Reading: The Square of Opposition

Last time, we were wrapping up our discussion on Aristotle's ethics, particularly focusing on the doctrine of the golden mean. This concept suggests finding a balance between two extremes to determine the correct approach to permissible actions in life. It's crucial to understand that this doesn't imply finding a "golden mean" for actions like stealing or murder. If we recognize certain actions as intrinsically wrong, we set them aside. For permissible actions, however, we seek a balance between the most extreme behaviors.

One point I'd like to emphasize relates to Plato's perspective on the good life, which he believed meant imitating God — embodying the highest form and striving for rationality. Yet, this doesn't provide a tangible standard. Plato presents the idea of a universal form — an embodiment of goodness, beauty, and truth — but how does that guide one in making concrete decisions? Aristotle, on the other hand, offers a more tangible approach. While Plato emphasizes the imitation of God but leaves the "how" vague, Aristotle posits that the ultimate aim of ethics is happiness. Individual happiness, he argues, depends on our actions and the life we lead. This happiness is rooted in making rational choices in line with virtue. To determine the virtuous action, we should apply the golden mean. By consistently practicing this and employing reason, we achieve happiness through virtuous living. For Aristotle, virtue becomes a habit, something ingrained through consistently doing what's right.

Aristotle's approach feels more comprehensive and grounded, with happiness as its tangible goal. In contrast, Plato's perspective might resonate more with those who lean towards religious sentiments, as it suggests imitating God. But in Plato's framework, how do flawed humans strive to emulate such a lofty ideal? This differs from Christian teachings which encourage followers to imitate Christ, who provided clear examples of forgiveness, love, justice, and more.

Shifting gears, I want to introduce a new topic: the problem of knowledge. Under what conditions can we be certain of our beliefs? Some beliefs inherently carry uncertainty. For instance, believing that tomorrow will be sunny comes with inherent doubt. However, there are certainties like "one plus one equals two."

Now, concerning knowledge and certainty, Aristotle proposed that just as there are first principles of being (ontology), there are first principles of knowledge (epistemology). These principles in the realm of knowledge are typically self-evident. For instance, Aristotle emphasized the law of non-contradiction — the idea that something cannot be both true and false simultaneously in the same context. This law underpins logical reasoning and stands as a foundational principle of knowledge. Interestingly, even attempts to deny this law inherently use it. Logic relies on this principle, and various logical rules can be derived from it.

Aristotle warned against those who might playfully or perversely claim they can believe in both a statement and its contradiction. He asserted that it's impossible to genuinely hold such contradictory beliefs. This principle and others have been employed in theological arguments, like those aiming to prove the existence of God. While Aristotle himself did not use these principles for theological conclusions, many who followed in his footsteps did.

In summary, the principle of sufficient reason, which suggests that everything must have a causal explanation, is foundational in both ontology and epistemology.

Causes are always equal to or greater than their effects.

"Understand that they're always equal to or greater than their effects. There exists in my mind an idea of infinity. This doesn't mean infinitely big; it means infinitely perfect. So, we're back to that again. Remember, it's all about only perfections. There exists in my mind an idea of something with all and only perfections.

The conclusion is, therefore, there exists in reality an infinite cause of my idea of infinity. Think about that. The person is saying, 'I have in my mind an idea of an infinite being. What is that? It's the idea of a being with all perfections and only perfections.'

According to the first premise, the principle of sufficient reason, there has to be a cause of my idea. For everything, there must be a causal explanation. There must be some causal explanation for that idea in my mind. Remember Descartes' second premise: causes are always equal to or greater than their effects. So whatever caused my idea of infinity has to be equal to or greater than the idea of infinity. Well, nothing's greater, so it must be equal. Therefore, the conclusion is that there exists an infinite cause of my idea of infinity.

This argument was presented by one of the greatest mathematicians who ever lived, Rene Descartes, who invented analytic geometry and was a remarkable logician. He constructed this proof, which, frankly, is an embarrassment. Let me explain why.

Firstly, one could object to the principle of sufficient reason. Yet, we won't dwell on that. 'Causes are always equal to or greater than their effects.' How can you be sure of that? But instead of debating that, I want to highlight the main flaw. The argument states: 'There exists in my mind an idea of infinity. Therefore, there exists in reality an infinite cause of my idea of infinity.' This suggests that the cause of this idea must be infinite because it's an idea of infinity. But this is deceptive. An idea of infinity is not inherently infinite. My idea of a being with all perfections isn't a being with all perfections. If it were, it wouldn't be an idea; it'd be an actual being with all perfections. My idea is not infinite. The idea of a bicycle isn't a bicycle. The idea of an infinite being isn't an infinite being. This idea would require an infinite cause only if it were infinite, but it's not. This is another reason I believe trying to prove the existence of God is a pursuit Christians should avoid.

What about logic? Aristotle delves into this. When we reason, we make statements. A statement asserts that something is so. Aristotle identified four basic kinds of statements: All S is P, No S is P, Some S is P, and Some S is not P. Known as the square of opposition, it showcases how these statements relate to our self-evident experiences. For instance, if 'All horses are racehorses' is true, then 'No horses are racehorses' is false, and so on.

Furthermore, Aristotle noticed that there are ways to combine these statements. Over time, they've been labeled as A, E, I, and O type statements. When constructing an argument, the simplest form has two premises and a conclusion. If an argument uses the statement types A, E, I, and O respectively, then it's in a specific mood. Given the four types, we can deduce there are 64 possible moods for an argument."

The position of the middle term is figure one, and the argument is valid. If it's true that all horses are racehorses, and it's true that all racehorses are valuable, then it's true that all horses are valuable. Do you understand? Can you see that this is a valid argument? It means that if the premises were true, the conclusion would have to be true. In fact, the premises are not true, are they? Not all horses are racehorses. And it may not be true that all racehorses are valuable. But if those were true, they would lead to the conclusion that all horses are valuable. In other words, all horses would be valuable. The argument is in the form of "A number one".

That's the mood of the argument. Here's the figure of the argument. Together, that constitutes the form.

What one notices then is that the specifics don't matter. You don't have to be discussing horses, racehorses, or value. As long as the first statement is an A-type proposition and the common term is the predicate, and as long as the second statement is an A-type proposition with the common term as the subject, the argument is going to be valid. Such premises will logically entail the conclusion. And it's not just this discovery. Consider that there are four possible placements for the middle term: both can be positioned as the subject, both as the predicate, or in combinations of subject and predicate. Thus, we have figures 1, 2, 3, and 4. It turns out that when discussing an argument with "A" as the mood, figures 2, 3, and 4 are invalid. They don't necessitate the conclusion. Only figure 1 is valid. Given that there are four forms and 64 moods, there are 256 forms for this argument, which Aristotle named the "syllogism". A syllogism has two premises and a conclusion; it's the simplest argument form. Out of 256 forms, only 15 are valid.

Can you imagine the nature of debates before this was understood? Debaters would present their arguments, with one side stating, "This, this, and this, therefore this is true", only to have the opposing side retort, "Even if your premises are true, your conclusion doesn't have to be." They'd then stand there, with no concrete method to evaluate the validity of the argument. Aristotle's approach was to break the argument down into syllogisms, determine its mood, figure out its form, and then see if it's among the 15 valid ones. If not, discard it; if so, engage with it. Nothing was the same after Aristotle's innovation.

Remember, of the 256 syllogism forms, only 15 are valid. Instead of memorizing these 15 valid forms, scholars derived four or five summarizing rules. These allowed for a quick validity check. The evaluation criteria included whether the middle term was distributed and whether two negative premises could yield any conclusion. The list goes on. However, I won't delve deeper into that, as this isn't a logic course. What's crucial to understand is that Aristotle's method dominated logical discourse for nearly two millennia. In fact, in the 11th century, when Raymond Lull advanced beyond Aristotle's framework, he dismissed his own findings simply because they contradicted Aristotle's. It wasn't until the early 19th century that logical theory evolved further. Up until then, Aristotle's approach was the primary means to evaluate arguments.

To show you what's changed since, consider the early 20th-century works of Whitehead and Russell. They sought to establish a logical foundation for mathematics that was more comprehensive than Aristotle's syllogisms. Aristotle's method was limited to categorical statements, which have a singular subject and predicate. Often, we want to make more complex assertions, using conjunctions like "either-or", "if-then", or "both x and y". Whitehead and Russell developed a system capable of handling these compound statements, moving beyond Aristotle's limits.

And now, what it means to set up an argument to check whether it's valid—remember, "valid" means that if the premises are true, the conclusion must be—well, what we would do would be something like the following:

Suppose I take the premises of the argument my opponent is giving me here, and I set them up, and it turns out that they look like this. He's saying that if either R or S is true, then T and U are true. He's also affirming that if not R were true, then V would imply not V. He asserts, furthermore, a third premise: not T. Notice that the little tilde here means "No."

So here we have "either or," "if then," "both and," and "not." Now, he says the conclusion is "not V is true." There's a way now to tell whether this conclusion actually follows from those premises. Watch closely, because what I'm about to do is what's called a logical proof.

For Step Four of the argument, I write down "not T or U." I arrive at that because I use the rule of addition on line three. Now I'm going to use a rule called De Morgan's theorem. It says that "not T, or U" is the same as saying "not T and U." Then, I derive "Not, not R or S," and that's using a rule called modus tollens.

Then, after a series of other logical steps, I reach the conclusion "V." The process involves changing this by De Morgan's again, deriving that V is a tautology, and applying the rule of implication. Each step is systematic and uses a specific rule of logic.

This is the system that they came up with, which was able to handle much more than just Aristotelian syllogisms. By the 1930s, this had been expanded to include what we call propositional functions. This is known as the predicate calculus, and it has been used to express a variety of logical relationships.

For instance, a fun example I showed students says, "Given any X, if X is a place, then, given any Y, if Y is a home, then X is not like Y." If you interpret it, it reads, "There's no place like home." These symbolic notations allow for some playful expressions!

To conclude, what Aristotle started has been carried forward and developed way beyond anything he could have imagined. We have methods today to take very complex arguments, express them in symbols, and then check them systematically. There's also a method to prove an argument is not valid. The beauty of this method is that it often reveals underlying assumptions that the original argument might have missed. It allows us to more fully understand and evaluate arguments, showing us the depth and structure of logical thinking.

This concludes our brief excursion into Logic, which began with Aristotle's invention of the syllogism and ways to check the validity of arguments. We'll continue with his views in future discussions.