Download Rules of Inference in Propositional Logic: A Comprehensive Guide and more Assignments Mathematics in PDF only on Docsity!
Rules of Inference Introduction: 1.1:Proofs: Proofs in mathematics are valid arguments. 1.2:Argument: By an argument, we mean a sequence of statements that end with a conclusion. 1.3:Valid Argument: 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. That is, an argument is valid if and only if it is impossible for all the premises to be true and the conclusion to be false. 1.4:Rules of Inference: To deduce new statements from statements we already have, we use rules of inference which are templates for constructing valid arguments. Rules of inference are our basic tools for establishing the truth of statements. 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.
However, this can be a tedious approach. For example, when an argument form involves 10 different propositional variables, to use a truth table to show this argument form is valid requires 2 10 =1024 different rows. Fortunately, we do not have to resort to truth tables. Instead, we can first establish the validity of some relatively simple argument forms, called rules of inference. These rules of inference can be used as building blocks to construct more complicated valid argument forms. We will now introduce the most important rules of inference in propositional logic.
2.1: Modus Ponens:
The tautology (p∧ (p → q)) → q is the basis of the rule of inference called Modus Ponens, or the Law of Detachment. (Latin for "mode that affirms by affirming”). The symbol ∴ denotes “therefore”. Modus Ponens tells us that if a conditional statement and the hypothesis of this conditional statement are both true , then the conclusion must also be true. Example 1: Suppose that the conditional statement “If it snows today, then we will go skiing” and its hypothesis, “It is snowing today,” are true. Then, by modus ponens, it follows that the conclusion of the conditional statement, “We will go skiing,” is true. p p → q ……………. ∴ q 2.2: Modus Tollens:
q: I cannot go to work r: I will not get paid using Hypothetical Syllogism, we have, p → q q → r
∴ p → r 2.4: Disjunction Syllogism: A disjunctive syllogism means ''mode that affirms by denying'', is a syllogism in which the two premises are mutually exclusive and they cannot be both true nor can they be both false. For example: First: premise in which we include a disjunctive conjunction: My friend's name is Ali or Raza. Second: premise denied: My friend's name is not Ali. Conclusion: My friend's name is Raza. Example: “ I will choose soup or I will choose salad. I will not choose soup. Therefore, I will choose salad.” Solution: p : I will choose soup.
q: I will choose salad. Using Disjunction syllogism, we have, p q p q OR p q q p Exercise State which rule of inference is the basis of the following argument: 1: “If you have a current password, then you can log onto the network.” Solution: p: you have a current password q: you can log onto the network p p → q ……………. ∴ q
tomorrow. Therefore, if it rains today, then we will not have a barbecue tomorrow. Solution: P: it rains today Q: we will not have a barbecue today R: we will not have a barbecue tomorrow p → q q → r
∴ p → r 5: If it snows today, the university will close. The university is not closed today. Therefore, it did not snow today. Solution: P: it snows today Q: the university will close
q
p → q ……………. ∴
p
6: Steve will work at a computer company this summer. Therefore, this summer Steve will work at a computer company or he will be a beach bum. Solution: P: Steve will work at a computer company this summer Q: he will be a beach bum Using Rules of Inference to Build Arguments Example 6: Show that the hypotheses “It is not sunny this afternoon and it is colder than yesterday,” “We will go swimming only if it is sunny this afternoon,” “If we do not go swimming, then we will take a canoe trip,” and “If we take a canoe trip, then we will be at home by sunset” lead to the conclusion “We will be home by sunset.” Solution: Let p: “It is sunny this afternoon.” q: “It is colder than yesterday.” r: “We will go swimming.”
Home work: Example 7: Show that the hypotheses “If you send me an email message, then I will finish writing the program,” “If you do not send me an email message, then I will go to sleep early,” and “If I go to sleep early, then I will wake up feeling refreshed” lead to the conclusion “If I do not finish writing the program, then I will wake up feeling refreshed.”
