W3

roll
exam! See Mock Exam (syllabus ... click on this week!)

I. Some old issues:

A. Tables: Memorize! Remember:

 

 

 

 

B. Unless:

  • I said: use 'v' for unless. ("We win unless they score": WvS and anything logically equivalent. Like ~S>W.)
  • More ideas? A better example?
  • Me: Another Example

 

 

 

 

 

C.The following lists a number of equivalences. We've done most already. For now, look at triple-bar.

Equivalent English Forms (Each table element -- i.e., box -- below gives English forms instances of each of which can be symbolized by a sentence of any SL form on its right. Please note that there are many more English forms than can be covered below.) Equivalent SL Forms (Each table element below gives SL sentence-forms to guide in translating English sentences of forms found on the left. Please note that this is an incomplete list of possible symbolizations.) Example Applications (Each of the table elements below shows a way to apply the table elements on their left.)

If P, then Q.
If P, Q.
Provided P, Q.
Were P to hold, Q would be true.
Should P be true, Q.
P only if Q.
P is a sufficient condition for Q.
P implies Q.

P>Q
~Q>~P
If there is fire, then there is Oxygen" or "There is fire only if there is oxygen" may both be symbolized as 'F>O' or equivalently as '~O>~F'.
P if Q.
P provided Q.
P is a necessary condition for Q.
Q>P
~P>~Q
"Water is a necessary condition for life" or "there's water if there's life" may both be symbolized as 'L>W' or equivalently as ~W>~L'.
P if and only if Q.
P just in case Q.
P is necessary and sufficient for Q.
P=Q
(P>Q)&(Q>P)
"An argument is sound if and only if it is both valid and has true premises" may be symbolized as either 'S=(V&T)' or '[S>(V&T)] &[(V&T)>S]

Both P and Q.
P and Q.
P but Q.
Q and P.
Q but P.
P however Q.
P although Q.
P moreover Q.

P&Q
Q&P
"Sandra is both brave and careful", "Sandra is brave, moreover she is careful' or "Sandra is brave but careful" can all be symbolized as 'B&C' or 'C&B'.
Either P or Q.
Either Q or P.
P or Q.
Q or P.
At least one of P, Q.

PvQ
QvP
"Either the other team will score and tie up the game, or we win!" can be symbolized as '(S&T)vW'.
P unless Q.
Q unless P.
Unless P, Q.
Unless Q, P.
PvQ
~Q>P
~P>Q
"We win unless the other team scores" can be symbolized as 'WvS'.
Neither P nor Q.
Not-P and not-Q.
~(PvQ)
~P&~Q
"They neither scored not tied the game" may be symbolized as either '~(WvT)' or '~W&~T'.
It's not the case that both P and Q.
Not both P and Q.
Either not-P or not-Q.
~(P&Q)
~Pv~Q
"Sandra is not both brave and careful" may be symbolized as either '~(B&C)' or '~Bv~C'.

 

Harder Symbolizations (2.4)

 

 

You do (Use B,C,L):

  1. Brazil wins a gold unless Canada does too.
  2. Brazil wins a gold provided Canada wins a gold.
  3. Neither Canada nor Luxemburg wins a gold.
  4. Provided Brazil doesn't win a gold, then both Canada and Luxemburg win.
  5. Brazil wins a gold only if neither of the other two win a gold.
  6. Brazil's winning a gold is a sufficient condition for both Canada and Luxembrug not winning.

 

D Informal Proofs

1. Suppose P is logically true, then ~P is logically false. 1,2,3,4,5

 

 

 

 

 

 

(You do...)

2. If P is logically true, then so is PvQ.

 

 

 

(We do...)

3. Show that PvQ can be logically true even though both it's components are logically indeterminate.

 

Hint:

 

 

 

 

 

 

 

 

II. Homework questions?

A. Syntax?

B. Symbolization? More help and problems in Chapter 4. You're ready for that now!

C. Arguments and indicators?

D. Tables and Truth Functionality

 

 

 

III. Tables and section 3.1

A. All our SL connectives are truth functional. That just means that the truth value of a compound sentence depends on the truth value of the components. So, ...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OK, now let's look at some of the 3.1 problems...and the special tool to help you think about them.

 

 

 

 

 

IV. 3.2's Full Tables

A. What if we don't know t-values? E.g., 'A&~B' about Art and Beth's being students? Then...

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

B. This is just doing 3.1, but for each way that the atomic sentences can be true or false.

 

C. You try: ~A>(B&A)

help?

 

 

D. Now, what about 'A&(B&C)'??? How do we get all the possiblities? Think about what could be true about Barb and Carol, IF Art is a student...and then IF he is NOT a student. Let's write this up in a table. Or just look at the finished product.

 

 

 

 

 

 

 

 

Now, just do the table... or just look.

 

 

 

 

 

 

 

Next Question: How many rows if we have 4 atomic sentences?

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Now, we need to do some more. We do.

And you do...