 1.5 Rules of Inference.

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1.5 Rules of Inference

Valid Arguments in Propositional Logic
By an argument, we mean a sequence of statements that end with a conclusion. By valid, we mean that the conclusion, or final statement of the argument, must follow from the truth of the preceding statements, or premises, of the argument. an argument is valid if and only if it is impossible for all the premises to be true and the conclusion to be false.OR If all the premises are true, then the conclusion must be true.

argument form. the validity of an argument follows from the validity of the form of the argument. when both Pq and p are true, then q must also be true. We say this form of argument is true (i.e the conclusion must also be true) whenever all its premises (all statements in the argument other than the final one, the conclusion) are true, If one of the premises is false, we cannot conclude that the conclusion is true. (Most likely, this conclusion is false.)

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Example: Consider the following argument involving propositions: “If we have 3+5=8 then 3+5=10”, “3+5=10”, Therefore, 3+5=8 ــــــــــــــــــــــــــــــــــــــــــــــــــ

DEFINITION 1 An argument in propositional logic is a sequence of propositions. All but the final proposition in the argument are called premises and the final proposition is called the conclusion. An argument is valid if the truth of all its premises implies that the conclusion is true. An argument form in propositional logic is a sequence of compound propositions involving propositional variables. An argument form is valid if no matter which particular propositions are substituted for the propositional variables in its premises, the conclusion is true if the premises are all true. the conclusion is true if the premises are all true

From the definition of a valid argument form we see that the argument form with premises
PI , P2 , , Pn and conclusion q is valid, when (PIᴧP ᴧPn )  q is a tautology. The key to showing that an argument in propositional logic is valid is to show that its argument form is valid.

Rules of Inference for Propositional Logic
We can always use a truth table to show that an argument form is valid. We do this by showing that whenever the premises are true, the conclusion must also be true. Truth Table Formal Informal

modus ponens, or the law of detachment.
This tautology is the basis of the rule of inference called modus ponens, or the law of detachment. This tautology leads to the following valid argument form,

”If you are a student then you will get an ”ID
Therefore, You will get an ID The general form of this argument is: If P then Q P Therefore Q If a student failed in mathematics, she either obtained a degree less than 60 or that the absence proportion exceeded 25%, The student does not have any of the previous cases Therefore, Student succeeded The general form of this argument is: If P then Q Not Q Therefore Not P Either team A or Team B will win the match  Team B lost Therefore Team A won The general form of this argument is: Either P or Q  Not P Therefore Q If John graduated ,then he will get a job If John gets a job, he will marry. Therefore, if graduated, , he will marry. The general form of this argument is: If P then Q If Q then R Therefore If P then R

Example: Determine whether the following argument is valid or invalid (Hint: use the truthe table)

Solution: 1/ We make all premises true: “pq”=T P= T, 2/ See in the table where is p=T & “Pq”=T, You see it in the raw 1 so this is the critical row, 3/ Now see what aboute the conclusion? The conclusion is true, Therefore, the argument is valid.

Example: Determine whether the following argument is valid or invalid (Hint: use informal method) Pq ⌐p ــــــــــــــــــــــــ :. ⌐ q

Solution: put all premises true: “Pq”= T , ⌐p=T, then p=F Now, put q= T, So, we still have “Pq” =T but q =T, Then ⌐ q=F, The conclusion is Fale whereas all primisis are true, Therefor, the argument is invalid.

Example: Determine whether the following argument is valid or invalid (Hint: use informal method) Pq q(pr) P _____________ :. r

Solution: put all premises true: “pq” = T ① “q(pr)”=T ② Put Q= (pr) then we have qQ=T P=T, ③ ـــــــــــــــــــــــــــــــــــــــــــــــــــــــــــــ Since p=T and “pq” = T, then q=T,④ Now, from ④ & ② q=T & “q(pr)”=T then Q= (pr)=T, ⑤ From ③ & ⑤ p=T & (pr)=T, then r must be true. Therefore r=T And the argument is valid

Example: Determine whether the following argument is valid or invalid (Hint: use informal method) P ⌐r r(pq) rp _____________ :. ⌐p⌐q

Solution: “P ⌐r”=T * P=T⌐r=T ** P=F⌐r=T *** P=F⌐r=F put all premises true: “P ⌐r”=T ① “r(pq)”=T ② “rp”=T ③ ـــــــــــــــــــــــــــــ :. ⌐p⌐q

“r(pq)”=T ② “rp”=T ③ ** ① By * *** “P ⌐r”=T ⌐r=T then r=F By ②
(pq) must be true,i.e (pq)=T But by * P=T (then ⌐p=F) Then q must be True , i.e q=T, then ⌐q=F Therefore: ⌐p⌐q=T “P ⌐r”=T * P=T⌐r=T ** *** “r(pq)”=T ② “rp”=T ③

“r(pq)”=T ② “rp”=T ③ ** P=F⌐r=T By ** ① ⌐r=T then r=F ***
But by ** P=F (then ⌐p=T) By ② (pq) either true or false,i.e (pq)=T or (pq)=F 1/ if (pq)=T then either q=T or q=F, # If q=T then ⌐q=F Therefore: ⌐p⌐q=F and the argument is invalid “P ⌐r”=T * ** P=F⌐r=T *** “r(pq)”=T ② “rp”=T ③

Homework: Determine whether the following argument is valid or invalid: