Day 28 in Mat115

Today:

• Announcements:
  • The Northern Kentucky University online course evaluation collection period is now open.

    You should have received an email with a link to fill out your confidential evaluations for each class in which you are enrolled for this session.

    Access the student login link at http://eval.nku.edu

  • Your trios are due: but let's look at some more examples before handing them in. You can tidy yours up, perhaps, as we go.

  • Next week you're going to be describing your projects in class. Your final project due date is 12/9: by then, it should be in an electronic form that I can put on the web.

    Make sure that you pay attention to the writing. These should be coherent, well-constructed, 51st great mathematical ideas. Make sure that you can be proud of them!

  • Last time we had a lottery to see who goes first next week. Here's how it came out:

    Tuesday    Thursday  
    adkins     acree     
    bertram    ball      
    brooks     brownfield
    burns      colbert   
    condit     frye      
    fulmer     gasser    
    hayes      gudgeon   
    helbig     hancock   
    reed       huelsman  
    riley      isler     
    simpson    lotscher  
    steelman   north     
    tindall    pugliano  
    xi         riggs     
               smith     
    

    I believe that it was adjusted as follows (correct me if I'm wrong!):

    Tuesday	Thursday
    adkins	acree
    brooks	brownfield
    burns	colbert
    condit	frye
    fulmer	gasser
    hayes	huelsman
    helbig	hancock
    ball-riley	isler
    simpson	bertram-gudgeon
    steelman-pugliano	north
    tindall	reed-lotscher
    xi	riggs
    	smith

• Today: Last Logic of Lewis Carroll, and other logical bits
  • Logic, as seen through the eyes and ideas of Lewis Carroll (as described in his book Lewis Carroll in Numberland), who thought that symbolic logic can be "an entertainment for children to develop their powers of logical thought."
    
  • Elements of The Game of Logic
    • An example:
      1. Babies are crazy;
      2. Crazy people should be in an asylum.

      What conclusion can we reach? If you reached a correct conclusion, then you've properly performed a logical calculation.

    • Fundamentals: tools to help us "calculate" logically:
      • Definition of the universe of discourse:
        the set of all things under discussion.

      • Definition of a statement (or proposition):
        A sentence with truth value (either true of false)

      • Definition of a syllogism:
        a group of statements (called premises) that lead to a conclusion

      • sorites (so-right-ease): A form of argument in which a series of incomplete syllogisms is so arranged that the predicate of each premise forms the subject of the next until the subject of the first is joined with the predicate of the last in the conclusion. An example is the "crazy babies" problem above.

      • The method: eliminating the common term

        So, in the baby example, the common term "crazy" is eliminated.

      • Three types of propositions:
        1. Some new cakes are nice
        2. No new cakes are nice
        3. All new cakes are nice

        Examples (from p. 175, Lewis Carroll in Numberland)

      • What is negation? x' = "not x". Let's try it in the case of each of the three types of propositions:
        1. (Some new cakes are nice)'
        2. (No new cakes are nice)'
        3. (All new cakes are nice)'

      • The board:
        • Bilateral
        • Trilateral
        • Counters:

          Let us agree that a Red Counter, placed within a Cell, shall mean "This Cell is occupied" (i.e. "There is at least one Thing in it").

          Let us also agree that a Red Counter, placed on the partition between two Cells, shall mean "The Compartment, made up of these two Cells, is occupied; but it is not known whereabouts, in it, its occupants are." Hence it may be understood to mean "At least one of these two Cells is occupied: possibly both are."

          Our ingenious American cousins have invented a phrase to describe the condition of a man who has not yet made up his mind which of two political parties he will join: such a man is said to be "sitting on the fence." This phrase exactly describes the condition of the Red Counter.

          Let us also agree that a Grey Counter, placed within a Cell, shall mean "This Cell is empty" (i.e. "There is nothing in it").

    • Examples: let's use the board
      • Crazy babies
      • Here's a classic:
        1. Aristotle is a man.
        2. All men are mortal.
        3. Aristotle is mortal.

    • One of Lewis Carroll's favorite examples

    • Let's look at another example or two from our "trios" sheet (which you should be using for homework)

    • A More Complicated Example:
      1. Babies are illogical
      2. Illogical persons are despised
      3. Nobody is despised who can manage a crocodile
      Solution:
      • Using Venn Diagrams (and their problems), and comparing
      • Carroll's method, and
      • A modification of the board

    • Some paradoxes, logical connundrums, and other strange examples
      • From a falsehood anything follows: Bertand Russell is the Pope.
      • Russell's paradox
      • Epimenides the Cretan (from the island of Crete!) said this: All Cretans are liars. Is this a paradox or not? That is, are either of
        1. Epimenides says "All Cretans are liars."
        2. Epimenides is a Cretan.
        3. Epimenides is a truth-teller.
        or
        1. Epimenides says "All Cretans are liars."
        2. Epimenides is a Cretan.
        3. Epimenides is a liar.

        logically consistent?

      • The dilemma of Protagoras and Eualthus

    • Further examples:
      • "No son of mine is dishonest; People always treat an honest man with respect."
      • "All cats understand French; Some chickens are cats".
      • Trios of Concrete Propositions (and a selection of Trios of Concrete Propositions for homework)