Categorical Logic 1.0

Recap

As you know by now, validity is an important concept in this course. Equally important, in so far as they help us wrap our minds around validity, are the notions of logical truth and logical falsity. Recall that a sentence is logically true if and only if it is not possible for the sentence to be false. Here are two examples of logically true sentences:

• "Either Yuri is a tricycle or Yuri is not a tricycle."
• "It’s false that I am a banana and I am not a banana."

In the first sentence, it is obvious that either it is the case that 'Yuri' refers to a tricycle or it's not. That's just it. There are no other options. Since or-sentences, formally called disjunctions, are true as long as one of the disjuncts is true, then it's true no matter what. In the second sentence, we have the denial of a contradiction. If I were to tell you that "I am a banana and I am not a banana", you'd know two things. 1. I'm probably losing my mind; and 2. it is not possible to both be a banana and not be a banana. This is a contradiction. It's literally not possible for this sentence to be true. In other words, this is a logically false sentence. But notice that the example bulleted above is the negation of this contradiction. This means that, in effect, the sentence is saying that this contradiction is false, which is true. It will always be true that a contradiction is false. This is why we label the full sentence, "It’s false that I am a banana and I am not a banana", logically true.

The world, of course, is not this simple. If only it were that all sentences were either always true or always false. In fact, most sentences are logically indeterminate. Put another way, the truth or falsity of a logically indeterminate sentence does not hinge on the logical words, like "not" and "or", that it contains. In other words, logically indeterminate sentences are neither logically true nor logically false.

The last concept we'll recap here is that of logical entailment, for which we will use the symbol "⊨". This symbol, which is read as 'double turnstile', simply asserts that given a set Γ of sentences, Γ logically entails a sentence if and only if it is impossible for all the members of Γ to be true and that sentence false. I know this sometimes looks scary to students. Let me try that again.

Γ is simply the Greek letter gamma. It stands for a set of sentences like, {“Ann likes swimming”, “Bob likes pudding”, “Carlos hates Dan”}.¹ This set (Γ) entails the following sentence: “Carlos hates Dan”. Why? Well clearly because it is a member of the set. It's right in there. We will learn about more complex methods of entailment, but this is the basic idea. A set of sentences logically entails a sentence if and only if it is impossible for all the members of the set to be true and that sentence false. Written out, it looks like this:

{“Ann likes swimming”, “Bob likes pudding”, “Carlos hates Dan”} ⊨ “Carlos hates Dan”

 

 

Special cases of logical concepts

The Jargon of Induction

Before delving into deduction for good, we should briefly look at the language used to assess inductive arguments. An inductive argument is said to be strong when, if the premises are true, then the conclusion is very likely true. An inductive argument is cogent when a. it is strong, and b. it has true premises.

During the middle of the 19th century there were great advancements in formal deductive logic (which will be covered in this course), as well as a revival in inductive logic spearheaded by the British philosopher John Stuart Mill (pictured below), who is mostly known for his Utilitarian system of ethics. All this to say that the study of inductive reasoning is an extremely worthwhile task. As previously mentioned, inductive logic will unfortunately not be covered in this course, but Mill’s collected works, including his A System of Logic Ratiocinative and Inductive are available online (see also Hurley 1985, chapter 9).

 

 

 

Storytime!

Previously, I had stated that Aristotle devised the science of logic ex nihilo. This is true, since no one in the Western tradition had explicitly worked out the study of logical concepts the way Aristotle did. This is not to say, however, that thinkers before this time weren't making use of logical concepts in their argumentation. All of us naturally use logical language. That is, we tend to make use of logical words such as "and", "or", "if...then", etc. Developmental and comparative psychologist Michael Tomasello even has a theory as to when sapiens developed the capacity to use logical language, a step which required new psychological capacities (see Tomasello 2014: 107).

We all even naturally make use of the important concept of logical consistency. Simply think about a time when someone was blatantly contradicting themselves. It bothered you not merely because they were probably lying, but because inconsistencies (typically) trigger some mental discomfort, alerting us that something is wrong.

So it shouldn't surprise us that philosophers before Aristotle were making use of logical concepts even without the formal study of logic having been initiated. I could list basically any philosopher here, but, in an introductory course, there is no better example than Socrates.

Most of what we know about Socrates comes from his two most famous students: Plato and Xenophon. From what we can piece together, Socrates clearly saw the discipline of philosophy as a way of life. He taught, contrary to the traditions of his time, that virtue wasn’t associated with noble birth; rather, it is something that is learned, a form of moral wisdom. His conversations with others, at least according to Plato, would help people search for the right answers to life's most fundamental question: how should I live? Repeatedly, Socrates stressed the need to develop moral strength, the requirement that we use our powers of reason to reflect on our lives and the lives of those around us, and, ultimately, that we devote ourselves to wisdom, justice, courage, and moderation.

In many cases, Socrates first had to get the people he had conversations with to realize a deficiency in their view. Only then can he help them move towards more sound positions. In one of Plato's dialogues, Euthyphro, Socrates discusses the nature of piety with Euthyphro, the latter of which is mired in inconsistencies. This dialogue is sometimes painful to read since Euthyphro is clearly reeling. Socrates, as a character in Plato's dialogue, is making use of logical techniques for showing inconsistencies. All this, remember, was before the science of logic had been explicitly begun. This is a testament to the intellectual profundity of Socrates, a trait he was able to impart on his students, Plato and Xenophon.

 

 

Important Concepts

Clarifications

As you learned in the Important Concepts above, Aristotle focused on the logical form of sentences and realized that there are only four types of categorical sentences. I'll reproduce them here:

Let's make some clarifications. Aristotle believed that all declarative sentences can be put into this form. Of course, there are other types of sentences with other linguistic functions, e.g., exclamations. But only declarative sentences can be true or false, and so those are the ones we are concerned with. Remember, logic is concerned with truth-preservation. Exclamations are neither true nor false. It would be a category mistake to label a sentence like "Golf sucks!" or "SUSHI!!!" or "¡Vamos Pumas!" as either true or false. They are more so expressions of emotion on the part of whoever uttered the sentence. So, declarative sentences are what Aristotle had in mind when he was thinking of logical form.

Also, you must've noticed in the Important Concepts a clarification about the word "some". This is important to learn now, so you don't make mistakes later on. "Some" means "at least one" to logicians. Students often tell me that "some", to them, means "more than one". While you're engaging in assessing arguments for validity, though, you must use the logician's sense of "some".

Here's a related issue. What do you intuit if someone says, "There are some tamales that are spicy"? You might interpret this as saying both "There are some tamales that are spicy" and "There are some tamales that are not spicy". This might especially be the case if you don't like spicy food. You hear the first sentence as implying the second one. However, in this class, we will use a logically rigorous definition of implication, which we won't really dive into until Units III and IV. The long and short of it is that if you see a sentence like "There are some tamales that are spicy", this does not imply the sentence "There are some tamales that are not spicy". The only time you can assume that there are tamales that are not spicy is if you are given the sentence "There are some tamales that are not spicy".

 

 

Standard Form in Categorical Logic

With the clarifications out of the way, let's practice putting sentences into their proper logical form. We will refer to this as the standard form for sentences in categorical logic. The rules are simple. Make sure that the premises/conclusion are in the following order:

1. The quantifier (which is either "all", "no", or "some")
2. Subject class (led by a noun)
3. Copula (i.e., the word "are")
4. Predicate class (also led by a noun)

Here are various examples. After a few of them, try to rewrite them yourself before you see how I put them into standard form. It's ok if our subject and predicate classes aren't the same, since we are likely to use different nouns to start off our categories.

 

 

Three Types of Categorical Reasoning

Although Aristotle's categorical reasoning has now been formalized into a robust mathematical framework (see Marquis 2020), Aristotle began with just three types of categorical reasoning. First, an immediate inference is an argument composed of exactly one premise and one conclusion immediately drawn from it. For example:

1. All Spartans are brave persons.
2. So, some brave persons are Spartans.

A mediate inference is an argument composed of exactly two premises and exactly one conclusion in which the reasoning from the first premise to the conclusion is “mediated” by passing through a second premise. This type of argument is also called a categorical syllogism. For example:

1. All goats are mammals.
2. All mammals are animals.
3. Therefore, all goats are animals.

Lastly, a sorites is a chain of interlocking mediate inferences leading to one conclusion in the end. For example:

1. All Spartans are warriors.
2. All warriors are brave persons.
3. All brave persons are strong persons.
4. So, all Spartans are strong persons.

 

 

Looking ahead

The whole point of deductive logic at this early stage was to see which conclusions necessarily followed from their premises. In other words, it was simply the study of validity. Hence, Aristotle devised different ways for assessing the three types of categorical reasoning for validity. These will occupy us for the rest of this unit. We will begin with immediate inferences, or one-premise categorical arguments. After that, we will move on to mediate inferences, which are the main challenge of this unit. In fact, mediate inferences will be the reason why you might dream of circles in the nights to come. After that, we will take a brief look at assessing sorites for validity.

But before we can run, we have to crawl. We'll begin with immediate inferences. Aristotle developed two methods for assessing these one-premise arguments: 1. the Square of Opposition and 2. the laws of conversion, obversion, and contraposition. We will focus on the Square of Opposition. This will require that we learn how to use a "paper computer". Stay tuned.

 

 

FYI

Homework!

• Categorical Logic 1.0 Homework

Advanced Material—

• Book Chapter: Merrie Bergmann, James H. Moor, Jack Nelson, and James Moor, The Logic Book, Chapter 1

 

Footnotes

1. Note that we use the "{" "}" symbols to capture the content of a set of sentences.

2. There is a reason for why the four sentences of categorical logic are labeled "A", "E", "I", and "O". The universal and particular affirmative are labeled "A" and "I" since these are the first two vowels in the Latin word affirmo, which means "I affirm". The universal and particular negative are "E" and "O" for the vowels in the Latin word nego, which means "I deny".