One if the interesting things about this blog is that, while most of what I talk about is politics and culture, the post with the most continuing popularity is an older post I did on the difference between traditional and modern logic. I was going to continue that discussion in later posts but not only got rather busy, but ran into some conceptual problems in the next section of what is basically a pamphlet I wrote a few years ago that I have yet to resolve to my satisfaction.
In the meantime, here is a discussion of the differing views on truth conditionality that address some of the same issues I addressed in the earlier article.
One of the questions I get rather often from students and logic instructors about traditional logic is why it doesn't teach truth tables. Modern logic, the most common kind of logic encountered in high school and college, uses them, so why does traditional logic ignore them?
Many people encounter a smattering of logic in high school math courses, which teach a few of the rudiments of modern logic. Here, more than likely, they will encounter simple truth tables. Truth tables were invented by Ludwig Wittgenstein, perhaps the the 20th century's most influential modern philosopher. He invented them to accompany the calculus into which modern analytic philosophers had transformed logic. They were seen a way to quickly solve for the truth of simple and complex logical propositions in the modern system.
Let's take the statement, "There are seven days in the week and twenty-four hours in a day." In the modern system of logic we would want to immediately reduce this down to its formal elements. Let's say that P = "there are seven days in the week" and that Q = "there are twenty-four hours in a day." If we did this, then we could represent the statement as follows:
P and Q
How do we find out whether the statement "P and Q" is true? In modern logic, the truth value of this statement is determined by its elements--in this case, the statements signified by "P" and "Q". We know as a matter of simple common sense that the statement "P and Q" is true only when both the statements represented by "P" and "Q" are themselves true--in other words, if it is true to say that there are seven days in the week and if it is true to say that there are twenty-four hours in a day. If either one or both of these statements are false (in other words, if a week is made up of something other than seven days or if a day is made up of something other than 24 hours--or both), then the statement would be false.
Using truth tables, we would set forth all of the truth possibilities of P and Q so we could see clearly when "P and Q" is true and when it is false:
P Q P and QT T TT F FF T FF F F
We don't really need to go to all this trouble to verify that a simple statement like "P and Q" is true. But what if you had a statement like "P and (Q or (If R, then S))"? When statements become this complex, truth tables can be an easier way to calculate their truth value.
So if truth tables make the determination of the truth of statements more easy to calculate, then why doesn't traditional logic teach them?
There are several answers to this question. The first is practical, the second is theoretical.
The Practical Usefulness of Truth Tables is Overstated
The first reason is that, although truth tables have certain technical applications, they are not practically useful in actual argument or discussion, since most statements used in everyday speech and even in academic conversation never get to the level of complexity that would require a truth table to figure them out. They are certainly helpful in certain scientific applications and for computer computer programming (modern logic's most practical application), but outside those fields, they are seldom needed.
I have not only taught logic, but engaged in private and public debate for over 25 years. While I have made use of William of Sherwood's traditional mnemonic verse of the 19 valid syllogism forms and the procedure for backing into missing premises repeatedly (both of these are covered in my Traditional Logic, Book II), I have never had to resort to a truth table.
This is partly the result of the fact that most real life argumentation is conducted in or reducible to categorical reasoning on which you cannot use truth tables anyway. This is because categorical reasoning operates on the basis of the relations between individual terms (which are neither true nor false, since only full statements can be true or false) and truth tables work only with hypothetical reasoning, which operation on the basis of relations between statements. In addition, even though modern logical techniques were developed primarily to deal with complex philosophical and scientific problems in an academic context, the vast majority of the reasoning you encounter even there consists simply in chain arguments (strings of simple arguments strung together) that don't require any advanced calculus to solve.
The Faulty Metaphysics Behind Modern Logic
The second reason for the absence of truth tables from traditional logic has to do with the philosophical differences between the traditional and modern systems of logic. To state it baldly, traditional logic doesn't believe in truth tables.
The reason they are used in one system and not the other has to do with a concept called truth functionality. What is truth functionality? “A compound proposition,” said Edward Simmons, “is said to be truth-functional when its truth as a whole depends solely upon the truth values of its component parts.” In other words, the truth or falsity of its parts will tell us the truth or falsity of the whole.
In the statement above, "P and Q", we can tell its truth from its component parts. "P and Q" is called a "conjunctive proposition"--it conjoins P and Q. Traditional logicians believe that conjunctive statements are the only kind of statements whose truth can be "solved" in a truth table--the only kind of statements, in other words, that are truth functional. No other kinds of logical statements ("P or Q", "If P, then Q", etc.) are truth functional in this way.
The reason traditional logicians deny the truth functionality of hypothetical propositions has to do with the underlying assumptions about language and reality. To illustrate this, let's take another simple statement, this time a conditional statement (This is where the problem with modern logic's assumptions become very clear):
If it rains, then my dog will get wet
In modern logic, we would "solve" for the truth of this statement using a truth table:
P Q If P then QT T TT F FF T TF F T
This kind of statement is considered true in every possible case except when P is true and Q is false (the second line). Let's say my dog is an outside dog and has no protection from the rain. In that case, when it rained my dog would get wet--both P and Q would be true, and therefore it would be true to say (as on the first line of the truth table) that the entire statement, "if it rains my dog gets wet" is true.
But let's say it was raining, but my dog was in the garage, dry and cozy. In that case, it would be true to say that it was raining, but false to say that if it rains, then my dog gets wet (as indicated on the second line of the truth table). It rains, but my dog does not get wet. The statement would therefore be false.
But what about the other two possibilities, in other words, when it is not raining at all (when P is false and Q is either true or false--the last two lines of the truth table)? Why, as indicated in the truth table, do modern logicians say the statement would be true in those cases?
As someone who has heard the explanations of why this is the case--as well as having tried to explain it to his own students in class--I can testify to the difficulty in trying to understand this.
But the fact is that modern logic's treatment of the conditional statement (particularly its treatment of conditional statements in which the antecedent is false) is problematic not because it is complicated; it is problematic because it is problematic.
In the traditional system a conditional statement is considered true only if the fact that your dog gets wet really occurs as a result of the rain—in other words, if the statement asserts what is called a valid sequence. To put it another way, there must be a real logical relation between the rain and your dog getting wet. The fact of it raining must, in some way, materially imply that your dog will get wet.
In the modern system, however, there need be no real connection all. All that is required is that, as a matter of fact, the consequent (my dog will get wet) is not false when the antecedent ( it rains) is true. Unless this is the case, the statement is considered true. Therefore, in the modern system, statements such as:
If the moon is made of green cheese, then ducks can swim
are considered true statements, since their antecedents ( in this case, "the moon is made of green cheese") is false at the same time that the consequent ("ducks can swim") is false. In fact the antecedent is false and the consequent true, therefore (according to the modern logician) it is a true statement.
While modern logic considers this statement true, traditional logic sees it, again, for what it is: nonsense. The moon being made of green cheese clearly has no relation (logical or otherwise) to the fact that ducks are able to swim.
In the traditional system, conditional statements are considered to assert a necessary connection between their elements (the antecedent and the consequent), while in modern logic the only connection has to do with the happenstance coincidence of the truth or falsity of the elements. There must be either a cause and effect or ground-consequent relation between the antecedent and the consequent. The rain and the dog getting wet are to be seen as having a fundamental metaphysical relation (in this case a cause-effect relation) to one another. The assumption behind modern logic is that such necessary connections either do not exist or that they do not need to be accounted for in our system of logic.
The underlying problem here is that modern logic is concerned with the attempt to quantify reality. It wants to turn logic into a kind of calculus. This was the dream of philosopher Gottfried Leibniz, who hoped that one day man could create what he called a "calculus ratiocinator"--a logic machine for the "solution" of logical problems. In many ways Leibniz was Aristotelian in his thinking (traditional logic is Aristotelian), but he would have had to have had very non-Aristotelian assumptions in order to believe that this was even possible.
Traditional logic does not attempt to reduce logic to a quantitative calculus, largely because it views logic as a linguistic and metaphysical art, not a technical mathematical calculus. Traditional logicians recognize a distinction between what is called extension and comprehension--on other words, that any comprehensive view of human reasoning would have to recognize both the quantitative aspects of human language, but also the qualitative. It rejects modern logics reduction of all human reasoning to extensionality.
Traditional logicians reject the idea that language can be quantified in the way that modern philosophers believe it can. Logic, according to the traditionalists, is inherently qualitative and logocentric (centered on the Word), and attempts to quantify logical language can only serve to distort the process of reasoning.
Behind the idea of such a calculus is a view of the world fundamentally at odds with traditional metaphysical beliefs. Ultimately, the only way logic can be made into a calculus is by denying the essential metaphysical nature of the world that logical language attempts to portray.
This, of course, is not a problem for the logical positivists who developed modern logic because they did not believe in traditional metaphysics, although, of the three people who wrote the book that put modern logic on the academic map (Bertrand Russell, Alfred North Whitehead, and Wittgenstein, the latter of whom greatly influenced, but did not actually author the book) both Whitehead and Wittgenstein later repudiated it--for different reasons.
Their progenitor is David Hume, the 18th century British empiricist philosopher who went so far as to question the rationality of the belief in cause and effect. What modern logic has done is to create a system of logic that honors Hume's positivism by ignoring metaphysical reality: You can "solve" an "If P then Q" statement by ignoring the metaphysical implication in it and taking account solely of the "truth value" of its elements.
In the modern view, in other words, "If, ... then" statements do not posit either a cause/effect or ground/consequent relation. They operate basically like conjunctive statements, ignoring the very relation that those who use them actually mean to assert (cause/effect or ground/consequent).
It is logic for Humeans.
In other words, the question over truth conditionality--in addition to anything else that might be wrong with it--is the logical consequence of a faulty view of metaphysics.