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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 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 $(\exists y)P(y)$.
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
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