ABSTRACT
Let me first introduce some of the logical connectives that will be
used in the following:
Inputs Inclusive disjunction Implication Conjunction
p q p ∨ q p → q p ∧ q 0 0 0 1 0
0 1 1 1 0
1 0 1 0 0
1 1 1 1 1
Here, 0 and 1mean false and true, respectively, and p, q are arbitrary propositions. To these connectives we should add the negation ¬ that inverts the truth value of proposition (to deny a true proposition
is false and vice versa). The equivalence↔will be introduced below. Let us now make some elementary consideration. Suppose that
we know that a certain proposition p is true or we hypothetically assume that it is true. Logically speaking, we can also deny it
Mechanical Logic in Three-Dimensional Space Gennaro Auletta Copyright c© 2014
September 13,
propositions obviously denote alternative state of affairs. We can
express this by building a disjunction of the two. The resulting
proposition (p ∨ ¬p) is obviously always true. We can reverse this order of consideration and start from the latter proposition
as a composed one and allowing the possibility to deny each of
its components. By denying ¬p we obtain p whilst by denying p we obtain ¬p. In other words, we can obtain the two atomic propositions by denying one of the components of the proposition
p ∨ ¬p. However, it is in principle also possible to deny more than one component of a molecular proposition. However, in such a case
(inwhichwe only have p and¬p) we run immediately (after the first step) into the contradiction p ∧ ¬p (which is indeed the negation of p ∨ ¬p). Reached this point, we can again reverse this order of consideration and start from the proposition p∧¬p and build other propositions by affirming one of its components. In this case, we
again obtain the propositions p and ¬p. We can depict this very elementary logical space as in Fig. 1.1.
