 Now before going to the rules of inference we will take a look at the connective that we have discussed in the previous lecture namely implication the connective implication is denoted by an arrow like this or an arrow like this we will be using the former symbol to denote implications suppose p and q are two propositions then the statement p implies q is denoted by right arrow q what we have seen is that this proposition namely p implies q is false only when p is true but q is false the truth table of p implies q is written as now we will introduce some terminologies here the left hand side proposition in the compound statement p implies q namely p is called premise or hypothesis or antecedent the right hand side expression or proposition q is called conclusion or consequent so we can reward what we have said just now that is the statement p implies q is false only when the antecedent is true but the consequent is false we now move on to another type of implication which is called a logical implication a proposition p logically implies a proposition q that is q is a logical consequence of p implication p implies q is true for all possible assignments of truth values of p and q in other words p implies q is a tautology here let us recall that a tautology is a proposition which is always true irrespective of the values of the propositional variables involved in it now we define the word inference the word inference is used designate a set of premises accompanied by suggested conclusions regardless of whether conclusion is a logical consequence of the premise and inference is valid if the implication is a tautology otherwise it is faulty let us have a relook at what I have stated suppose we are having some propositions which we call premises and denote by p 1 p 2 p n and what we can do is that we can combine them all with a conjunction to get a single premise p 1 and p 2 and so on and p n so this is the combined premise that we have and we write another proposition what we call is a conclusion or a consequent and we join them by a implication symbol now this total thing will be called an inference we can write this also in a in another form which is writing p i is one below the other in a list and then write a horizontal line and then write therefore q both these notations will mean the same thing now we will call an inference valid if the statement given here that is the implication is a tautology otherwise we say that it is not valid as a rule of inference we will have a list of valid inferences and we will also have some very common faulty inferences or fallacies we will move on to right now so let us go to the next page here we have rule of rules of inference related to the language of proposition the first one is called addition here we see that it is essentially implies p union q now p q are propositions p or q which is true if p is true therefore if p is true then p or q is always true and thus the implication p implies p or q is always true and therefore this implication is a tautology and hence a valid inference we can quickly draw a truth table and check this fact the last column is corresponding to the proposition p implies p or q and we see that it is always true second p and q therefore p in tautological form this is p and q implies p this is called simplification now we can again draw a truth table to check that it is indeed a tautology for that we take all the possible values of p and q which is f f t t f and t t and p and q and finally p and q implies p p and q is of course f f f and t and here we see that if the antecedent is false then the implication is always true and at the end antecedent is true and consequent is also true therefore the implication is true and therefore we see that p and q implies p is a tautology third one the third rule is p and p implies q therefore q in the tautological form this is p and p implies q implies q this is called modus ponens let us check that it is indeed a tautology by using a truth table we have p q again as before and we have p implies q and lastly we have p and p implies q and then p and p implies q implies q we list down the truth values of pq then p implies q is true true false and true p and p implies q is false false false and true and here p and p implies q and q it is implies q is whenever this is false this is true so we have to only check the case when where p and p implies q is true and then q is true this is true therefore the truth value is true and thus we see that modus ponens is indeed a valid inference for negation of q p implies q therefore negation of p this is written in the tautological form as negation of q and p implies q implies negation of p this is called modus ponens fifth rule is p or q negation of p therefore q in the tautological form this is p or q negation of p implies q this is called disjunctive syllogism I am not showing that these rules are indeed tautologies like I have shown for the first three cases but using the exactly the same process it can be verified that these are tautologies there are few more common rules of inferences left which I will list right now the sixth one is p implying q q implying r therefore p implies r in the tautological form this will look like p implying q and q implying r implies p implies r this is called hypothetical syllogism then seventh is p q therefore p and q this is simply conjunction 8 p implies q and r implies s p or r therefore q or s this is written as p implies q and r implies s and p or r implies q or s this is called constructive dilemma 9 p implies q and r implies s negation of q or negation of s therefore negation of p or in the tautological form this is this is called destructive dilemma and finally there are some equivalence equivalences which are very important one is the de Morgan's law which states that given any two propositions p and q p or q negation is equivalent to negation of p and negation of q p and q negation of the whole is equivalent to negation of p or negation of q these together are called de Morgan's laws and we have another equivalence which is called the law of contrapositive which states that p implies q is equivalent to negation of q implies negation of p once we have seen a long list of valid inferences we will end today's lecture by looking at some faulty inferences so we move on to some common fallacies the first fallacy is the fallacy of affirming the consequent which is written as p implying q q therefore p let us look at the tautological form and try to build the truth table this means that p implies q and q this whole implies p now the corresponding truth table will be now we come to the point where we have problem because in the first line p implies q and q is false and p is false therefore the implication is true in the second line p implies q and q is true whereas p is false therefore the implication is false and we stop here because of this we see that the inference that we stated above is not valid an example of this fallacy is like this suppose p is the proposition the price of gold is rising and q is the proposition inflation is surely coming then p implying q is if the price of gold is rising then inflation is surely coming the second statement is inflation is surely coming now from these two statements we are tempted to write that therefore the price of gold is rising but what we have learned from the fallacy of affirming the consequent is that this inference is not logically valid this is not a valid inference the next fallacy that we will study is called fallacy of denying the antecedent this fallacy is written in this way p implies q negation of p therefore negation of q now we have already stated that p imply q is equivalent to negation of q implies negation of p therefore we can replace this inference by negation of q implies negation of p negation of p therefore negation of q and we see that this is precisely the previous fallacy that is fallacy of affirming the consequent the third and last fallacy that we will look at in this lecture is called non sequitur fallacy this fallacy is essentially taking two propositions which are not connected and claiming that one implies the other for example suppose I say that Socrates is a man and conclude that therefore Socrates is mortal is a fallacy because Socrates is a man and Socrates is a mortal they are essentially two different propositions here I am not using a proposition that a man is mortal or anything that is something outside the system for example although this looks reasonable I could have taken two different propositions such as a triangle has three sides therefore a triangle is a square so obviously this look this looks meaningless but they are both same because they are essentially trying to connect two different propositions so non sequitur fallacy is of this type p therefore q where pq are two different propositions with this we come to the end of this lecture thank you