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Truth-table: A matrix made up of a top-row in which there is
a formula of
propositional logic and
2n rows, where n = the number of distinct propositional
variables in the formula, and the rows
in the leftmost column are made up of all possible assignments of T and
F (respectively: true and
false) to the variables in the formula. The
column under the formula has a T or an F in any of the rows depending on
whether the formula is respectively true or false given the assigments
made to the propositional variables in the leftmost column on that row. An n*m
matrix
is simply an arrangement made up of n rows with m columns.
Here is an example of a truth-table that defines the properties of the
connective '&' read as 'and':
| |
P&Q |
| v(P)=T v(Q)=T |
T |
| v(P)=T v(Q)=F |
F |
| v(P)=F v(Q)=T |
F |
| v(P)=F v(Q)=F |
F |
Usually this is replaced by:
| P |
Q |
P&Q |
| T |
T |
T |
| T |
F |
F |
| F |
T |
F |
| F |
T |
F |
which in the first two columns gives all possible assignments of T
and F to all possible variables in the formula.
Here is a similar truth-table for the properties of '~' read as 'not':
And for 'V' read as 'or':
| P |
Q |
PVQ |
| T |
T |
T |
| T |
F |
T |
| F |
T |
T |
| F |
T |
F |
Note that each truth-table can be equivalently replaced
by a set of equations like the following one, that is equivalent to the
last truth-table: v(PVQ)=T iff v(P)=T or v(Q)=T
v(PVQ)=F iff v(P)=F and v(Q)=F.
One can also use truth-tables to represent cases with more than two
truth-values. The difference is that for v distinct variables in the
formula, and t distinct truth-values,
there will have to be tv rows for all the possible distinct
valuations of the formula.
Above there are the truth-tables for the logical terms
'~' (not), '&' (and), and 'V' (or). The two other standard propositional
connectives are '-->' (implies) and 'iff' (if and only if) and have
these truth-tables
| P |
Q |
P-->Q |
| T |
T |
T |
| T |
F |
F |
| F |
T |
T |
| F |
T |
T |
| P |
Q |
PiffQ |
| T |
T |
T |
| T |
F |
F |
| F |
T |
F |
| F |
T |
T |
And the following table shows that in terms of
truth-tables the formula '(p&q)-->p' i.e. '(p and q) implies p' is a
tautology:
| p |
q |
|
(p&q) |
(p-->q) |
((p&q) --> q) |
| T |
T |
|
T |
T |
T
T T |
| T |
F |
|
F |
F |
F
T F |
| F |
T |
|
F |
T |
F
T T |
| F |
F |
|
F |
T |
F
T F |
|
