Rules of Replacement in Propositional Logic: Formal Proof of Validity

Rules of Replacement in Propositional Logic: Formal Proof of Validity


In my notes titled "Rules of Inference in Propositional Logic: Formal Proof of Validity"
(look for these notes in the search engine of Studypool), I discussed the way in which
arguments are proven valid using the 10 rules of inference. In these notes, I will discuss
the 10 rules of replacement as another method that can be used to justify steps in the
formal proof of validity.

Rules of replacement are logical equivalences or logically equivalent sentence forms,
which allow us to replace or substitute one member of a pair in the process of proving
the validity of arguments.

But why the rules of replacement?

They are important because there are cases wherein the 10 rules of inference may not
be employed in proving or demonstrating the validity of arguments. Hence, when cases
like this occur, the rules of replacement may be the best, if not the only, method that
can be employed in proving the validity of arguments.

Rules of Replacement vs Rules of Inference

It might be worthwhile at this point to briefly sketch the major differences between
rules of replacement and rules of inference before we proceed to discuss in great detail
the nature and dynamics of the 10 rules of replacement.

For one, the rules of inference are forms of valid arguments, while the rules of
replacement are forms of equivalent propositions. This is the reason why we have the
symbol ∴ (read as "therefore") in rules of inference, while in rules of replacement, as I
will show later, we have the equivalent sign ≡ (read as "if and only if") between two
parts or propositions.

In my previous post on rules of inference, we can notice that the rules of inference only
work in one direction and can hardly be used within larger compound propositions,
while the rules of replacement, as I will show below, work in either direction and can be
applied whenever larger compound propositions occur.

Lastly, the rules of inference can have two premises and, for this reason, the justification
may have two numbers before the abbreviated rule (please see my post on rules of
inference through this link: http://philonotes.com/index.php/2018/03/28/rules-of-

, inference/). Rules of replacement, on the other hand, on only have one proposition,
which is equivalent to another proposition; thus, the justification will always have only
one number in front of the abbreviated name.

Let me now proceed to the discussion on the 10 rules of replacement in order to fully
understand the points I have just made above.

Rules of Replacement

There are 10 rules of replacement, namely:

1) Double Negation (D.N.),
2) Commutation (Comm.),
3) Association (Assoc.),
4) De Morgan's Theorem (D.M.),
5) Material Implication (M.I.),
6) Transposition (Trans.),
7) Distribution (Dist.),
8) Material Equivalence (M.E.),
9) Tautology (Taut.), and
10) Exportation (Exp.).

1. Double Negation

The form of double negation is as follows:

p≡~~p

In double negation, we can replace ~ ~ p with p and vice versa because ~ ~ p
is absolutely the same with p. Hence, ~ ~ p is equivalent to p.

Consider the proposition below.

It is not true that Melbert is not studying. (p)

As we can see, the proposition above is a simple proposition (because there is no other
component proposition) with two negation signs "not". Thus, if we symbolize the
proposition, then we have to symbolize the negation signs accordingly. So, if we let p
stand for "It is not true that Melbert is not studying", then the proposition is symbolized
as follows: