## Logical Errors in a Logic Textbook, Alas!

1. Early on in his Socratic Logic, Peter Kreeft offers what I’ll take to be two arguments in favor of a thesis that traditional, or Aristotelian, logic is superior to modern, or symbolic or mathematical, logic, that, indeed, modern logic is, as I’ll put it for him, a defective logic. The matter is of some importance in logical theory, for, if he is right, the past century and a half or so of progress in logic has been an illusion.

2. I will critique the first of the two arguments in the present post, leaving the second for a post to come. The first argument can be formulated as:

Any logic accepting an invalid argument as a valid argument is a defective logic.
Modern symbolic/mathematical logic is a logic accepting an invalid argument as a valid argument.
Therefore, modern symbolic/mathematical logic is a defective logic.

Kreeft and I can easily agree that the argument is a perfectly valid argument, in that, if its two premises are true (or were true), then the conclusion must be true (or would have to be true). But then the question arises of whether the argument is more than just a valid argument, but a sound argument, having both of its premises true. Now, though Kreeft leaves the first premise unexpressed, I am fully confident that he would take it as true, without any “ifs,” “ands,” or “buts.” We disagree, however, about the second premise, for while Kreeft holds that it is true, I hold that it is simply false.

3. To see that this is the case, let’s turn, then, to Kreeft’s formulation of the argument in question. It starts out well enough (p. 24).

A logically valid argument is one in which the conclusion necessarily follows from its premises. In a logically valid argument, if the premises are true, then the conclusion must be true. In an invalid argument this is not so. “All men are mortal, and Socrates is a man, therefore Socrates is mortal” is a valid argument. “Dogs have four legs, and Lassie has four legs, therefore Lassie is a dog” is not a valid argument. The conclusion (“Lassie is a dog”) may be true, but it has not been proved by this argument. It does not “follow” from the premises.

So far, so good.

Kreeft continues (Ibid.).

Now in Aristotelian logic, a true conclusion logically follows from, or is proved by, or is “implied” by, or is validly inferred from, only some premises and not others. The above argument about Lassie is not a valid argument according to Aristotelian logic. Its premises do not prove its conclusion. And common sense, or our innate logical sense, agrees.

Still, so far, so good.

But then the argument goes off the rails, for Kreeft goes on to claim in the next sentence:

However, modern symbolic logic disagrees.

That is, according to Kreeft, while Aristotelian logic and common sense agree that the argument about Lassie is not valid and that its premises do not prove its conclusion, “modern symbolic logic disagrees.” The disagreement is not over whether the argument about Lassie is an argument, but over whether it is valid. Kreeft is therefore committed to saying modern logic holds that the argument about Lassie is a valid argument and therefore that modern logic holds that some invalid argument about Lassie is a valid argument.

4. Kreeft could not be more wrong. In no way would the so-called “predicate calculus,” that branch of modern logic by which the argument at hand would be assessed, pronounce it valid. On the question of whether or not the argument about Lassie is a valid argument traditional logic and modern logic in full agreement.

5. Let me set the argument forth both as it would be presented in traditional logic and as it would be presented in the “predicate calculus.” In traditional logic, stating the argument even a bit more carefully than Kreeft does, the argument would read:

All dogs are animals having four legs.
Lassie is an animal having four legs.
Therefore, Lassie is a dog.

As, as we have seen, Kreeft has said:

The conclusion (“Lassie is a dog”) may be true, but it has not been proved by this argument. It does not “follow” from the premises.

And no logician will or even can come up with a sequence of inferences acceptable to traditional logic that would begin from the two premises and those two premises alone and logically arrive at the proffered conclusion. For, if all that we are given to know is what we know through the argument’s premises, it may well be, for all we know, that Lassie is, not a dog, but a turtle.

On the other hand, of course, we could introduce the premise that Lassie is a dog into the argument. The argument would then read:

All dogs are animals having four legs.
Lassie is an animal having four legs.
Lassie is a dog.
Therefore, Lassie is a dog.

Then, however, we would have a different argument, one in which the conclusion that Lassie is a dog would certainly follow from the premises. But this argument is precisely that, a different argument.

6. Now, in the modern logic, the argument would be stated as:

For any existent x, if x is a dog, then x is an animal having four legs.
Lassie is an animal having four legs.
Therefore, Lassie is a dog.

The judgment that Kreeft advanced about the Aristotelian formulation applies equally well here.:

The conclusion (“Lassie is a dog”) may be true, but it has not been proved by this argument. It does not “follow” from the premises.

And no logician will or even can come up with a sequence of inferences acceptable to modern, symbolic or mathematical logic, that would begin from the two premises and those two premises alone and logically arrive at the proffered conclusion. For, if all that we are given to know is what we know through the argument’s premises, it may well be, for all we know, that Lassie is, not a dog, but a turtle.

On the other hand, of course, we could introduce the premise that Lassie is a dog into the argument. The argument would then read:

For any existent x, if x is a dog, then x is an animal having four legs.
Lassie is an animal having four legs.
Lassie is a dog.
Therefore, Lassie is a dog.

Then, however, we would have a different argument, one in which the conclusion that Lassie is a dog would certainly follow from the premises. But this argument is precisely that, a different argument.

7. If the foregoing holds up, I have shown that the first of the two arguments that Kreeft has offered on behalf of the thesis that modern symbolic/mathematical logic is a defective logic, fails, in that he has not shown that that argument’s premise, that:

Modern symbolic/mathematical logic is a logic accepting an invalid argument as a valid argument.

is true.

In a post to come, I will offer a critique of the second of the two arguments Kreeft advances on behalf of the thesis that modern symbolic/mathematical logic is a defective logic.

Until next time.

Richard

* Peter Kreeft, Socratic Logic. A Logic Text Using Socratic Method, Platonic Questions, and Aristotelian Principles. Edited by Trent Dougherty (Edition 3.1; South Bend, Indiana: St Augustine’s Press, 2014 )

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