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Arguments with Quantified Statements

This guide explores the principle of Universal Instantiation in formal logic, which asserts that if a property is true for all elements in a domain, it is also true for any particular element. By examining examples like Socrates' mortality and the properties of even numbers, we illustrate how to apply Universal Modus Ponens and Modus Tollens to derive valid conclusions. We also discuss the importance of form in logical arguments, providing insight into errors such as the Converse and Inverse Errors, and the significance of valid argument forms in ensuring logical consistency.

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Arguments with Quantified Statements

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  1. Arguments with Quantified Statements M260 3.4

  2. Universal Instantiation • If some property is true for everything in a domain, • then it is true of any particular thing in the domain.

  3. Example • All human beings are mortal. • Socrates is a human being. • Therefore Socrates is mortal.

  4. Simplify rk+1 r • rk+1 r = rk+1 r1 • = r(k+1)+1 • = rk+2 • Uses universal truths

  5. Universal Modus Ponens • For all x, if P(x) then Q(x) • P(a) for a particular a • Therefore Q(a)

  6. Example • If a number is even, then its square is even. • K is a particular number that is even • Therefore k2 is even.

  7. Formal version • hummm

  8. Prove that the sum of two even integers is even do it here

  9. Universal Modus Tollens • For all x, if P(x) then Q(x). • ~Q(a) for some particular a • Therefore ~P(a)

  10. Example • All human beings are mortal. • Zeus is not mortal. • Therefore Zeus is not a human being.

  11. Valid Argument Form • No matter what predicates are substituted for the predicate symbols in the premises, if the premise statements are all true then the conclusion is also true. • An argument is valid if its form is valid.

  12. Use Diagrams for Zeus and Felix • All human beings are mortal. • Zeus is not mortal. • Therefore Zeus is not a human being.

  13. Use Diagrams for Zeus and Felix • All human beings are mortal. • Felix is mortal. • Therefore Felix is a human being.

  14. Converse Error • For all x, if P(x) then Q(x). • Q(a) for a particular a. • Therefore P(a) • INVALID

  15. Inverse Error • For all x, if P(x) then Q(x). • ~P(a) for a particular a. • Therefore ~Q(a) • INVALID

  16. No • No polynomials have horizontal asymptotes • This function has a horizontal asymptote. • Therefore this function is not a polynomial

  17. Rewrite • For all x, if x is a polynomial, then x does not have a horizontal asymptote. • Use Universal Modus Tollens.

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