 today's topic is theory of inference for statement calculus the learning outcome will comprehend the theory of inference after at the end of the session the contents are we'll see what are the different terminologies then we'll introduce introduction validity using truth table and the example what are the terminologies? proofs proofs in mathematics are valid arguments and establish the truth of mathematical statements an argument is a sequence of statements and end with a conclusion the argument is valid if the conclusion or the final statement follows from the given preceding statements or given premises or given proofs the main function of the logic is to provide rules of inference or principles of reasoning the theory associated with such rules is known as inference theory if it is concerned with the reference of conclusion from certain premises conclusion derived from the certain premises by using accepted rules of reasoning the process of derivation is called as a formal proof or deduction rules of inference are criteria for determining the validity of an argument validity using truth table we say that b logically follows from a or b is valid conclusion or the consequence of the premise a it can be said that if and only if conditional b is tautology then a tautologically implies b we say that a premise h1, h2, h3 and hm and a conclusion c follows logically and only if h1 and h2 and hm conditional c if suppose that is equal to tautology then we say that h1 and h2 and hm logically follows or c logically follows from the h1 to hm given a set of premises conclusion it is possible to determine whether conclusion logically follows from the given premises by constructing the truth tables as follows let p1 and p2 pn be the all atomic variables appearing in the premises that is h1 to hm and the conclusion c if all possible combinations of truth values are assigned to p1 to p3 or p1 to pn if the truth values are h1 to hm and c are interred in the table we look for the rows in which h1 to hm has got the truth value t if for every such row c has the value true then it holds h1 to hm logically i mean c logically follows from h1 to hm example determine whether the conclusion c follows logically from the given premises what are the h1 and h2? h1 is p conditional q this is p conditional q and h2 is p and conclusion is q let us construct truth table on the paper h1 is p implies q then h2 is p and conclusion is q let us construct truth table for these premises premises are h1 and h2 so p and q as usual t, t, f, f then t, f, t and f p implies q this is h1 that is t, f, t and t and this is p already known to this is h2 this is h1 so h1 and h2 we have to find h1 and h2 we have to find in a truth table h1 and h2 this is h1 and h2 t and t is nothing but t f and t is nothing but false t and f is false t and f is false now let last one is h1 and h2 conditional c c is nothing but q this is q means h1 and h2 conditional q t conditional t is tautology f conditional f is tautology f conditional t is also tautology and f is also tautology so what we see that this all has got true value hence we can say that q logically follows logically follows from h1 and h2 let us consider the next example demonstrate that r is a valid inference from the premises premises are the first premise second premise and third premise p conditional q is the first premise second is q conditional r and third is p now this can be this can be proved this can be proved pi without constructing truth table with a derivation in a derivation in a derivation these are the step numbers these are the premises we are using the numbers of premises and this is actually premise there are rule p and rule t is there rule p is nothing but introduction of any premise in the given derivation and rule t is nothing but after using the two premises if you are writing some conclusion that is called as a rule t so let us see let us use rule p and rule t over here so the first equation in the first step we will consider one other premise in the derivation p conditional q say rule p then consider second premise in the second step suppose q conditional r say that is again using rule p rule p now we have a rule i13 implication rule is there p implies q q implies r that implies p implies r using this i13 rule we can write using these two premises if I am able to write the conclusion pr so I will write conclusion intermediate conclusion pr so this is called as a rule t and for that I am using i13 for that this is step number 3 for that I am using premise 1 and 2 premise 1 and 2 for that conclusion is p conditional r now uhh my conclusion is r but I am getting pr so one more is left so this is used this is also used one more is left that is p third let us introduce the third premise in the derivation at step number 4 and step this is called as a premise number the order of the premise is 4 means 1, 2 and 4 this is again using rule p now using these two using these two I can write conclusion as r using rule t and for that I am using the equation if p, p implies q is there for that I can write conclusion as p implies uhh p implies pr r I can write I can write r as a conclusion so this is called as a rule i11 implication rule i11 so for that it is step number 5 and for that I am using pr and p means that 1, 2, 4 can be used so this is how I have proved that r is a valid conclusion valid conclusion for these three premises I hope uhh now this can be solved this can be solved with truth table but when the number of variables are going to increase at that time at that time the the size of the truth table will go on increasing and it will become a tedious job so that is why we are using this uhh particular derivation by which we can solve we can solve the same uhh or we can show r as valid inference r as valid inference for the given premises p conditional q q conditional r and p without constructing truth table this is without constructing truth table and the earlier example this example this example was with a truth table so there are two methods there are two methods by which you can solve we can solve uhh or we have two different solutions with truth table or without truth table let us go to ppt think and write here determine whether the concluency follows logically or not the answer is no q does not logically follows from the given premises these are the references I have used I hope you have understood thank you