Categorical Logic

This is Unit Three of Logic, Part Two: the Categorical Syllogism. Part One handled statements one at a time. This part puts two of them together and asks the question the whole system was built for: does a third statement follow? Before anything else, the chart Part One asked you to memorise, because two of the five rules here consist of nothing but checks against it. Statements about a whole class make a claim covering their first term. Statements that deny make a claim covering their second term. That is the whole chart, and it decides more verdicts in this part than anything else. You get two independent methods for testing validity — one visual, one a checklist — and they should always agree. When they do not, one of them was run carelessly, and finding out which is the exercise. By the end you will be able to do five things. One: name a syllogism by its two coordinates, and identify its three terms by the role each plays. Two: test validity by diagram — and read a verdict of invalid for what it actually claims, which is less than people assume. Three: apply the five rules, and say which one changes answer depending on the standpoint. Four: diagnose the two most common failures by name. Five: take an argument out of rough prose and deliver a verdict under both standpoints.

The object is tightly specified: two premises, one conclusion, and exactly three terms with each appearing twice. That count is not a stylistic preference. A fourth term breaks the machine, and the usual way a fourth term sneaks in is a synonym swapped quietly for another.

The object.

Logic 092 The Categorical Syllogism

Two supporting statements, one conclusion, and three class terms each used twice. A fourth term ends the analysis before it starts — including one hidden by swapping in a synonym.

The three roles.

Logic 093 Major, Minor, and Middle Terms

The conclusion's two terms take one role each, and the third appears only in the premises. That third one is the engine: it joins the other two and then disappears from the result.

Every possible syllogism has a name made of two coordinates, and the reason to learn them is practical: naming a form is how you look up whether it is one of the small number that work. Out of two hundred and fifty-six arrangements, fifteen survive on the modern reading.

The two coordinates.

Logic 094 Mood and Figure

One coordinate is the sequence of the three statement forms; the other is where the connecting term sits across the two premises. Together they name every arrangement — the medieval logicians gave the working ones names of their own.

Watch the naming run.

Logic 095 Identifying Mood and Figure

Put it in the standard order, read the three letters off, then find the connecting term in each premise to fix the position. The trap is ordering: which premise is which is decided by the terms it holds, never by which one happens to be written first.

The first of the two methods. Three overlapping circles, both premises drawn onto them, and then nothing further — you do not draw the conclusion. If it is already there, the premises produced it. And when it is not there, the verdict says something narrower than it sounds.

The test.

Logic 096 The Venn Test for Validity

Put both premises on three circles and then look. Valid means you never had to add anything. One ordering rule matters: draw the universal premises first, because blanking a region decides where a mark is forced to go.

A verdict of yes.

Logic 097 Venn Test: Valid Case

Three circles, and the first premise blanks the part of the connecting class lying outside the predicate. The second blanks more. By the time both are drawn, the region the conclusion needs empty is already empty — nothing was added.

And a verdict by refusal.

Logic 098 Venn Test: Invalid Case

Both premises go on, and the region the conclusion needed emptied is left partly open. Read that correctly: an open region is not a claim that anything is in there. It records that the premises left the question unanswered.

The second method is a checklist of five, and the first two are pure distribution checks — which is why Part One insisted on the chart. Run them against it mechanically; neither requires any judgment.

Rules one and two.

Logic 099 Rules Method: Distribution Rules

The connecting term has to be covered by at least one premise. And any term the conclusion covers must have been covered in the premise it came from. The second rule inspects the conclusion's terms only — premises may cover whatever they like.

Two more rules, both about affirming and denying. The first is intuitive once stated. The second cuts in both directions, and forgetting the second direction is a common way to pass an argument that should fail.

Rules three and four.

Logic 100 Rules Method: Quality Rules

Two premises that both exclude never establish a connection — that is the third rule in a sentence. The fourth pairs a denying premise with a denying conclusion, and it fails equally hard read the other way round.

The fifth rule is the one where Part One's standpoint question comes back and actually changes an answer. Two premises that speak only about whole classes cannot deliver a conclusion asserting that something exists — on the modern reading. On the older one it can pass, under a condition.

Rule five.

Logic 101 The Existential Fallacy

Horned unicorns, dangerous horned things, therefore some dangerous unicorns — something was conjured that neither premise supplied. The older standpoint permits it when the class is known to have members, so settle the standpoint before giving a verdict.

Two violations are common enough to carry their own names, and both have everyday disguises worth recognising away from the page — one is the engine of guilt by association, and the other is how a conclusion quietly claims more than the premises paid for.

The first named failure.

Logic 102 Undistributed Middle

Cats are mammals, dogs are mammals, so cats are dogs. The connecting class is never covered by either premise, so nothing joins. Outside the classroom this is reasoning that two things sharing a category are the same thing.

The second.

Logic 103 Illicit Major and Illicit Minor

A conclusion covers a term that its own premise never covered. Which of the two terms offends gives the failure its particular name, and the diagnosis is the same in both cases: the conclusion is claiming more than the premises purchased.

Real prose rarely stops at one syllogism. It runs several together and leaves the intermediate conclusions unstated, which makes a long chain feel like a single confident stride.

The compressed chain.

Logic 104 Sorites

Two or more syllogisms with their middle results left out. Test each junction separately, because one bad link fails the whole chain no matter how solid everything around it is.

The unit closes on one argument in rough prose — with a restricting word in it, so Part One's translation trap is waiting at the very first step — run through translation, naming, both test methods, and a verdict stated under each standpoint.

The capstone. Licensed contractors, and county work.

Logic 105 Capstone: Full Syllogism Analysis

Translate, reorder, name it, diagram it, run the five rules, and give the verdict under both readings. The diagram agreeing with the checklist is worth something only when both were actually run — working one and assuming the other proves nothing.

Five things you can now do, and Unit Three is complete. You can name a syllogism by its two coordinates and identify its three terms by role — with the connecting term the one that joins the other two and then leaves. You can test validity by diagram: both premises on three circles, universals first, and nothing added. And you can read a failed test correctly — an open region means the premises left it open, not that something is in there. You can run the five rules: the connecting term covered at least once, no term over-covered in the conclusion, no pair of denials, denial matched on both sides, and no existence conjured from two universals. You can name the two common failures — the link that never forms, and the conclusion claiming more coverage than its premise bought. And you can take an argument out of prose and deliver a verdict under both standpoints, with two independent methods that had better agree.