1
A statement is about two circles
Draw one circle for the S things and one for the P things. They make four regions: S only, both, P only, neither. Every categorical statement says something about whether a region is empty or occupied.
Lesson · Arguments and Validity · Categorical logic
Categorical Statements and Immediate Inference
All, No, Some and Some-not are claims about which regions of a two-circle diagram are empty and which are occupied. Reading them that way turns conversion, obversion and contraposition into checks: a conclusion follows when the premise forces its regions.
All S are P: S outside P is empty
No S are P: S inside P is empty
Some S are P: S inside P is occupied
Some S are not P: S outside P occupied
valid when the premise forces the regions
Four statements, four region claims
Core ideas
What this lesson establishes
The propositional connectives cannot see inside a sentence. Categorical statements are about the things a sentence is about, and regions are the cases.
2
Universal statements remove, particular ones require
All and No say that a region is empty. Some and Some-not say that a region is occupied. An unmarked region is left open, which is not the same as occupied.
3
A one-step inference is a test of forcing
Conversion, obversion and contraposition each take one statement and write another. The new statement follows when every pattern of empty and occupied regions that the first allows also makes the second true.
The material
Statements as regions
Four forms, each a claim about one region, and then the three rewritings that logicians have always tried on them.
Definition · Categorical statement
A categorical statement relates two terms, a subject S and a predicate P, in one of four forms: A, All S are P; E, No S are P; I, Some S are P; and O, Some S are not P.
form statement what it claims about regions
---------------------------------------------------------
A All S are P S outside P is empty
E No S are P S inside P is empty
I Some S are P S inside P is occupied
O Some S are not P S outside P is occupied
This reading is the Boolean one: a universal statement says that a region has nothing in it, and does not say that any other region has anything in it. “All S are P” is true if there are no S at all. The next lesson sets this beside the older reading and says what each costs; for now the lab shows the Boolean one.
Example · All squares are rectangles
Let S be the squares and P the rectangles. The claim empties one region only: the squares that are not rectangles. It says nothing about the region of rectangles that are not squares, which is where the two-by-one rectangle lives, and it says nothing about whether the region of squares that are rectangles has anything in it.
An immediate inference has one premise. The test is the same as for a larger argument: list the patterns of empty and occupied regions the premise allows, and check the conclusion in each. With two terms there are four regions and sixteen patterns; a premise that fixes one region, empty or occupied, leaves the other three open, so eight patterns survive it, and that is the count the lab reports. If the conclusion fails in some allowed pattern, that pattern is the counterexample; if it fails in none, the inference is valid.
Conversion swaps subject and predicate. No S are P converts to No P are S, because both empty the single region where the circles overlap. The same holds for Some, which occupies that region. But All S are P empties S outside P, and All P are S empties P outside S, a different region. Some S are not P occupies S outside P, and its converse needs P outside S: the same failure, with occupied in place of empty.
Contraposition swaps the terms and takes the complement of each, written non-P. All S are P becomes All non-P are non-S, and both empty the same region, S outside P. Obversion changes the quality and takes the complement of the predicate: All S are P becomes No S are non-P. Again one region is emptied.
Theorem · Which one-step inferences are valid
Conversion is valid for E and I, invalid for A and O. Contraposition is valid for A and O, invalid for E and I. Obversion is valid for all four. Each verdict is read off which region the premise and conclusion constrain, and nothing is memorised.
The lab draws two circles, shades what the premise forces empty, and puts a cross in any region it forces occupied. When the inference is invalid it draws the counterexample, the smallest pattern that satisfies the premise and fails the conclusion.
Interactive
Categorical logic by regions
Every pattern of empty and occupied regions
Method
Testing a one-premise inference
Five steps; the second is where most errors enter.
1
Name the terms, and their complements
Pick the two terms and write non-S and non-P where they occur. A complement is a term in its own right: it is everything outside the circle.
2
Turn each statement into a region claim
A and E empty a region. I and O occupy one. Write which region, using the four names S only, both, P only and neither.
3
List what the premise forces
Mark the regions the premise empties or occupies. Leave every other region open, because an unmarked region may be empty or occupied.
4
Ask whether the conclusion's region is forced
If the conclusion empties a region, the premise must empty it. If it occupies one, the premise must occupy it.
5
If it is not forced, describe the counterexample
Say which region the premise leaves open that the conclusion needs. That region, emptied or occupied the wrong way, is the case the conclusion does not survive.
Regions and rewritings
Score 0 / 0Choose the answer you can justify.
Squares and rectangles, converted
Let S be the squares and P the rectangles. The premise is All squares are rectangles, and the conversion offered is All rectangles are squares.
premise squares outside rectangles: empty
conclusion rectangles outside squares: must be empty
premise leaves that region open
counterexample: only rectangles outside squares
The conclusion needs a region the premise never mentions. The pattern with only that region occupied satisfies the premise (no square is missing from the rectangles) and fails the conclusion (there is a rectangle that is not a square). In the lab this is the convert-a preset, and the counterexample it draws is the same pattern. A premise can force one region without forcing its mirror image.
Common mistakes
1
Converting “All S are P” to “All P are S”The first empties S outside P; the second empties P outside S. All squares are rectangles is true, and all rectangles are squares is false of the two-by-one rectangle. The lab draws exactly that pattern as the counterexample, a region occupied that the premise never touches.
2
Reading “Some” as “some but not all”Some S are P requires one occupied region and is silent about the rest. It is compatible with every S being a P, and Some S are P and Some S are not P can both be true at once. “Some” here means at least one.
3
Taking an unshaded region to be occupiedA shaded region is empty, and a cross is occupied; a blank region is open. A premise that says nothing about a region allows it to be either, which is what the counterexample uses. Blank is permission, and the lab tests every way of using it.
Check your understanding
✓
Finish when you can name the region that decides each inference.
Given one categorical premise and a conclusion, write each as a claim about empty or occupied regions, say whether the premise forces the conclusion's region, and describe the counterexample pattern when it does not.
Note: Every statement here is read the Boolean way, with no assumption that a term has members. The next lesson makes that assumption a switch and shows which inferences depend on it.