Categorical Logic

This is Unit Three of Logic: Categorical Logic. The unit runs long, so it comes in two parts. This is the first, and it is about single statements. Categorical logic is the oldest formal system there is, and it works by a severe restriction: every statement it handles relates exactly two classes, saying that one falls inside the other or stays outside it, either entirely or in part. That restriction is the source of the power. Because there are only four ways to relate two classes, there are only four statement forms — and once a sentence is in one of them, everything about it can be worked out mechanically. Part One builds the forms, the properties that identify them, the diagrams, the relations between them, and the translation rules that get real sentences into the system. Part Two puts two of them together into a syllogism. By the end you will be able to do five things. One: name any categorical statement's form from how much of its subject it covers and whether it affirms or denies. Two: say which of a form's terms are distributed — the chart that two later rules run on entirely. Three: diagram all four forms, keeping the two kinds of mark distinct. Four: say which relations between the forms survive the modern reading and which ones need an extra assumption. Five: translate ordinary sentences into standard form, including the pair that reverses the terms.

Begin with the object. A statement qualifies for this system when it puts two classes into a relation of inclusion or exclusion — and the restriction to exactly two is not a limitation the system apologises for. It is the reason the system can be tested at all.

The object.

Logic 066 Categorical Statements

Every senator being a citizen puts one whole class within another. Two classes per statement is the hard constraint, and it is what makes the machinery possible.

The two slots.

Logic 067 Subject and Predicate Terms

The class named first, then the class named second. Careful here: what fills a slot is the class noun on its own, not the verb phrase wrapped around it.

There are exactly four ways two classes can be related under this system, and they get four letters that came out of Latin — two vowels for the affirmations, two for the denials. Everything in the rest of the unit is bookkeeping on these four.

The four, and where the letters came from.

Logic 068 The Four Standard Forms

Two Latin verbs supply them: one for affirming, one for denying, each contributing two vowels. Four forms carry the entire system, so learn them as a set.

The first form.

Logic 069 A Statements

No whale gets out of being a mammal. Note what this does not say: whether any whales exist is a separate question, and a surprisingly loaded one.

The second.

Logic 070 E Statements

Reptiles and mammals share no members at all. The exclusion is symmetric, so the sentence works identically read from either end.

The third.

Logic 071 I Statements

One doctor who flies is enough to make it true. In this system the word means at least one — never several, and never some-but-not-all.

The fourth.

Logic 072 O Statements

One employee who is not a manager makes it true. And this form does not quietly promise its opposite: saying some are not carries no commitment that some are.

Two questions identify any of the four without memorising them individually. Ask how much of the subject class the statement covers, and ask whether it puts things in or keeps them out. Those two answers pin the form exactly.

The first question.

Logic 073 Quantity

Two of the forms speak about the entire subject class and two about a portion. Watch the exclusion form — it is about every member, saying of each that it stays outside.

The second question.

Logic 074 Quality

Two forms place members inside the predicate class and two keep them out. Combine the two answers and the form is fixed — nothing here concerns how good anything is.

Now a property that sounds technical and pays off enormously later. Two of the five syllogism rules in Part Two consist of nothing but distribution checks, so the small chart in this topic is worth committing to memory rather than looking up.

What the property means.

Logic 075 Distribution

Saying every senator is a citizen makes a claim covering each senator, and no claim covering each citizen. So the property attaches to a term as it sits in a particular statement, never to the term by itself.

The chart.

Logic 076 Distribution Across the Four Forms

There is a two-line shortcut worth carrying: statements about a whole class cover their first term, and statements that deny cover their second. Memorise it cold.

Two circles and two kinds of mark handle all four forms. The two marks say fundamentally different kinds of thing — one reports that a region holds nothing, the other reports that something is in there — and trading one for the other is the error that ruins the diagrams later.

The two marks.

Logic 077 Venn Diagrams for Statements

Denying any overlap means blanking out the shared region. Asserting an overlap means putting a mark in it. Emptiness and existence are not interchangeable claims.

Watch all four go onto the same frame.

Logic 078 Diagramming All Four Forms

Two circles, the subject on the left. For the universal affirmation, blank out the part of the subject lying beyond the predicate, because the claim says nothing lives there. Mixing the two marks is the standing error.

Four relations hold between the forms, arranged on a diagram that has been taught for two thousand years. Exactly one of them is unconditional. The other three depend on an assumption the next topic will make explicit, and modern logic declines to make it.

The one that always holds.

Logic 079 Contradictories

Pair each universal with the particular that denies it, and the two can never agree — one true forces the other false, in both directions. This relation survives every interpretation the unit will consider.

The second relation.

Logic 080 Contraries

Total approval and total disapproval cannot hold together, and when opinion is split neither holds. Note the fine print: the modern reading drops this relation entirely.

The third is its mirror.

Logic 081 Subcontraries

Some approving and some not sit together comfortably. Same fine print as the relation before it.

The fourth sends truth downward.

Logic 082 Subalternation

If the claim about every whale holds, the claim about at least one follows. Falsity travels the other direction. On the modern reading nothing travels at all.

Here is what those three relations were resting on. When a statement covers a whole class, does it quietly promise that the class has members? The older reading says yes. Boole says no, and the choice decides which relations you are allowed to use.

The two standpoints.

Logic 083 Existential Import

A sign warning that trespassers will be fined is true, useful, and hopefully about nobody at all. That was Boole's point. Modern work assumes his reading unless a problem says otherwise — so check the standpoint before answering.

Three ways to rewrite a statement into a different one. Each is legal for some forms and illegal for others, and the operations are equally easy to perform either way — nothing about doing one wrongly feels wrong, which is why this topic is about the permissions rather than the mechanics.

The first operation.

Logic 084 Conversion

Legal for the exclusion form and the partial-inclusion form. The famous wreck is doing it to a universal affirmation: every dog being a mammal does not make every mammal a dog.

The second is legal everywhere.

Logic 085 Obversion

Change whether it affirms or denies, then replace the second term with everything outside it. The complement means literally everything outside the class — not the everyday opposite. Not-happy is not sadness.

The third takes both moves at once.

Logic 086 Contraposition

Legal for the universal affirmation and the particular denial. Note which operations serve which forms — one serves two of them, the other serves the other two, and swapping the pairings is this unit's most reliable mistake.

One statement, run through several rewrites in sequence. Each link has to be checked separately, because a chain that starts from a true statement can end anywhere at all if one step was not permitted for that form.

Watch a chain run, from sparrows.

Logic 087 Applying Immediate Inferences

Start from a true universal, flip its quality and complement the second term, then take the next legal move. The difficulty is not performing the operations — it is that performing an illegal one feels identical to performing a legal one.

None of this machinery touches a sentence until the sentence is in standard form, and real sentences never arrive that way. The procedure is mechanical; two English words defeat it reliably, and one of them is the most-missed item in the whole subject.

The general procedure.

Logic 088 Translating into Standard Form

Decide how much and whether it affirms, then force the sentence into the frame using plural class nouns. The result reads awkwardly. That is the trade being made on purpose: elegance for testability.

The pair that reverses.

Logic 089 "Only" and "The Only"

A sign restricting entry to members tells you everyone entering is a member — not that every member enters. The near-identical phrasing puts the terms the other way round. Ask what the sentence rules out.

And statements about one person.

Logic 090 Singular Statements

Treat the individual as a class containing only him, and a sentence about Socrates becomes a universal affirmation. Artificial, and it is what lets machinery this old handle individuals at all.

Five sentences taken through the procedure, and the pair from the previous topic appears here doing exactly what it does to students — placing the terms in opposite positions while looking like the same instruction.

Watch the procedure run.

Logic 091 Translation Walkthrough

Ask what the sentence asserts about which class, then rebuild it with an explicit quantifier, the two class nouns, and the linking verb. Where a restricting word appears, it marks the term after it as the second slot, not the first.

Five things you can now do — and Part Two builds syllogisms out of all of it. You can name any categorical statement's form by asking two questions: how much of the subject class it covers, and whether it puts members in or keeps them out. You can say which terms a form distributes, using the shortcut worth carrying: statements about a whole class cover their first term, and denials cover their second. You can diagram all four forms, keeping emptiness and existence apart — blanking a region and marking one are different claims and never substitute. You can say which square relations hold unconditionally and which need the older standpoint, and you know to check which standpoint a problem is using before answering it. And you can translate ordinary sentences into the system, including the pair of restricting phrases that place the terms in opposite positions.