Department of Philosophy (PHI)

Statements Equivalent to Conditionals

Once you have learned to recognize a variety of valid and invalid forms of conditional argument, you can evaluate many unfamiliar forms of argument by recasting them as equivalent conditional arguments. Any disjunction is equivalent to a conditional, and so is any negated conjunction. Any conditional is equivalent to another conditional (its contrapositive). Thus, whenever we find conditionals and/or disjunctions and/or negated conjunctions in an argument, we can substitute equivalent conditionals for them and check for familar valid and invalid conditional argument forms. For instance, the confusing argument "Not both P and not Q; either not Q or R; therefore if not R, then not P" is equivalent to an argument that is very familiar and not at all confusing. Read on...

1. Conditionals equivalent to conditionals (contrapositives).

If you negate both components of a conditional, and switch them around, the resulting conditional (called the contrapositive) is equivalent to the original conditional.

If R then S = if not S then not R Example. If it rained, it's steamy = if it's not steamy, it didn't rain

When you negate an already negative component, the double negation should be cancelled.

If R then not S = if not not S then not R = if S then not R

If not R then S = if not S then not not R = if not S then R

If not R then not S = if not not S then not not R = if S then R

Contraposition involves two operations: we negate both components and switch them around. Neither operation by itself gives an equivalent conditional.

If P then Q ≠ If Q then P

If P then Q ≠ If not P then not Q

Contraposition changes the form of a conditional argument but does not alter its validity.

2. Disjunctions equivalent to conditionals.

You can construct a conditional equivalent to a disjunction. Take either disjunct as the consequent, and the negation of the other disjunct as the antecedent.

Either R or S = if not R then S = if not S then R

Example. Either Rog or Sal will help = If Rog won’t help, Sal will = If Sal won’t help, Rog will

When you negate an already negative component, the double negation should be cancelled.

Either not R or S = if R then S = if not S then not R

Either R or not S = if not R then not S = if S then R

Either not R or not S = if R then not S = if S then not R

3. Negated conjunctions equivalent to conditionals.

You can construct a conditional equivalent to a negated conjunction (a "not both" type compound). Take either conjunct as antecedent, and the negation of the other conjunct as consequent.

Not both R and S = if R then not S = if S then not R

Example. Rog and Sal won’t both help = If Rog helps, Sal won’t = If Sal helps, Rog won’t

When you negate an already negative component, the double negation should be cancelled.

Not both R and not S = if R then S = if not S then not R

Not both not R and S = if not R then not S = if S then R

Not both not R and not S = if not R then S = if not S then R

Applications: arguments deciphered.