II. Vocabulary and Logic

In philosophy, we take precise definitions of words very seriously. Many philosophers have spent a good chunk of their lives arguing about what a particular word or phrase means. In this spirit, I am going to introduce several technical terms that have particular meaning in logical philosophical discourse. These words may have different colloquial uses, so be conscientious about using them properly in your philosophy papers.

Statement. A statement is a sentence that can either be true or false. Example: "It will rain tomorrow." We say that this sentence has a truth-value (either true or false), and perhaps we can even know what the truth-value is. Not all sentences in English are statements-questions, commands, and propositions, for example. Some non-statement sentences can, however, be transformed into statements with some re-wording. Example: "Be a doctor!" can be usefully transformed into "You should be a doctor" if the context permits.

Conditional Statement. A conditional statement is also a sentence that can be either true or false. They are special, though, because they occur so often in arguments. A conditional statement has two parts: the antecedent and the consequent. A conditional statement generally has the form of an "if, then" statement, in which we have "If [the antecedent], then [the consequent]." Consider the following example:
If you earned an A on the final paper, then you earn an A in the class.
The entire statement is the conditional, "you earned an A on the final paper" is the antecedent, and "you earn an A in the class" is the consequent.

Argument. An argument is a set of statements, one of which is the conclusion, and the others are premises. The premises provide support for the conclusion. In other words, the conclusion asserted to be true on the basis of the premises. Example:

Premise: Either it will rain tomorrow, or it will be sunny tomorrow.
Premise: It will not rain tomorrow.
Conclusion: It will be sunny tomorrow.

An argument can be good or bad based on (1) how well the premises support the conclusion, and (2) whether the premises are actually true.

Validity. A valid argument is one in which it is not possible for the conclusion to be false if the premises are true. This is a very bold statement, not about what is actually the case, but about what could possibly be the case. It is helpful, when considering validity, to consider the notion of possible worlds. I can imagine a possible world in which grass is blue, and I can imagine a possible world in which trees are blue. But consider the following argument:

Premise: If grass is blue, then trees are blue.
Premise: Grass is blue.
Conclusion: Trees are blue.

There is no possible world in which the premises are true, but the conclusion false. Thus, this is a valid argument.

Conversely, an invalid argument is one in which it is possible for the premises to be true and the conclusion false. Consider the following argument:

Premise: If you earned an A on the final paper, then you earn an A in the class.
Premise: You earn an A in the class.
Conclusion: You earned an A on the final paper.

It is possible for the premises to be true, but the conclusion false, if there are other ways to get an A in the class, and you achieved one of them. For example, it may also be a policy in the class that if your homework average is an A, then you earn an A in the class. In this case you can earn an A in the class without earning an A on the final paper.

On the other hand, the following argument is valid:

Premise: If you earned an A on the final paper, then you earn an A in the class.
Premise: You earned an A on the final paper.
Conclusion: You earn an A in the class.

Here, if the premises are true, then the conclusion must be true.

There are some valid argument forms that are so common that they have been given names. The first example above has the form of modus ponens:

If A, then B.
A.
So, B.

In that example, A stands for "grass is blue," and B stands for "trees are blue."

A similar argument is the following:

Premise: If grass is blue, then trees are blue.
Premise: Trees are not blue.
Conclusion: Grass is not blue.

This argument has the form of modus tollens:

If A, then B.
Not B.
So, not A.

*It is important to note that, given our definitions, a statement cannot be valid or invalid, and an argument can neither be true nor false.

Soundness. A sound argument is a valid argument in which all the premises are actually true in our world. This means that any argument that is either invalid, or valid with at least one false premise, is unsound. Consider this example again:

Premise: If grass is blue, then trees are blue.
Premise: The grass is blue.
Conclusion: Trees are blue.

This is a valid but unsound argument because at least one of the premises is not actually true.

Consider again the invalid argument from above:

Premise: If you earned an A on the final paper, then you earn an A in the class.
Premise: You earn an A in the class.
Conclusion: You earned an A on the final paper.

This argument is unsound because it is invalid, regardless of whether the premises are actually true.

Strength. Not all unsound arguments are bad; an invalid argument may be a good argument. Consider the following argument:

Premise: 90% of Americans are afraid of snakes.
Premise: Jane is an American.
Conclusion: Jane is afraid of snakes.

This argument is invalid because it is certainly possible that Jane is part of the 10% of Americans who are not afraid of snakes. Thus, it is possible for the premises to be true and the conclusion to be false. However, it is unlikely that Jane is a part of the 10% rather than the 90%, so it is unlikely that the conclusion would be false if the premises are true.

A strong argument, then, is an invalid argument in which is likely that the conclusion is true, given that the premises are true. Unlike validity, strength can come in degrees. Consider a similar argument:

Premise: 99% of Americans are afraid of snakes.
Premise: Jane is an American.
Conclusion: Jane is afraid of snakes.

Here, it is even more likely that the conclusion is true given that the premises are true. And since it is more likely, we say that this argument is stronger than the first, although they are both considered strong.

Conversely, a weak argument is an invalid argument in which it is not likely that the conclusion is true, given the truth of the premises. Consider the following argument:

Premise: 30% of Americans speak French.
Premise: Jane is an American.
Conclusion: Jane speaks French.

While it is possible that Jane is part of the 30% of Americans who speak French, it is more likely that she is part of the 70% who do not. Thus, this is a weak argument.

Strong arguments are not always those with premises that assert percentages. Many (if not all) scientific laws are actually the conclusions of strong arguments, the premises of which are assertions about what we have experienced so far, and the conclusions of which are assertions about what we will continue to experience in the future. These are often referred to as arguments by induction.

In addition, many arguments by analogy are strong arguments. This type of argument is one we come across quite frequently in philosophy. The premises generally are (1) that two situations are analogous (or alike in some important respects), and (2) that certain things are true of one situation. The conclusion is then that those same things will be true of the second situation. The strength of arguments by analogy depend on how good the analogy is for the purposes of the argument.

Cogency. Just as we can evaluate valid arguments in terms of the actual truth or falsity of their premises, we can evaluate invalid arguments. A cogent argument is a strong argument in which all the premises are actually true in our world. This means that any argument that is either weak, or strong with at least one false premise, is uncogent. Consider the following argument:

Premise: All swans observed so far are white.
Conclusion: All swans are white.

This is quite a strong argument, but, unfortunately, black swans have now been observed in Australia. Thus the premise is false, and the argument is uncogent.

Formal Fallacy. An invalid argument may be a bad argument, but not because it is weak or uncogent; rather it may be bad because the premises do not support the conclusion. A formal fallacy is an argument that has a similar form to one of the valid forms we have named, but is not valid. One example we used above has the form of affirming the consequent:

Premise: If you earned an A on the final paper, then you earn an A in the class.
Premise: You earn an A in the class.
Conclusion: You earned an A on the final paper.

Although this argument seems to resemble the form modus ponens, it is actually invalid.

Another formal fallacy is denying the antecedent:

Premise: If you earned an A on the final paper, then you earn an A in the class.
Premise: You did not earn an A on the final paper.
Conclusion: You did not earn an A in the class.

Although this argument resembles the form modus tollens, it is actually invalid because it is possible that the premises could be true while the conclusion false. Recall the example above: it may also be a policy in the class that if your homework average is an A, then you earn an A in the class. In this case you can earn an A in the class without earning an A on the final paper.

Informal Fallacy. We have seen that some invalid arguments may be good, and some may be bad, depending on whether the premises adequately support the conclusion. Similarly, while some valid arguments may be good, not all valid arguments are good arguments. Those with clearly false premises are bad, of course. However, there are some particular arguments that seem to have a valid form and possibly true premises, but, upon examination of the content of the argument, clearly exemplify errors in reasoning. Click on this link to see some examples of informal fallacies.