09.02 Lesson Plan: Semantics

Course policies

Syllabus

Idea of logic:

Sentence  --- represents ---> state of affairs

What do we mean by a "state of affairs"?

- George Bush is President of the United States at 11:30am on Sept 2, 2008.

- On January 20th, 2009, either John McCain or Burak Obama will be
	President of the United States.

* We can think of a state of affairs as a set of possible worlds.
* Likewise: we can think of every "well formed" sentence as being
either TRUE or FALSE in each possible world.

LOGICAL REASONING: determining what must follow from a sentence (or
set of sentences)

HYPOTHESIS:

- On January 20th, 2009, either John McCain or Burak Obama will be
	President of the United States.

(DRAW WORLDS)

CONCLUSION:

- On January 20th, 2009: If John McCain is not President, then Burak Obama is
	President.

(CIRCLE WORLDS)

Question: can we also conclude:

- On January 20th, 2009: If John McCain is President, then Burak Obama is
	not President.

(DRAW A WORLD WHERE BOTH ARE PRESIDENT)

Why not?  We must add something to our hypothesis, e.g.

- Only one person can be President at any particular time.

Are we done?  Consider a world where John McCain's nickname is "Burak
Obama".

- John McCain and Burak Obama are different people.

"Semantic" == "model theoretic" Reasoning:
directly determine that if state of affairs H holds, then state of
affairs C must hold:
    H  |= C       H ENTAILS C

What is the relationship between the SET of worlds for H and the SET
of worlds for C?
- the worlds for H must be a SUBSET of the worlds for C.

This may seem backwards to you at first -- the conclusion has MORE
WORLDS than the hypothesis === the conclusion has NO MORE INFORMATION
than the hypothesis.

Consider a case where the hypothesis has two sentences:

H1: The sun is shining (at a particular time -- implied from now on)
H2: The birds are singing (some particular birds)

C1: The sun is shining.

Clearly, the worlds in which H1,H2 are true is a strict subset of the
worlds where C1 is true.

CAN ADDING TO THE HYPOTHESIS EVER RULE OUT A CONCLUSION?

No?  Why not?

This is a property of all classical logics: they are "monotonic".  In
fact, late in the course we will look at the attempt to create logics
that do have this property.  Some logicians do not consider
"nonmonotonic logics" to be real logics at all!

Propositional logic:

English is a bad language for logical reasoning:
- ambiguous
- wordy
- hard to tell if a sentence is "well formed" in the manner needed to
truely describe a state of affairs.

Propositional logic is the most basic formal, or mathematical logic.

Ingrediants of a sentence: two kinds of symbols:
- propositions
- logical connectives
- parentheses

Examples with symbols such as P, Q.

The INTENDED MEANING of a proposition is OUTSIDE of logic itself.
You might write:

P means "John McCain is President"
Q means "Barak Obama is President"

Then the sentence

~ (P & Q)

means (to you!) that "It is not the case that both John McCain and
Barak Obama are President".

and write

P & (~(P&Q)) |= ~Q

to represent the corresponding (correct) argument.
BUT logic doesn't care if you instead INTEND

P means "Paris Hilton is President"
Q means "The price of oil is $50 a barrel"

SO FAR:
Logic
Possible worlds
Syntax of propositional logic

Next idea:

Inference = a truth-preserving way of manipulating sentences