Chapter 5.5: Traditional Square of Opposition

Traditional Square of Opposition•The traditional square of opposition is an arrangement of lines that illustrates logically necessary relations among the four kinds of categorical propositions.•A collection of logical relationships traditionally embodied in a square diagram.

Opposition •Standard-form categorical propositions having the same subject and terms and the same predicate terms may differ from each other in quality, or in quantity, or in both. Any kind of differing has been traditionally called opposition. The term is used even when there is no apparent disagreement between the propositions.

Four relations in the traditional square of opposition

1. Contradictory= opposite truth values2. Contrary= at least one is false (not both

true)3. Subcontrary= at least one is true (not both

false)4. Subalternation= truth flows downward,

falsity flows upward

Contradictories

• Two propositions are contradictories if one is the denial or negation of the other- that is, if they cannot both be true and cannot both be false.• They are opposed in both quality (one affirms, the other denies) and quantity (one refers to all, and the other to some). Of the pair, exactly one is true and exactly one is false.• A and O are contradictories, as E and I.

• Examples:A & O All judges are lawyers

Some judges are not lawyers E & I No politicians are

idealists Some politicians are

idealists

Contraries

•Two propositions are said to be contraries if they cannot both be true- that is, if the truth of one entails the falsity of the other

•The traditional account of categorical propositions held that universal propositions (A and E) having the same subject and predicate terms but differing in quality (one affirming, the other denying) were contraries.•Same subject and predicate, both universal, differs in quality

•If A is given as true, the corresponding E proposition is false (because at least one must be false), and if an E proposition is given as true, the corresponding A proposition is false.

A & E All cats are animals. No cats are animals.

All poets are dreamers.No poets are dreamers

E & A No cats are dogs.

All cats are dogs.•But if an A proposition is given as false, the corresponding E proposition could be either true or false, in this case the E proposition has logically undetermined truth value. Similarly, if an E proposition is given as false, the corresponding A proposition has logically undetermined truth value.

•Contraries cannot both be true, but they can both be false

All animals are catsNo animals are cats

Subcontraries

•Two propositions are said to be subcontraries if they cannot both be false, although they may both be true•Expresses partial opposition•Same subject and predicate, both particular, differs in quality.

•The traditional account held that particular propositions (I and O) having the same subject and predicate terms but differing in quality (one affirming the other denying are subcontraries).

•If I proposition is given as false, the corresponding O proposition is true (because at least one must be true), and if an O proposition is given as false, the corresponding I proposition is true. But if either an I or an O proposition is given as true, then the corresponding proposition could be either true or false, thus in this case the corresponding proposition would have logically undetermined truth values.I&O Some cats are dogs

Some cats are not dogs

Some diamonds are precious stonesSome diamonds are not precious stones

O&I Some cats are not animalsSome cats are animals

Subalternation

•When two propositions have the same subject and the same predicate terms, and agree in quality (both affirming or both denying) but differ in quantity (one universal, the other particular), they are called corresponding propositions. A proposition has a corresponding I proposition, and the E proposition has a corresponding O proposition.•This opposition between a universal proposition and its corresponding particular proposition is known as subalternation.

A&I All spiders are eight-legged animals Some spiders are eight-legged animals

All politicians are liarsSome politicians are liars

E&ONo whales are fishesSome whales are not fishes

No soldiers are cowards.Some soldiers are not cowards.

•The universal proposition (A or E) is called the superaltern, and the particular is called the subaltern (I or O).•The downward arrow transmits only truth, and the upward arrow only falsity.

•If an A proposition is given as true, the corresponding I proposition is true also, and if an I proposition is given as false, the corresponding A proposition is false.

•But if an A proposition is given as false, this truth value cannot be transmitted downward, so the corresponding I proposition will have logically undetermined truth value. Conversely, if an I proposition is given as true, this truth value cannot be transmitted upward, so the corresponding A proposition will have logically undetermined truth value. The same rule applies to E and O propositions.•To remember the direction of the arrows for subalternation, imagine that truth descends from “above,” and falsity rises up form “below.”

InferencesAn inference is said to be mediate when more than one premise is relied upon (syllogism) because the conclusion is drawn from first the first premise through the mediation of the second. But when a conclusion is derived from only one premise the inference is said to be immediate.

Examples of very useful immediate inferences from the Traditional Square of Opposition

•If an A proposition is our premise, then once can validly infer that the corresponding O proposition is false.•If an A proposition is our premise, the corresponding I proposition is true.•If an I proposition is our premise, its corresponding E proposition, which contradicts it must be false.

There are also immediate inferences that can be derived given the truth or the falsehood of any one of the four standard-form categorical propositions.A is given as true: E is false; I is true; O is false.E is given as true: A is false; I is false; O is true.I is given as true: E is false; A and O are undetermined.O is given as true: A is false; E and I are undetermined.A is given as false: O is true; E and I are undetermined.E is given as false: I is true; A and O are undetermined.I is given as false: A is false; E is true; O is true.O is given as false: A is true; E is false; I is true.

I is false

Using the contradictory, determined that E is true

Both the subalternation and the contrary tell us that A is false

The contradictory, subcontrary, and subalternation, all give us O being true.

•We will not always get clear answers. In this example, knowing that E is false give us that I is true, but the truth values for A and O.•If we know that E is false we can use the contradictory that would make I true. But truth does not go upward, and if E is false A could still be false. And if I is true then O could also be true, since both of them could be true.•O and A in this particular case have undetermined truth values.