# Inference Theory of the Predicate Calculus

MathematicsComputer EngineeringMCA

To reach on 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 of InferenceName
$$\begin{matrix} \forall x P(x) \\ \hline \therefore P(y) \end{matrix}$$

Rule US: Universal Specification

$$\begin{matrix} P(c) \text { for any c} \\ \hline \therefore \forall x P(x) \end{matrix}$$

Rule UG: Universal Generalization

$$\begin{matrix} \exists x P(x) \\ \hline \therefore P(c) \text { for any c} \\ \end{matrix}$$

Rule ES: Existential Specification

$$\begin{matrix} P(c) \text { for any c} \\ \therefore \exists x P(x) \end{matrix}$$

Rule EG: Existential Generalization

• 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

Published on 09-Aug-2019 07:01:33