Yet More Logic in Scripture

bible_with_books_med[1]Logic students regularly struggle with immediate inferences, and (as is often the case when students have more than usual difficulty) they can begin to wax philosophical about the value of learning this particular concept. As an initial response to such students, I want to give a couple of examples of immediate inferences used in the Bible. Two equivalent immediate inferences for categorical statements are obverse and contrapositive.

Obverse changes the quality of the statement, and takes the complement of the predicate. It gives equivalent statements for all four forms of categorical statement:

All S is P  ≡  No S is non-P
No S is P  ≡  All S is non-P
Some S is P  ≡  Some S is not non-P
Some S is not P  ≡  Some S is non-P

Jesus uses the obverse in Mark 2:22, where He says,

“No one pours new wine into old wineskins. Otherwise, the wine will burst the skins, and both the wine and the wineskins will be ruined. No, they pour new wine into new wineskins.”

Contrapositive switches the subject and predicate of the statement, and changes both to their complements. It gives equivalent statements for universal affirmative and particular negatives:

All S is P  ≡  All non-P is non-S
Some S is not P  ≡  Some non-P is not non-S

Paul uses something like the contrapositive in Romans 11:6 when he argues,

“And if by grace, then it is no longer of works; otherwise grace is no longer grace. But if it is of works, then it is no longer grace; otherwise work is no longer work.”

This is more obviously the contrapositive when the conditional statements are translated into categorical form.

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