p q r
T T T
F T T
T F T
F F T
T T F
F T F
T F F
F F F
A proposition expresses both agreement and disagreement with the truth values of elemental propositions. This makes those truth values the conditions that determine whether the proposition is true or not.
It immediately strikes one that introducing elemental propositions provided the vasis for understanding all other kinds of sentences. Indeed, elemental propositions are palpably required to understand general propositions.
Given n elemental propositions, we can calculate the L(n) ways a particular proposition can both agree and disagree with their truth values. In an additional column dedicated to that proposition, we can denote that the proposition agrees with the combination of values in that row of the truth table by assigning the mark 'T' to the row. The absence of this mark denotes disagreement.
Now, expressing agreement with the truth values of the elemental propositions expresses the truth values of the proposition itself. Thus, the proposition is the expression of its truth values. The sign that results from adding the correlating marks to the truth table is its propositional sign.
It is clear that the complex of signs 'F' and 'T' in the truth table has no object (or complex of objects) corresponding to it, just as there is none corresponding to the horizontal and vertical lines or to the brackets. There are no 'logical objects'. Of course the same applies to all signs that express what truth tables express.
F T T
F F T)
If the order of the truth table is fixed once and for all by a combinatory rule, then the last column by itself will express the truth-conditions. If we now write this column as a row, the propositional sign will become
'(TT-T) (p,q)' or more clearly '(TTFT) (p,q)'.
(Note that the number of places in the left-hand pair of brackets is determined by the number of terms in the right-hand pair.)