Logic 4.3

A and E propositions are considered to have existential import by Aristotle if their subject terms denote existing things. The Boolean (George Boole) position is that no universal propositions imply existential import. Basically, existential import is a limiting factor on what counts as a proposition.

“All dogs are animals” has existential import
“All unicorns are wild” has no existential import.

Venn Diagrams: A diagram to indicate the distribution of the terms of a proposition. No E import for A and E (there is no X).

The Modern Square of Opposition
A and O propositions are exactly opposite, since A asserts that the entire subject class is part of the predicate class, while O asserts that there is something of the subject class that is not in the predicate class. A and O are contradictories. E and I propositions are also contradictories, since E asserts that the overlap is empty, while I asserts there is at least one thing in the overlap.

Contradictory relationships establish the modern square of opposition:


Contradictory relations necessarily have the opposite truth-value. If A is true, then O is false and vice versa; If E is true, then I is false and vice versa. No other inferences, however, are possible; they are logically undetermined.

Immediate Inferences have one proposition and follow necessarily from the logically determinable opposition.
“Some students are gifted at math.
Therefore, it is false that all students are gifted at math.”

Immediate inferences are unconditionally valid regardless if they assert anything about existing things. Once we assume the truth or falsity of a proposition, we can immediately infer the truth value of its claimed inference.

If we use a Venn diagram to show the relationship between a proposition and its immediate inference, then if the two diagrams are the same (or the conclusion shows at least as much as the premise), then the inference is valid; otherwise it is not.

Existential import is key if we are to make the case for a valid immediate inference from all to some. From the Boolean standpoint, since we cannot assume Existential Import, A to I and E to O inferences commit the Existential Fallacy.