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DeMorgan’s Laws

If you have taken a course in logic, you are probably familiar with these formulas. Their validity is intuitively clear: The conjunction A&B is false when either, or both, of its parts are false. This is precisely what ~A or ~B says. And the disjunction A or B is false only when both A and B are false, which is precisely what ~A and ~B says.

You will rarely get an argument whose main structure is based on these rules—they are too mechanical. Nevertheless, DeMorgan’s laws often help simplify, clarify, or transform parts of an argument.

Example-1: (DeMorgan’s Law)


It’s not the case that the senator will be both reelected and not acquitted of campaign fraud.

Let R stand for “the senator will be reelected,” and let A stand for “acquitted of campaign fraud.” Using these symbol statements to translate the argument yields


~(R & ~A)


which by the first of DeMorgan’s laws is equivalent to


~R or ~(~A)


This in turn can be reduced to


~R or A


This final diagram tells us that the senator either will not be reelected or will be acquitted, or both.

Example-2: (DeMorgan’s Law)


It is not the case that either Bill or Jane is going to the party.

This argument can be diagrammed as ~(B or J), which by the second of DeMorgan’s laws simplifies to (~B and ~J). This diagram tells us that neither of them is going to the party.

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