| no premises |
Comment. Another special case is a valid sequent
without premises. In this case, validity requires that
there be no lines of the TT on which the conclusion
is false, since no premises are present to be
considered. |
|
| P Q | |- P -> (~P -> Q) |
| T T | T F T |
| T F | T F T |
| F T | T T T |
| F F | T T F |
Table 2.6 A valid sequent without premises. |
|
| P Q R | |- (P <-> Q) -> (P v ~R) |
| T T T | T T T F |
| T T F | T T T T |
| T F T | F T T F |
| T F F | F T T T |
| F T T | F T F F |
| F T F | F T T T |
| F F T | T F F F |
| F F F | T T T T |
Table 2.7 An invalid sequent without premises. |
| tautology |
Definition. A sentence f is a TAUTOLOGY (or, is
TAUTOLOGOUS) when the sequent that has no
premises and has f as its conclusion is valid. |
|
Comment. When a sentence is a tautology, it cannot
be false: its TT has only Ts in the column for the
sentence. Some sentences have only Fs appearing in
their column of a TT; others have both Ts and Fs.
The sentence appearing in table 2.6
is a tautology. |
| inconsistent and contingent |
Definition. A sentence that has only Fs in its column
of a TT is INCONSISTENT. A sentence that is
neither tautologous nor inconsistent is CONTINGENT. |
|
Comment. The sentence appearing in table 2.7 is
contingent. |
|
| P Q | ((P -> Q) -> P) ->P |
| T T | T T T |
| T F | F T T |
| F T | T F T |
| F F | T F T |
Table 2.9 ((P -> Q) -> P) -> P is tautologous. |
|
| P Q | (P & Q) <-> (~P v ~Q) |
| T T | T F F F F |
| T F | F F F T T |
| F T | F F T T F |
| F F | F F T T T |
Table 2.10 (P & Q) <-> (~P v ~Q) is inconsistent. |
| Exercise 2.3 |
Use TTs to establish that all the theorems considered
in chapter 1 are tautologies. |