Modus TollensModus Tollens is a rule of inference pertaining to the connective.Assume we have:P Q: "If it is raining, then there are clouds in the sky." P (it is raining) is called the antecedent.Q (there are clouds in the sky) is called the consequent.

Modus TollensModus tollens sates that if the consequent is false, then the antecedent is false.Now if we know for a fact that there are no clouds in the sky, we can safely conclude that it is not raining. If the consequent ("There are clouds in the sky") is false, then the antecedent ("It is raining") must also be false, by Modus Tollens.

Modus TollensLet's write our steps formally:P Q: "If it is raining, then there are clouds in the sky." ~Q: "There are no clouds in the sky." ---------- ~P: "It is not raining." The conditional P Q and the given ~Q are above the line of dashes, and the conclusion ~P obtained by applying Modus Tollens is below the line.P Q:~Q:---------- ~P:

DeMorgan's Law DeMorgan's law is a rule of inference pertaining to the NOT, AND, and OR connectives.DeMorgan's law is used to distribute a negative to a conjunction or disjunction.Let us consider the following statement: "It is not true that he took both Database and Networking." Formally, we would write:~(C ^ P): "It is not true that he took both Database and Networking.In this expression, C refers to the phrase "He took Database" and P refers to the phrase "He took Networking."

DeMorgan's LawDeMorgan's law states that this expression can be converted into another expression, completely equivalent to the original:~C v ~P: "He did not take Database or he did not take Networking."To understand why, let's first see what the original statement:"It is not true that he took both Database and Networking" means. It can mean three things:1. He took Database but not Networking. (C is true and P is false, or ~P is true).2. He took Networking but not Database. (P is true and C is false, or ~C is true).3. He did not take either Database or Networking. (C is false and P is false, or ~P and ~C are true).

DeMorgan's LawIf we look closely at these three conclusions, we see that in all of them either ~P is true, or ~C is true, or both ~P and ~C are true. This is an example of a disjunction. Formally, we would write the following, together with the original statement:~(C ^ P): "It is not true that he took both Database and Networking." ---------- ~C v ~P: "He did not take Database or he did not take Networking."That is exactly what DeMorgan's law means. The given expression ~(C^P) is above the line of dashes, and the new expression ~Cv~P formed by applying DeMorgan's law is below the line.

DeMorgan's LawIt can also work where a disjunction is converted into a conjunction with the negation of each member of the expression.~(P v Q): "It is not true that the book is boring or the newspaper is interesting." ---------- ~P ^ ~Q: "The book is not boring and the newspaper is not interesting."

Chain RuleThe Chain Rule is a rule of inference pertaining to the connective.The Chain Rule is used to combine two conditionals of the form P Q and Q R into P R.Imagine we are given two conditionals, P Q and Q R:P Q: "If Jane leaves home late, she will miss her train.

Chain RuleNow let's consider the second conditional, Q R. Note that it contains the same letter that we used in the first conditional, namely Q. This means that Q has to remain the same as in the first conditional.

Chain RuleQ R: "If Jane misses her train, she will be late for work."Given the two conditionals, It is perfectly natural for us to say: "If Jane leaves home late, she will be late for work."

Chain RuleFormally, we would write:P Q: "If Jane leaves home late, she will miss her train." Q R: "If Jane misses her train, she will be late for work." ---------- P R: "If Jane leaves home late, she will be late for work."The given conditionals are above the line of dashes, and the new expression P R formed by applying the Chain Rule is below the line.P Q:Q R:---------- P R:

Modus PonensModus Ponens is a rule of inference pertaining to the connectives.Modus Ponens states that if the antecedent of a conditional is true, then the consequent must also be true.Imagine we have the following conditional sentence: "If it is raining, then there are clouds in the sky." Formally, we would write:P Q: "If it is raining, then there are clouds in the sky."

Modus PonensIn this expression, "If it is raining" is the antecedent and "There are clouds in the sky" is the consequent.Now if we know for a fact that it is raining, then we have to conclude that there are clouds in the sky. If the antecedent ("It is raining") is true, then the consequent ("There are clouds in the sky") must also be true, by Modus Ponens.

Modus PonensLet's write our steps formally:P Q: "If it is raining, then there are clouds in the sky." P: "It is raining." ---------- Q: "There are clouds in the sky." The conditional P Q and the given p are above the line of dashes, and the conclusion Q obtained by applying Modus Ponens is below the line.P Q:P:---------- Q: