Valid & invalid arguments
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The document discusses valid and invalid arguments in propositional logic. It defines arguments and their forms, and validity. It explains how to test an argument form for validity using a truth table. It then discusses several rules of inference for propositional logic including modus ponens, modus tollens, generalization, simplification, disjunctive syllogism, hypothetical syllogism, and provides examples of applying these rules. Finally, it discusses arguments with quantified statements and the rules of universal instantiation, universal modus ponens, and universal modus tollens.
Valid & invalid arguments
- 2. Valid & Invalid Arguments oArgument is a sequence of statements ending in a conclusion. oDetermination of validity of an argument depends only on the form of an argument, not on its content. “If you have a current password, then you can log onto the network.” p=“You have a current password” q=“You can log onto the network.” p → q p ∴ q where ∴ is the symbol that denotes “therefore.”
- 3. Valid & Invalid Arguments oAn argument is a sequence of statements, and an argument form is a sequence of statement forms(have proposition var.). o All statements in an argument and all statement forms in an argument form, except for the final one, are called premises (or assumptions or hypotheses). oThe final statement or statement form is called the conclusion. The symbol ∴, which is read “therefore,” is normally placed just before the conclusion.
- 4. Valid & Invalid Arguments oTo say that an argument form is valid means that no matter what particular statements are substituted for the statement variables in its premises, if the resulting premises are all true, then the conclusion is also true. Conclusion q is valid, when (p1 ∧ p2 ∧ · · · ∧ pn) → q is a tautology. oTo say that an argument is valid means that its form is valid.
- 5. Valid & Invalid Arguments oThe truth of its conclusion follows necessarily or by logical form alone from the truth of its premises. When an argument is valid and its premises are true, the truth of the conclusion is said to be inferred or deduced from the truth of the premises. If a conclusion “ain’t necessarily so,” then it isn’t a valid deduction.
- 6. Testing an Argument Form for Validity 1. Identify the premises and conclusion of the argument form. 2. Construct a truth table showing the truth values of all the premises and the conclusion. 3. A row of the truth table in which all the premises are true is called a critical row. If there is a critical row in which the conclusion is false, then it is possible for an argument of the given form to have true premises and a false conclusion, and so the argument form is invalid. If the conclusion in every critical row is true, then the argument form is valid.
- 7. Testing an Argument Form for Validity op →q ∨ ∼r oq → p ∧ r o∴ p →r Hence this form of argument is invalid
- 8. Testing an Argument Form for Validity “If 101 is divisible by 3, then 1012 is divisible by 9. 101 is divisible by 3. Consequently, 1012 is divisible by 9.” Although the argument is valid, its conclusion is incorrect, because one of the hypotheses is false (“101 is divisible by 3.”). If in the above argument we replace 101 with 102, we could correctly conclude that 1022 is divisible by 9.
- 9. Rules of Inference for Propositional Logic oAn argument form consisting of two premises and a conclusion is called a syllogism. o The first and second premises are called the major premise and minor premise, respectively. oThe most famous form of syllogism in logic is called modus ponens.
- 10. Modus Ponens oThe modus ponens argument form has the following form(“method of affirming”): If p then q. p ∴ q If the sum of the digits of 371,487 is divisible by 3, then 371,487 is divisible by 3. The sum of the digits of 371,487 is divisible by 3. ∴ 371,487 is divisible by 3. oif a conditional statement and the hypothesis of this conditional statement are both true, then the conclusion must also be true.
- 11. Modus Tollens o Modus tollens(“method of denying” (the conclusion is a denial)) has the following form: If p then q. ∼q ∴ ∼p (1) If Zeus is human, then Zeus is mortal; and (2) Zeus is not mortal. Must Zeus necessarily be nonhuman? Yes! If Zeus is human, then Zeus is mortal. Zeus is not mortal. ∴ Zeus is not human. Because, if Zeus were human, then by (1) he would be mortal. But by (2) he is not mortal. Hence, Zeus cannot be human.
- 13. Rules of Inference for Propositional Logic oWhen an argument form involves 10 different propositional variables, to use a truth table to show this argument form is valid requires 210 = 1024 different rows. oRules of inference. First establish the validity of some relatively simple argument forms. A rule of inference is a form of argument that is valid. Thus modus ponens and modus tollens are both rules of inference.
- 14. Rules of Inference for Propositional Logic oGeneralization The following argument forms are valid: a. p b. q ∴ p ∨ q ∴ p ∨ q if p is true, then, more generally, “p or q” is true for any other statement q. Anton is a junior. ∴ (more generally) Anton is a junior or Anton is a senior. oAt some places with the name Addition It is below freezing now. Therefore, it is below freezing or raining snow.
- 15. Rules of Inference for Propositional Logic oSpecialization The following argument forms are valid: a. p ∧ q b. p ∧ q ∴ p ∴ q These argument forms are used for specializing. Ana knows numerical analysis and Ana knows graph algorithms. ∴ (in particular) Ana knows graph algorithms. oAt some places with the name simplification It is below freezing and snowing. Therefore it is below freezing.
- 16. Rules of Inference for Propositional Logic oElimination The following argument forms are valid: a. p ∨ q b. p ∨ q ∼q ∼p ∴ p ∴ q oThese argument forms say that when you have only two possibilities and you can rule one out, the other must be the case. x − 3 =0 or x + 2 = 0. If you also know that x is not negative, then x ≠ −2, so x + 2 ≠ 0. By elimination, you can then conclude that ∴ x − 3 = 0. oAt some places with the name Disjunctive Syllogism
- 17. Rules of Inference for Propositional Logic oTransitivity o The following argument form is valid: p →q q →r ∴ p →r If 18,486 is divisible by 18, then 18,486 is divisible by 9. If 18,486 is divisible by 9, then the sum of the digits of 18,486 is divisible by 9. ∴ If 18,486 is divisible by 18, then the sum of the digits of 18,486 is divisible by 9. oAt some places with the name Hypothetical Syllogism
- 18. Rules of Inference for Propositional Logic-Example o“If it rains today, then we will not have a barbeque today. If we do not have a barbeque today, then we will have a barbeque tomorrow. Therefore, if it rains today, then we will have a barbeque tomorrow.” p: “It is raining today.” q: “We will not have a barbecue today.” r: “We will have a barbecue tomorrow.” So the argument is of the following form:
- 19. Rules of Inference for Propositional Logic oProof by Division into Cases:The following argument form is valid: p ∨ q p →r q →r ∴ r oIf you can show that in either case a certain conclusion follows, then this conclusion must also be true. x is positive or x is negative. If x is positive, then x2 > 0. If x is negative, then x2 > 0. ∴ x2 > 0.
- 20. Rules of Inference for Propositional Logic oConjunction p q ∴ p ∧ q oResolution p ∨ q ¬p ∨ r ∴ q ∨ r
- 21. Rules of Inference for Propositional Logic-Example o“It is not sunny this afternoon and it is colder than yesterday,” “We will go swimming only if it is sunny,” “If we do not go swimming, then we will take a canoe trip,” and “If we take a canoe trip, then we will be home by sunset” lead to the conclusion “We will be home by sunset.” p: “It is sunny this afternoon ” q: “It is colder than yesterday.” r: “We will go swimming .” s:” we will take a canoe trip” t: “We will be home by sunset” Step Reason 1. ¬pΛq Premise 2. ¬p Simplification using (1) 3. r → p Premise 4. ¬r Modus tollens using (2) and (3) 5. ¬r → s Premise 6. s 7. s→t Modus ponens using (4) and (5) Premise 8. t Modus ponens using (6) and (7)
- 22. Rules of Inference for Propositional Logic-Example
- 23. Arguments with Quantified Statements All men are mortal. Socrates is a man. ∴ Socrates is mortal. oUniversal Instantiation If some property is true of everything in a set, then it is true of any particular thing in the set. For all real numbers x, x1 = x. universal truth r is a particular real number. particular instance ∴ r 1 = r.
- 24. Universal Modus Ponens o Could be written as “All things that make P(x) true make Q(x) true,” in which case the conclusion would follow by universal instantiation alone.
- 25. Universal Modus Ponens If an integer is even, then its square is even. k is a particular integer that is even. ∴ k2 is even. oMajor premise can be written as ∀x, if x is an even integer then x2 is even. oLet E(x) be “x is an even integer,” let S(x) be “x2 is even,” and let k stand for a particular integer that is even. Then the argument has the following form: ∀x, if E(x) then S(x). E(k), for a particular k. ∴ S(k). oThis argument has the form of universal modus ponens and is therefore valid.
- 27. Universal Modus Tollens All human beings are mortal. Zeus is not mortal. ∴ Zeus is not human. Solution The major premise can be rewritten as ∀x, if x is human then x is mortal. Let H(x) be “x is human,” let M(x) be “x is mortal,” and let Z stand for Zeus. The argument becomes ∀x, if H(x) then M(x) ∼M(Z) ∴ ∼H(Z). This argument has the form of universal modus tollens and is therefore valid.
- 28. Universal Modus Tollens All human beings are mortal. Zeus is not mortal. ∴ Zeus is not human. Solution The major premise can be rewritten as ∀x, if x is human then x is mortal. Let H(x) be “x is human,” let M(x) be “x is mortal,” and let Z stand for Zeus. The argument becomes ∀x, if H(x) then M(x) ∼M(Z) ∴ ∼H(Z). This argument has the form of universal modus tollens and is therefore valid.