A Class-Room Introduction to Logic

May 4, 2009

Unit-VII: Immediate Inference-Square of Opposition

Filed under: Logic,Square of Oppsition — Dr. Desh Raj Sirswal @ 8:36 am

Inference: Inference is the act or process of deriving a conclusion based solely on what one already knows. Inference has two types: Deductive Inference and Inductive Inference. They are deductive, when we move from the general to the particular and inductive where the conclusion is wider in extent than the premises. In intelligence testing, mostly deductive inference ability is judged. Inference is studied within several different fields.

  • Human inference (i.e. how humans draw conclusions)is traditionally studied within the field of cognitive psychology.
  • Logic studies the laws of valid inference.
  • Statisticians have developed formal rules for inference (statistical inference) from quantitative data.
  • Artificial intelligence researchers develop automated inference systems.

IMMEDIATE INFERENCE:

Deductive inference may be further classified as (i) Immediate Inference (ii) Mediate Inference. In immediate inference there is one and only one premise and from this sole premise conclusion is drawn. Immediate inference has two types mentioned below: Square of Opposition , Eduction. Here we will study about Square of Opposition.

Square of Opposition:

Any logical relation among the kinds of categorical propositions (A, E,I and O) exhibited on the Square of Opposition. There are four ways in which propositions may be “observed” –as Contradictories, Contraries, Sub-contraries and sub-alternation. These are representing with an important and widely used diagram called the Square of Opposition. This is given below:

square

The four corners of this diagram represent the four basic forms of propositions recognized in classical logic:

A propositions, or universal affirmatives take the form: All S are P.
E propositions, or universal negations take the form: No S are P.
I  propositions, or particular affirmatives take the form: Some S are P.
O propositions, or particular negations take the form: Some S are not P.

Given the assumption made within classical (Aristotelian) categorical logic, that every category contains at least one member, the following relationships, depicted on the square, hold:

Contradictory:

Propositions are contradictory when the truth of one implies the falsity of the other, and conversely. A and O propositions are contradictory, as are E and I propositions. Here we see that the truth of a proposition of the form All S are P implies the falsity of the corresponding proposition of the form Some S are not P. For example, if the proposition “all industrialists are capitalists” (A) is true, then the proposition “some industrialists are not capitalists” (O) must be false. Similarly, if “no mammals are aquatic” (E) is false, then the proposition “some mammals are aquatic” must be true.

Contrary:

Propositions are contrary when they cannot both be true; if one is true, then other must be false. They can both be false. A and E propositions are contrary. An A proposition, e.g., “all giraffes have long necks” cannot be true at the same time as the corresponding E proposition: “no giraffes have long necks.” Note, however, that corresponding A and E propositions, while contrary, are not contradictory. While they cannot both be true, they can both be false, as with the examples of “all planets are gas giants” and “no planets are gas giants.”

Subcontrary:

Propositions are subcontrary when it is impossible for both to be false; if one is false then other must be true. They can both be true. I and O propositions are subcontrary. Because “some lunches are free” is false, “some lunches are not free” must be true. Note, however, that it is possible for corresponding I and O propositions both to be true, as with “some nations are democracies,” and “some nations are not democracies.” Again, I and O propositions are subcontrary, but not contrary or contradictory.

Subalternation:

Two propositions are said to stand in the relation of Subalternation when the truth of the first (“the superaltern”) implies the truth of the second (“the subaltern”), but not conversely. A propositions stand in the Subalternation relation with the corresponding I propositions. The truth of the A proposition “all plastics are synthetic,” implies the truth of the proposition “some plastics are synthetic.” However, the truth of the O proposition “some cars are not American-made products” does not imply the truth of the E proposition “no cars are American-made products.” In traditional logic, the truth of an A or E proposition implies the truth of the corresponding I or O proposition, respectively. Consequently, the falsity of an I or O proposition implies the falsity of the corresponding A or E proposition, respectively. However, the truth of a particular proposition does not imply the truth of the corresponding universal proposition, nor does the falsity of a universal proposition carry downwards to the respective particular propositions.

Inferences from Square of Opposition:

A number of very useful immediate inferences may be readily drawn from the information embedded in the traditional square of opposition. Given in the truth, or the falsehood, of any one of anyone of the four standards form categorical proposition, it will be seen that the truth or falsehood of some or all of the others can be inferred immediately.

A being given as True: E is false; I is true; O is false.

E being given as True: A is false; I is false; O is true.

I being given as True: E is false; A and O are undetermined.

O being given as True: A is false; E and I are undetermined.

A being given as  False : O is true , E and I are undetermined.

E being given as False: I is true; A and O are undetermined.

I  being given as False: A is false; E is true; O is true.

O being given as False: A is true; E is false; I is true.

For example:

What can you infer about the truth or falsity of the following if you assume “ Some reptiles are not poisonous” is false?

(1)    All reptiles are poisonous. – True

(2)   No reptiles are poisonous. –  False

(3)   Some reptiles are poisonous.- True

What is the name of the opposition relation in which the categorical statements differ:

  1. In quantity only?
  2. In both quality and quantity
  3. Between A and I.
  4. Between I and O.
  5. Between E and I.
Advertisements

Blog at WordPress.com.