 of inference for quantified propositions now let us discuss briefly whatever we have done in the previous lectures as a recall so we have we know what a proposition is and we will be denoting propositions by small letters PQR and so on so these are propositions which are essentially statements having a truth value now we also know what are predicates or open propositions predicates will be denoted by Px Qx Rx and so on where x varies over the universe of discourse or simply universe we have seen some rules of inferences so rules of inference involving propositions we list some rules and refer to them as fundamental rules so fundamental rule 1 we have seen this under the name of modus ponens which states that P and P imply Q therefore Q the tautological form of this rule is P and P implies Q implies Q and we have already checked that this rule is a indeed a tautology and therefore is a valid inference the next rule is fundamental rule 2 which is what we have studied in the name of hypothetical syllogism this rule states that P implies Q Q implies R therefore P implies R or in the tautological form P implies Q and Q implies R P implies R as before we can check that this is indeed a valid inference now the fundamental rule 3 this is known as de Morgan's laws de Morgan's laws state that not of P and Q is equivalent to not of P and not of Q not of P or Q is equivalent to not of P and not of Q finally the last law that we state here involving the propositions is fundamental law 4 fundamental rule 4 which is the law of counter positive the law of counter positive states that P implies Q is equivalent to not Q implies not P therefore in order to prove P implies Q we may as well prove not Q implies not P what is seen at this point is that if we move on to the first order logic from the propositional logic then we will have predicates and quantifiers in particular existence existential and universal quantifiers in this framework of the first order logic the rules of inference that we have derived for propositional logic are not sufficient therefore we need additional rules in what follows I will state four additional rules which are used in propositional logic along with the rules of the four additional rules which are used in predicate logic or the first order logic along with the rules that we have derived from the propositional logic so we move on to additional rules of inference required to prove assertions in involving predicates and quantifiers fundamental rule 5 this rule is called universal specification universal specification states that if a statement for all x Px assumed to be true then the universal quantifier can be dropped to obtain PC is true for an arbitrary object C in the universe apparently this rule is simple it just says that if we have a statement for all x Px where Px is a predicate and if we assume that it is true then we will be able to given any arbitrary object C in the universe the proposition PC that we get by replacing x by C in the predicate Px is going to be true in symbols we can represent this rule as for all x Px therefore PC for all C next we move to the fundamental rule 6 this is called universal generalization universal generalization states that if a statement PC is true for each element C in the universe then the universal quantifier can be fixed and we obtain the proposition for all x Px again in symbols we have PC for all C therefore for all x Px we move on to two more rules involving predicates and quantifiers fundamental rule 7 this is called existential specification existential specification states that if there exists x Px is assumed to be true then there is an element C in the universe such that PC is true by using symbols like before we can present this rule as there exists x Px therefore PC for some C lastly we look at fundamental rule 8 this rule is called existential generalization PC is true for some element C in the universe then there exists x Px and again in symbols we can write PC for some C therefore there exists x Px these are the rules that we will be using to prove propositions involving predicates and quantifiers let us look at some examples represent the following arguments symbolically decide whether they are valid the arguments are all men are fallible men are fallible kings are men therefore all kings are fallible now this is one argument and another argument that we will discuss is lions are dangerous animals there are lions therefore there are dangerous animals this is another argument now let us look at the first argument let mx is the predicate x is a man and kx is the predicate x is a king fx is a predicate x is fallible now we symbolize the argument the first argument in this way for all x mx implies fx for all x kx implies mx therefore for all x kx implies fx now we see the formal proof of this argument step wise so we write at each step an assertion and the reason one for all x mx fx so this is premise one that is this one mc implies fc now this is what we get by fundamental rule five and step one step one rule five three for all x kx implies mx now this is premise two for kc implies mc this is step three and rule five five now here we see that we can use disjunctive syllogism I am sorry we can use the rule two that is hypothetical syllogism on step two and step four so by using that we have kc implies fc this is step two and four along with rule two which is hypothetical syllogism and then we use rule six to obtain for all x kx implies fx step five rule six now we move on to the next argument involving lions and dangerous animals here in symbols we write let lx is a predicate x is a lion and dx is a predicate that x is a dangerous animal now the statements that we have namely lions are dangerous animal when we write by using predicates and universal quantifier we can say this that for all x lx implies dx and we can use existential quantifier saying there exists lx that is there exist animals for which lx is true and in the consequence therefore there exist x dx now let us see whether this is a valid argument or not again we go on to formal proof assertion reasons step one there exists x lx this is premise two and we use rule seven to write la step one rule seven step three we now use premise one that is for all x lx implies dx premise one and four I use rule five to write la implies da step three and rule five five da step step two and four rule one rule one is modus ponens so I am combining step two and step four by using modus ponens and write this and then in six there exist dx x dx this is step five and rule eight thus we have proved the validity of the argument that we have stated just now this is the end of today's lecture thank you