Inference Theory of the Predicate Calculus


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

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