Inference Theory of the Predicate Logic

To reach a conclusion on quantified statements, there are four rules of inference which are collectively called as Inference Theory of the Predicate Calculus.

Table of Rules of Inference

• Rule US: Universal Specification - From $(x)P(x)$, one can conclude $P(y)$.

• Rule US: Universal Generalization - From $P(c)$, one can conclude $xP(x)$ consider the fact that c is not free in any given premises. if x is free in a step resulted from Rule ES, then any variable introduced by Rule ES should be free in P(c).

• Rule US: Existential Specification - From $(\exists x)P(x)$, one can conclude $P(c)$ consider the fact that c is not free in any given premises and also not free in any prior step of derivation.

• Rule US: Existential Generalization - From $P(c)$, one can conclude $(\exists y)P(y)$.

Example

Consider the following argument popularly known as "Socrates argument".

• All men are mortal

• Socrates is a man

• Therefore Socrates is mortal

Let's use the Predicate formulae the above statements.

• H(x) : x is man

• M(x) : x is mortal.

• s: Socrates.

Now the above statements can be represented as −

• All men are mortal - $ (x)(H(x) \rightarrow M(x)) $

• Socrates is a man - $ H(s) $

• Socrates is mortal - $ M(s) $

As a statement, we need to conclude −

$ (x)(H(x) \rightarrow M(x)) \land H(s) \implies M(s) $

Solution

• (1) $ (x)(H(x) \rightarrow M(x)) $ - Hypotheses

• (2) $ H(s) \rightarrow M(s) $ - Rule US using (1)

• (3) $ H(s) $ - Hypotheses

• (4) $ M(s) $ - Simplification