PHL 313K

1. PHL 313K Logic, Sets and Functions Fall 2002 Prof. Robert Koons

2. Text and Software • Logic, Sets and Functions by Daniel Bonevac, Nicholas Asher & Robert Koons (Kendall/ Hunt, 2001). • Plato (for Windows and Macintosh). Available with text and on the Plato web site: • http://www.utexas.edu/courses/plato/

3. Grading • The course grade will be based on the following components: homework (5 assignments), four tests, and an optional final exam. The final exam can be used to replace one's lowest test grade. These components will be weighted in the following manner: • Homework: 20% • Four Tests: 80% • All four tests are mandatory

4. Course Web Site • http://www.utexas.edu/courses/phl313k • Contains syllabus, homework assignments, sample problems, practice tests.

5. Instructor • Prof. Robert C. Koons • rkoons@phil.utexas.edu • Phone: 471-5530 • Office: Waggener 405 • Office hours: • Wed. 3-4 • Thurs. 1-2, and by appointment.

6. Syllabus • Introduction to Logic Ch. 1 & 2 • Sentential Deduction Ch. 3 • Sentential Deduction Ch. 3 • HW #1, due Sept. 10 • Quantifiers Ch. 4 • First Test: Sept. 19 (sentential logic)

7. Top 10 reasons for taking PHL 313K • 10. Riveting lectures. • 9. Endlessly fascinating homework assignments. • 8. You get to say "modus ponens" over and over again. • 7. The kind of comraderie you missed by not enlisting and going to boot camp.

8. 6. It's your last chance to turn that lump of gray goo you call a "brain" into something capable of rubbing two thoughts together. • 5. Millions of trees will be saved because of all the mindless crap you won't be able to write anymore. • 4. It's guaranteed to make you rich and famous, or your money back. [Offer void in states containing a, e, i, o or u.]

9. 3. You'll be a big hit at parties, pointing out all the fallacies in everyone else's reasoning. • 2. When you're bouncing your grandkids on your knee, you'll be able to tell them, "I studied logic under Robert Koons". • 1. You won’t spend the rest of your life in a cubicle writing code.

10. Nature of Logic • Logic: both very old and very new • Old: Aristotle, 2500 years ago. • New: most work completed in last 100 years. Knowledge explosion.

11. Three parts of logic • Sentential logic. Logical “chemistry”: the logic of building molecular sentences with connective bonds: and, or, if, not. • Predicate logic. Logical “atomic physics”: the logic of simple sentences and clauses including quantifiers: all, some, none. • Set theory. The universal language of mathematics.

12. Basic concepts and terminology • DEFINITION. An argument is a finite sequence of sentences, called premises, together with another sentence, called the conclusion, which the premises are purported to support.

13. Conclusion indicator words • therefore, thus, hence, • so, consequently, it follows that, • in conclusion, • as a result, • then, • it must be that

14. Premise indicator words • because • for • since

15. Defining deductive validity • DEFINITION. An argument is deductively valid if and only if it's impossible for its premises all to be true while its conclusion is false.

16. Examples • Valid: • All men are mortal. • Socrates is a man. • Therefore, Socrates is mortal. • Invalid: • Grass is green. • Therefore, snow is white.

17. Three possibilities for valid arguments • True premises, true conclusion. • All men are mortal, Socrates is a man, therefore Socrates is mortal. • False premises, false conclusion. • No Democrat hunts, Bush is a Democrat, therefore Bush doesn’t hunt. • False premises, true conclusion. • All Democrats hunt, Bush is a Democrat, therefore Bush hunts.

18. What is impossible: • A valid argument with true premises and a false conclusion. • This is excluded by the definition of validity.

19. Some imperfections in this definition. • According to our definition, the following arguments are “valid”: • 2+2=5. Therefore, snow is green. • Bush is a Democrat. Therefore, 2+2=4. • In the first case, it is mathematically impossible for the premise to be true. • In the second case, it is mathematically impossible for the conclusion to be false.

20. What’s wrong with our definition • The logical validity of an argument should depend only on the logical form of the argument, not on whether it is “possible” in some non-logical sense for the premises to be true or the conclusion false. • We could say that an argument is valid if it’s logically impossible for the premises to be true and the conclusion false at once.

21. Validity applies to logical forms • An argument is valid if it has a logical form that is valid. • A logical form is valid if no argument having that form has true premises and a false conclusion.

22. Different kinds of logical form • Ch. 2 & 3 -- sentential logical form. Focus on words used to create compound sentences from simple ones: “and”, “or”, “if ... then”, “not” • Ch. 4 & 5 -- quantificational form. “Some”, “all”, “none”. Subject/predicate structure of sentences. • Ch. 6 -- the logic of identity. “is”, “=“

23. Soundness • DEFINITION. An argument is sound if and only if (a) it is valid and (b) all its premises are true.

24. We’re interested in validity, not soundness • We're not concerned with truth, falsity of sentences in logic. Logic pertains to all subject matters, logicians can't be experts on everything. • Some valid arguments are useful though not sound.

25. Two uses of valid arguments • 1. Start with known premises, deduce new truths. • 2. Start with doubtful premises, uncover hidden implications. • A special case of the second: reductio ad absurdum.

26. Reductio ad absurdum • A deductively valid argument with an obviously false conclusion. • The reductio reveals to us that at least one of its premises is false. • Usually, there is one premise (the “hypothesis”) whose truth is doubtful. • The reduction “reduces” the hypothesis to absurdity.

27. A classic example (Plato) • 1. If you give back what you owe, you always act rightly. (Hypothesis) • 2. If you return a deadly weapon to a temporarily insane friend, you give back what you owe. • Therefore, if you return a deadly weapon to a temporarily insane friend, you always act rightly. (Obviously false)

28. What is “obviously false”? • As logicians, there is only one thing that we can count as “obviously false”. • This is a contradiction: a statement of the form ‘A and not A’. • For example: snow is white, and snow is not white. Today is Wednesday, and today is not Wednesday.

29. Definition of “implication” • DEFINITION. A set of sentences, S , implies a sentence, A , if and only if it's impossible for every member of S to be true while A is false. • DEFINITION. A sentence, A , implies a sentence, B , if and only if it's impossible for A to be true while B is false.

30. Connection: validity & implication • If the argument “S1...Sn ˇ A” is valid, then {S1… Sn} implies A. • And vice versa: if {S1… Sn} implies A, then the argument “S1...Sn ˇ A” is valid. • The symbol ˇ separates the premises from the conclusion of an argument.

31. Equivalence • DEFINITION. A sentence, A , is equivalent to a sentence, B , if and only if it's impossible for A and B to disagree in truth value. • Equivalent: Nothing cheap is good = nothing good is cheap • Not equivalent: Everything expensive is good ± everything good is expensive.

32. Demonstrating lack of equivalence • Imagine 2 stores: • (1) everything is good, some cheap, some expensive. • (2) everything is expensive, some good, some bad. • In (1), everything is expensive is good, but not everything good is expensive. • In (2), conversely.

33. Logical properties of sentences • DEFINITION. A sentence is contingent if and only if it's possible for it to be true and possible for it to be false. • DEFINITION. A sentence is logically true if and only if it's impossible for it to be false.

34. DEFINITION. A sentence is contradictory if and only if it's impossible for it to be true. • DEFINITION. A sentence is satisfiable if and only if it's not contradictory.

35. Properties of sets • DEFINITION. A set of sentences is contradictory if and only if it's impossible for all of its members to be true. A set is satisfiable otherwise.

36. Some examples • Valid: • A rose is a rose. • You are what you are. • A man's gotta do what a man's gotta do. • Contradictory: • Yogi Berra: nobody goes there anymore: it's always too crowded. • Hegel: The moving body is both here and not here at the same time.

37. Inclusion relationships

38. Two very old logical principles • No satisfiable set of sentences can imply a contradiction. • Only a contradictory set of sentences can imply a contradiction.