In my previous post I said that in this one I would present the argument from evil, against the existence of an absolutely perfect god, as an illustration of how an argument can be such that its validity, but not its soundness, is perfectly evident. My plans have changed, but only slightly. That is, in this post I will spell out those rudiments of logic which are necessary to the ready seeing that the validity, but not the soundness, of the argument from evil is perfectly evident. In the next post, next but already posted, I will put that logic to use. If, therefore, you are familiar with the logician’s distinction between validity and soundness and with such types of argument as Modus Ponens, Modus Tollens, and Double Negation, you may want to pass over this post and proceed directly to the next one.
First, then, an argument is a set of propositions one of which, the conclusion, is said to be the case because the other or others, the premise or premises, is or are the case. For example:
If I am in North Andover, then I am in Massachusetts.
I am in North Andover.
Therefore, I am in Massachusetts.
I hasten to add that here the name “North Andover” refers to the North Andover that is a town on the Merrimack River, and that the name “Massachusetts” refers to the state in which at least the lower portion of the Merrimack is located. I do so in order to avoid the possible confusion with another North Andover and another Massachusetts, should such there be.
Next, a valid argument is an argument the conclusion of which follows necessarily from the premise or premises. More fully, a valid argument is an argument either (a) of which the conclusion must be true if the premise is or the premises are true or (b) of which the conclusion, though false, would have been be true if the premise or the premises had been true. The argument given above is an example of a valid argument, as is any argument which, like the following, has the same structure:
If you are in Cairo, then you are in Egypt.
You are in Cairo.
Therefore, you are in Egypt.
Arguments of this kind are known traditionally as Modus Ponendo Ponens (Modus Ponens, for short), the Way of Affirming by Affirming.
Another kind of valid argument is the one known traditionally as Modus Tollendo Tollens (Modus Tollens, for short), the Way of Negating by Negating, or Denying by Denying. Examples of Modus Tollens are:
If I am in North Andover, then I am in Massachusetts.
I am not in Massachusetts.
Therefore, I am not in North Andover.
If he is in Cairo, then he is in Egypt.
He is not in Egypt.
Therefore, he is not in Cairo.
On the other hand, an invalid argument, or a fallacious argument or, again, a fallacy, is an argument the conclusion of which can be false even if (a) the premise is or the premises are true or (b) could have been false even with (b) the premise or premises being true. For example:
If I am in North Andover, then I am in Massachusetts.
I am in Massachusetts.
Therefore, I am in North Andover.
Were I sitting in Winchester, Massachusetts, the two premises of this argument would still both be true while the conclusion would be false. The truth of the premises of an argument of this structure does not necessitate the truth of the conclusion.
Another example, with the same premises, but with a negative conclusion:
If I am in North Andover, then I am in Massachusetts.
I am in Massachusetts.
Therefore, I am not in North Andover.
As I am sitting, as I write, in North Andover, Massachusetts, the two premises here are both true while the conclusion is false. The truth of the premises of an argument of this structure does not necessitate the truth of the conclusion.
The latter two arguments are examples of the argument type known as the Fallacy of Affirmation of the Consequent; the “consequent” is the “then” clause of an “if…then…” proposition, in the last sample argument, the “I am in Massachusetts” of the first premise.)
Another type of fallacious argument is the argument type known as the Fallacy of Denial of the Antecedent; the “antecedent” is the “if” clause of an “if…then…” proposition, in the next sample argument, the “I am in North Andover” of the first premise:
If I am in North Andover, then I am in Massachusetts.
I am not in North Andover.
Therefore, I am not in Massachusetts.
Were I not in North Andover, the two premises of this argument would still both be true while the conclusion could be false. The truth of the premises of an argument of this structure does not necessitate the truth of the conclusion.
Another example of this fallacy, having an affirmation as its conclusion, is:
If I am in North Andover, then I am in Massachusetts.
I am not in North Andover.
Therefore, I am in Massachusetts.
Were I not in Massachusetts, the two premises of this argument would both be true while the conclusion would be false. The truth of the premises of an argument of this structure does not necessitate the truth of the conclusion.
A sound argument is a valid argument the premise of which is true or the premises of which are true; it follows that the conclusion will also be true. Given that, as I write, I am sitting in North Andover, the argument
If I am in North Andover, then I am in Massachusetts.
I am in North Andover.
Therefore, I am in Massachusetts.
is sound. The premises implying the conclusion are both true, as is, therefore, the conclusion.
An unsound argument is either a valid argument of which at least one premise is false or an invalid argument. Because I am at present writing in Massachusetts, an example of a valid argument of which at least one of the premises is false, and which is therefore unsound, is:
If I am in Cairo, then I am in Egypt.
I am in Cairo.
Therefore, I am in Egypt.
Another example, this one of an argument that is unsound because it is invalid, is:
If I am in Cairo, then I am in Egypt.
I am in North Andover.
Therefore, I am in Egypt.
An argument can be invalid and thus unsound even if the premises and the conclusion all happen to be true.
If I am in Cairo, then I am in Egypt.
I am in North Andover.
Therefore, I am in Massachusetts.
Though I am in Massachusetts, it is not because of the two conjoined facts that if I am in Cairo I am in Egypt and that I am in North Andover.
There are infinitely other types of valid argument. To illustrate, one is known as Contraposition:
If I am in Andover, then I am in Massachusetts.
Therefore, if I am not in Massachusetts, then I am not in Andover.
The Hypothetical Syllogism is an argument form that shows up often in ordinary reasoning:
If I am in Andover, then I am in Massachusetts.
If I am in Massachusetts, then I am in the U.S.A.
Therefore, I am in Andover, then I am in the U.S.A.
We’ll look at just two more, both versions of “Double Negation,” one of which will be used in the next post’s statement of the Argument from Evil.
Double Negation
I am in North Andover.
Therefore, it is not the case that I am not in North Andover.
and
It is not the case that I am not in North Andover.
Therefore, I am in North Andover.