w5

L1        (L2)

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EXAM: one week

This week: Ch 4 symbolizations, review

 

 

 

 

 

 

I. Concepts and Tables

  1. The Idea: Each deductive notion from chapter 1 (like "is valid") can be more precisely phrased when we say what is possible in terms of table rows.
  2. Easy example:





























  3. Homework Questions?



  4. Informal Proofs!






  5. You do:

    Give a proof that answers this question:
    Can a sentence of the form P&Q be logically true while either of P and Q are not logically true?

next....

 

 

 

 

 

 

 

 

 

 

 

II. Symbolizations

A. Expressive Completeness

  1. We have a bunch of connectives. Why not more?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

next...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2. We need to remember all those things on our list:

Don't forget our big list.

 

next...

 

 

 

 

 

 

 

 

 

 

 

 

3. Harder Symbolizations: remember our ideas:

    • commas mark the main connective (sometimes)
    • grouping rules

     

 

 

 

 

 

 

 

 

4. Examples using

A,B,C for Austria wins a gold, Britain wins a gold, Canada wins a gold

1. Britain wins a gold only if both Canada and Austria do.
2. If Britain wins a gold, then neither of the other two wins.
3. Provided Austria and Britian don't both win, at least one of Britain and Canada does win.
4. Either Britain's winning is a necessary condition for Canada to win or Britain's winning is a sufficient condition for Canada to win.

   fill in...

 

Homework

next (let's get started on some stuff from 4.3).

 

 

 

 

 

 

 

 

 

 

L2

roll

exam: next time

 

 

 

 

 

I. Symbolizations

A. Rember to break down into easier pieces. Example:

 

 

 

 

 

 

B. Counting

A,B,C for Austria wins a gold, Britain wins a gold, Canada wins a gold

 

1. At least one team wins a gold.
2. At least two teams win a gold.
3. Exactly one team wins a gold.
4. At most two teams win a gold.

    fill in...

5. Brute force method:

 

 

 

 

 

 

 

A,B,C for Austria wins a gold, Britain wins a gold, Canada wins a gold

 

6. You do:

  • At least one team wins.
  • At most one team wins.
  • Exactly two teams win.

next...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

II. Controversies

  1. "Or" of English vs. 'v' of SL ... I'll not add anything more. I've had my time and there is 4.3. Further thoughts?



  2. Is there an English Material conditional?
    • What does that mean?








    • So, back to our question...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   3. But think about one last example:

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

one last thing...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

III. Mock Exam

A. Which of the following are sentences? Which have main connective '>'? Which have form P>~Q?

  1. A~>B
  2. A>~B
  3. ~A>~B
  4. (L>J)>(P~R)
  5. (L>J)>~(P>~R)

 

B. Do some table tests.

  1. Is 'S>(K&S)' logically true?
  2. Is 'J=(J&K), ~K therefore ~J' valid?
  3. Is '~S>T' logically equivalent to 'S>~T'?

 

C. Symbolize

A,B,C for Austria wins a gold, Britain wins a gold, Canada wins a gold

  1. If Austria wins, then neither of the other two win.
  2. Unless Austria wins, both Britain and Canada win.
  3. Austria wins only if either Britain or Canada wins.
  4. Canada's winning is a sufficient condition for Britain to win.
  5. Canada wins if not both of Austria and Canada win.

 

 

 

D. Prove that if P&Q is logically true, then P is also logically true.