# Rules of Inference Involving Universal Quantifier

## Lecture description

The rules of inference universal instantiation, universal modus ponens, universal modus tollens, and universal generalization are introduced.

Students will learn how rules of inference involving the universal quantifier work.

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Although we know how quantified statements work let's explore how quantified statements can be used in arguments we've already seen how rules of inference work in statement calculus. In this lecture We'll look at some rules of inference that involve quantifiers. For example if I say all males are tall I can infer that John is tall assuming that John is male. We can represent this inference rule formally as for all x p of X Therefore P of G. In our example the domain of x is the set of all males. The predicate P of x is x is tall and J stands for John. In general the domain of x can be any set of objects. The predicate P of x can be any predicate and a constant symbol J can stand for any object in the domain of x. In general our rules of inference will look like this. For every x of X therefore PLB this rule of inference is called universal instantiation. OK now suppose I say every student is a coffee drinker. If we let piggybacks be X is a student and q of the x b x is a coffee drinker. Then we can express my statement here using the universal quantifier Bre-X paradox entails cubics OK. Now suppose Alex is a student then Evey is true. Where a. A stands for. Alex. Now since PLX aero Q of x is a predicate. Since that's a predicate we can apply a universal instantiation to this OK if we do that we get a day entails. Q The Ok by modus ponens we get. Q Of the in other words Alex is a coffee drinker. Thus we have the following rule of inference for every x picaxe entails Q of X Piya they are there for q of the K.. This is called Universal modus ponens. Now suppose Kimberlee is not a coffee drinker so not q. K where case stands for Kimberly. By applying universal instantiation to this we get p of K and tails. Q Okay okay so we have P of K entails Q of K and not. Q K by modus tollens we have not. P of k. In other words Kymberly is not a student. Thus we have the following rule of inference for every x x in Teoh's Q of X not q of k there or not PMK K. This is called Universo modus tollens universal instantiation allows us to infer from a universal statement a statement about a particular object in the domain of x. We can infer a universal statement for all x x. If we can show p of C for an arbitrary object see in the domain of x for example let the domain of x be the set of even integers. And let x be most x squared is a multiple of four OK. Let's see B an arbitrary even integer OK then C is equal to 2 k for some integer k. Ok then C squared is 2 k squared which is for k squared and thus C C squared. Is a multiple of four k so. P C is true. K since C is an arbitrary even integer. We can infer for all x. P of x K. The rule of inference just illustrated it has the following form P C for some arbitrary C and therefore for every x x K. This is called universal generalization.