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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.

Rule of Inference | Name |
---|---|

$$\begin{matrix}
\forall x P(x) \\
\hline
\therefore P(y)
\end{matrix}$$ | 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 $(\exists y)P(y)$.

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) $

(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

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