w5

L1 (L2)

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EXAM: one week

This week: Ch 4 symbolizations, review

I. Concepts and Tables

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.
Easy example:

Either Albert or Carola won the pick 3.
Albert didn't win.
Thus Carol won.

As far as this example goes there are only four sorts of possibility to consider.

They both win
He wins, but she doesn't
She wins but he doesn't
Neither of the two win.

And there's NO possiblity that premises are true and the conclusion false.

A C [A v C] , ~ A / C

T T
T F
F T
F F
Homework Questions?
Informal Proofs!
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?

If your answer is 'yes', then give an example. If your answer is 'no', prove it can never happen by supposing that P&Q is l.t.

next....

II. Symbolizations

A. Expressive Completeness

We have a bunch of connectives. Why not more?

For instance one could have a single connective to express "neither P nor Q". We might write this as 'P$Q' and could give it the following truth table.

	P	Q	P$Q
row one:	T	T	F
row two:	T	F	F
row three:	F	T	F
row four:	F	F	T

But we don't need this. We can just say '~P&~Q'.

Are there any connectives that we couldn't define from those we already have?

Back in chapter two, though, we saw that there were certain connectives, non-truth functional ones, which could not be symbolized in SL. Thus, there are limitations to the expressive power of SL.

However,...

it's worth pointing out that SL can express any truth function. What this means can best be seen by way of example.

Think about the following table for a "mystery" sentence-form '???':

P	Q	R	???
T	T	T	T
T	T	F	F
T	F	T	T
T	F	F	F
F	T	T	F
F	T	F	F
F	F	T	F
F	F	F	F

row 1 v row 3

next...

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

F unless G J if K L only if T K is a necessary condition for D J is a sufficient condition for R Not both J and K Both J and K are false Neither J nor K Either not-J or not-K

Don't forget our big list.

next...

3. Harder Symbolizations: remember our ideas:

commas mark the main connective (sometimes)

grouping rules

When one has a sentence of form:	Then:	Example (Hybrid):	Symbolization:
If P, then Q	Antecedent is P	If A and B, then C.	(A&B)>C
Either P or Q	First disjunct is P	Either both A and B, or C.	(A&B)vC
Both P and Q	First conjunct is P	Both if A then B, and C.	(A>B)&C

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.

Homework

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

roll

exam: next time

I. Symbolizations

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

If Sam runs numbers and ignores the IRS, then he either gets caught or gets lucky. Which is...

If R and I, then either C or L. Finally

We see '>' as the main connective and write

(R&I)>(CvL)

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.

5. Brute force method:

For 4 above there's the easy way to do it.

~(A&(B&C))

Or you can go through all the possibilities to make at most two teams win.

Row 2 or Row 3 or Row 4 ... .

But for many cases this "brute force" method will work just fine: 3. is an example.

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.

1. 2. 3.

next...

II. Controversies

"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?
Is there an English Material conditional?
- What does that mean?
  
  A material conditional is just an "if...then..." sentence that has the truth conditions of our '>': that is, it's false exactly when both its antecedent is true and its consequent is false.
- So, back to our question...
  
  does English have a material conditional?
  
  Remember that there are "were-would" conditionals that we called "counerfactuals". Their truth value doesn't depend on what is the case (like '&' and other truth functional connectives!) but on what would be the case.
  
  Examples?
  
  But there are statements in the "indicative mood" which are more plausibly truth functional. See the tutorial for some of the reasons here and why this is difficult.
  
  But for now...
  
  Here's another way to think about symbolizing the conditional. Suppose you and I are in a race. I say:
  
  (1) We won't both win.
  
  What does this mean? Just that
  
  (2) If one of us wins, the other loses.
  
  Now, notice that the only way for (1) to be false is for both you and I to win. So, the only way for the logically equivalent (2) to be false is for its antecedent to be true and its consequent false.
  
  Just what I asked for: an English material conditional.

~(Y&I) Y>~I I>~Y Then notice: these three are logically equivalent.

3. But think about one last example:

If G.W.Bush is a philosopher, then Halpin can jump over the moon.

O.K. Obviously false right?

Not so fast.

This one has false antecedent, "G.W.Bush is a philsopher" and false consequent, "Halpin can jump over the moon".

A material conditional is true in that case. So, this is a problem. What to do?

Note that a sentence can be unworthy (wrong to assert) without being false.
- it may be unworthy because it is useless ("All beatified trees are trees")
- it may be impolite ("you need a shower" said to a stanger)
- and, most importantly, it may be misleading but true ("Either George or Laura Bush is Human")
  
  So, just because a sentence seems strange, wrong, misleading, is not reason enough to think it wrong.
  Maybe our Bush is a philosopher example is like that...
It may help to think of our example

If G.W.Bush is a philosopher, then Halpin can jump over the moon.

in terms of another example:

If G.W.Bush is a philosopher, then I'm a monkey's uncle.

which is a back-handed way of saying that George is no philosopher. Notice that this requires the monkey's uncle conditional to be true when antecedent and consequent are false.
And, for one last sort of case, consider this schema:

If ..., then ...., else all bets are off.

This fits nicely with the material conditional.

one last thing...

III. Mock Exam

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

A~>B
A>~B
~A>~B
(L>J)>(P~R)
(L>J)>~(P>~R)

B. Do some table tests.

Is 'S>(K&S)' logically true?
Is 'J=(J&K), ~K therefore ~J' valid?
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

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

1. 2. 3. 4. 5.

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