THE LOGIC NOTES

Negation Glossary

Definition

The negation connective 'NOT' results in a formula NOTA which is true if and only if A is false.

Connectives with similar truth conditions in natural languages are also taken to express negation. The phrase 'it is not the case that' in English is the intended analogue of the formal connective.

Comments

By reversing the sense of a proposition, negation makes it possible to deny things by asserting the opposite. For instance, 'I am not guilty' is an assertion which constitutes a denial of guilt.

The truth table is:

NOT
0 1
1 0

In the natural deduction calculus, the elimination rule for negation says that A and NOTA together lead to absurdity. One notational convention for saying that a set of assumptions is absurd would be simply to leave the conclusion blank, but we prefer to write '⊥' to symbolise inconsistency. Hence what the NOTE rule amounts to is that ⊥ is an immediate consequence of A and NOTA. The introduction rule for negation says that if ⊥ follows from some set X of premises together with A, then NOTA follows from X.

Double negation: The introduction and elimination rules above are not quite sufficient to capture classical negation. We need to make special provision for double negation—i.e. for the fact that NOTNOTA is equivalent to A. Double negation elimination allows the inference from NOTNOTA to A.

We can avoid the use of the absurd constant ⊥ by running the elimination and introduction rules together, giving one rule RAA (reductio ad absurdum) which goes from A and NOTA to NOTB, optionally discharging assumptions of B if there are any.

Examples

  1. It is false that 12 is a multiple of 5, so it is true that 12 is not a multiple of 5.
  2. It is false that 15 is not a multiple of 5, so it is true that 15 is a multiple of 5.
  3. The following proof illustrates the rules for negation:
    SP {NOTp IMP q, NOTq ⊢ p} PL {1} {1} {NOTp IMP q} {A} PL {2} {2} {NOTq} {A} PL {3} {3} {NOTp} {A} PL {1,3} {4} {q} {1, 3 IMPE} PL {1,2,3} {5} {⊥} {2, 4 NOTE} PL {1,2} {6} {NOTNOTp} {5 [3] NOTI} PL {1,2} {7} {p} {6 NOTNOTE} EP

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