rules of inference in propositional logic formal proof of validity
formal proof of validity rules of inference
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Propositional Logic
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Rules of Inference in Propositional Logic: Formal Proof of Validity
Rules of inference are understood as elementary valid arguments that are used in
justifying steps in formal proofs.
In these notes, I will discuss the topic “Rules of Inference in Propositional Logic: Formal
Proof of Validity”. As is well known, a “formal proof of validity” is a series of
propositions, each of which follows from the preceding propositions by an elementary
valid argument form or, simply, rules of inference.
It is important to note that the rules of inference can be used as a method for
determining the validity of an argument. However, the main function of the rules of
inference is to prove the validity of an argument―thus the name “formal proof of
validity”. Normally, the arguments here are already valid, and what we will do is prove
that indeed the arguments are valid.
How do we prove the validity of an argument using the 10 rules of inference?
In doing so, we will construct a series of propositions based on the given argument using
the rules of inference. The goal here is to come up with a proposition that matches with
the conclusion of the given argument. In other words, we will extract from the premises
the conclusion of the given argument. Once we have successfully done this, then we can
say that we have proven the validity of the argument. Consider the illustration below.
I will fully explain this process later. But let me point here that the argument is now
proven valid because, as I already mentioned above, the last proposition in the new
series of propositions, that is, Proposition #7, which is ~ s, matches with the conclusion
of the given argument, which is ~ s.
, Now, there are 10 rules of inference that we can use in proving the validity of
arguments, namely, 1) Modus Ponens (M.P.), 2) Modus Tollens (M.T.), 3) Hypothetical
Syllogism (H.S.), 4) Disjunctive Syllogism (D.S.), 5) Constructive Dilemma (C.D.), 6)
Destructive Dilemma (D.D), 7) Conjunctive (Conj.), 8) Simplification (Simp.), 9) Addition
(Add.), and 10) Absorption (Abs.).
The forms of these rules are as follows:
Let me explain each rule below, including some of its variations.
In Modus Ponens, there are two premises, namely, p ⊃ q and p, and a conclusion, which
is q. Please note that the premises and conclusion are divided by a horizontal line. Note
as well that the “line” is also read as “therefore”. Thus, the form of Modus Ponens
reads: “if p then q, p, therefore, q.
In Modus Ponens, we have to take note that the second premise p affirms the
antecedent of the first premise p ⊃ q, and the conclusion q affirms the consequent of
the first premise p ⊃ q. Hence, if, for example, the antecedent of the first premise is
negated, that is, ~ p ⊃ q, then we need a ~ p for the second premise in order for us to
have a valid form of a Modus Ponens. See the illustration below.
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