# (Solved Homework): Using: Your familiar "un-quantified" rules of inference (modus ponens, disjunctive syllogism, etc). The quan…

Using: Your familiar “un-quantified” rules of inference (modus ponens, disjunctive syllogism, etc). The quantified versions of Modus Ponens/Tollens and Hypothetical Syllogism, all of which are presented in chapter 3.4 of our textbook and Axioms of logical equivalence (associativity, double negation, etc) show that the proofs depicted in table 4 are valid. Note that we are not asking you to use truth tables of any kind. We are asking you to use quantified (“universal”, in the terminology of Epp), rules of inference to deduce the conclusions themselves. For every step of the derivation, document which rule or axiom of equivalence you are using. We assume some well-defined domain D. Note that the last rule contains 3 (three) premises, the third of which is an existentially quantified disjunction.

1. The universal quantifier is Modus ponnens

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Example : All men are mortal

Socrates is a man.

$\therefore$ Socrates is mortal

Here let X implies universal statement

R(x)= All men P(x) =All mortal

Let a be the domain of X where a is considered as socrates

here q(a) is scocrates is men and P(a)= socrates being mortal.

2.The universal quantifier is modus tollens

Example : All good drivers are alert

people who drunk are not alert

$\therefore$ all people who drunk are not good drivers

From the above example R(x) =people who are good drivers and P(x)= all are alert

for a particular a belongs to X where q(a) is a set of people who are drunk

$\negation$~ P(a) $\negation\wedge$ q(a) =all people drunk are not alert

therefore ~R(a) implies that people who are drunk are not good drivers

3.Example: If it rains,you carry umbrella

If is it cloudy,it rains

neither Anil is carrying umbrella nor it is raining

$\therefore$ it is not cloudy

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