty years old. You are fifteen years old. Therefore I am older than you.

is an argument because the premises, (1) I am twenty years old and (2) You are fifteen years old support the conclusion, I am older than you. The word, “therefore” is an inference indicator – it indicates that the statement that follows it is a conclusion that is drawn from the preceding reason(s).

Soundness of an Argument

Two conditions must be met for an argument to be sound.

1. Truth of Reasons: The statement(s) given as reason(s) should be true.
2. Validity of Inference: The truth of the reason(s) should make likely the truth of the conclusions.

The second condition tests the relationship of the conclusions to the reason(s). The reason(s) and conclusion must be connected, or related to each other in such a way that the conclusion follows logically from the reason(s). In particular, the conclusion needs to be related to the reason(s) in such a way that the truth of the reason(s) would make the truth of the conclusion extremely likely. That is, the relationship must be such that if the reason(s) are (or were) true, then they would either guarantee, or at least make highly likely, the truth of the conclusion.

Both tests must be passed in order for the reasoning to successfully prove or explain its conclusion.

TRUTH OF REASON(S) + VALIDITY OF INFERENCE = SOUNDNESS OF REASONING

Using the sample argument I gave earlier, if it is true that I am twenty years old, and that you are fifteen years old, then it follows logically that I am older than you. This argument, then, passes both tests (its reasons are true, and the inference is valid), and hence, is a sound argument.

Deductive Validity

Reasoning in which the reason(s) are such that, if they are true, the truth of the conclusion is 100% guaranteed are said to be deductively valid.

If reasoning is deductively valid, then it is not even logically possible for the reason(s) to be true, and yet the conclusion be false.

Let’s again take my sample argument:

I am twenty years old. You are fifteen years old. Therefore I am older than you.

Here the conclusion follows with complete logical certainty from the reasons; the support that the reasons provide the conclusion is total, or 100%. Thus there is no logically possible or imaginable way in which the conclusion could be false if the reasons are true.

To test the argument's deductive validity, we must ask the questions:

Does the truth of the reasons guarantee, 100%, the truth of the conclusions?

or

Is there any way that the conclusion can be false if the reasons are true?

The answer to the first question is that, indeed, the truth of the reason guarantees, 100%, the truth of the conclusion: if it is true that I am twenty years old, and it is also true that you are fifteen years old, then it is true that I am older than you.

The answer to the second question (which is merely a different way of putting the first) is that there is, indeed, no way in which the conclusion can be false if the reasons are true: if it is true that I am twenty years old, and it is also true that you are fifteen years old, then there is no logical or imaginable way in which the conclusion (that I am older than you) can be false.

Given and Fixed Definitions

Note that during the evaluation of reasoning, the meaning of all the words in the reasoning are treated as given and fixed as they exist in the language. We should all agree on the definition of fifteen years old, and twenty years old, for instance.

Reference:
Thomas, Stephen Naylor. Practical Reasoning in Natural Language, 4th Ed., Prentice-Hall, NJ. 1997.