W2

L1    (L2 is below)

Roll
Homework: in later today or SOON
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Exam next week...see Syllabus, Calendar

Mock Exam

 

Chapter One and HW

I. Some review?

1. 'Possibility' is the key notion in deductive logic. For example, we say that an argument is (deductively) valid iff it's impossible for          What???       .

Next, what is the definition of consistency and what does it have to do with possibility?

 

 

 

 

2. Little boy/pool example (I did't go in the Berger's pool, my hair's not wet. Anyway, Billy made me do it!) This SET of sentences is consistent? inconsistent?

 

3. Informal Proof: If \ contains a logically false sentence, then \ is logically inconsistent.

 

 

II. Indirect Informal Proof

 

1. Example: 1. 2. 3.

 

 

 

 

 

 

2. Other Examples:

a. No argument with logically true conclusion is invalid.

b. Any two logically true sentences are logically equivalent.

 

 

III. Proof by Example

a. A valid argument can have a false conclusion.

b. Two logically indeterminate sentences can be logically equivalent.

 

 

 

IV. Homework: your questions or concerns...

Here's a statement of the homework. (You'll find these at the bottom of each chapter.)

 

 

 

Chapter Two: Our first symbolic language. AND the basis of everything else we do.

 

I. Think about 'And'. A simple "connective" of English. We'll symbolize it with the ampersand: '&'.

1. First note: there are lots of ways to say 'and':

Bob and Carol are students.
Bob is a student and Carol is a student.
Both are Bob and Carol are students.
Both are students.
Carol is, moreover so is Bob.

2. All these could be symbolized as

 

3. Terminology:

 

 

 

 

 

 

II. Semantics and Truth Tables (including matters of truth functionality and non-truth functionality):

 

1. When is a sentence true? Take our examples above.

These are true when Bob is a student and Carol is too.

OK, not too informative! But we'll make something of this. A sentence like

Bob is a student and Carol is a student

is true iff both it's components are true.

 

2. It's components could be true or false.

For example, if Bob is a student and Carol is not, then the whole sentence is True? False?

What are all the possibilities and in what possibilities is our sentence true. Easy to say, isn't it? (TRY!)

Now, let's re-express this in "truth table" form:

 

 

 

 

 

 

 

 

Now, think about some more connectives..."or", "not", "implies", "if and only if".

  1. 'not' is easy
    • think about examples: 'Sam is not a student' or 'Sam isn't a student' or 'it's not the case that Sam is a student'.
    • ~S
    • Table?






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      L2

      roll
      review and start where we left off:
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  2. 'or' is harder
    • examples: "either Sam or Bob is a student", "Sam is a student or Bob is a student"
    • v
    • Table??? (Note that "P" and "Q" are metavariables ranging over all sentenceds of SL. Just like in math when we say x+y=y+x, the variables don't name any particular thing. This allows generality.)
        P Q PvQ
      row one: T T ???
      row two: T F T
      row three: F T T
      row four: F F F

      We'll say that the question marks need to be filled in by 'T'
    • But how well does this represent English? Think about examples:
      • You can have soup or salad. (Implicit assumption: "not both")
        BUT This example is odd...








      • Chris will get promoted or, at least, will receive a raise.
        This shows...










      • Is there a better example? (More arguments later.)


  3. >
    • Examples:
      • "Sam's being a student implies that she gets a loan" (S>L)
      • "If Sam is a student, then so is Bob" (S>B)
      • "Sam is a student only if she applies and passes the boards". (S>(A&P))
    • Conditionals are difficult. Think about the above with "there's fire only if oxygen is present in mind". That is pretty clearly equivalent to "If there is fire then there is oxygen".
    • Truth Table? "If Sam is a student, then she gets a loan" is clearly FALSE on what possibility?










      The rest of the rows are controversial.
    • In fact, for some conditionals there is no definite way to fill in for the '?'.
      • These are called "counterfactuals" and are discussed at length in 2.3.
      • The idea is that something like "If Al Gore had won the 2000 election, he would have cut taxes on the middle class", if true, has nothing to do with the actual truth values (F,F) but depends on the way the world would have been if Gore had gotten credit for a few more FL votes.
      • Because the truth value of counterfactual does not depend only on its component parts, we can write no truth table for it. We will say that it's not truth functional.
    • But other examples allow the truth table to be completed.







     

     













  4. = represents "if and only if" or "just in case". We'll later be able to motivate it's truth table. For now just see it as asserting equality of truth value:
  5.   P Q P=Q
    row one: T T T
    row two: T F F
    row three: F T F
    row four: F F T

 

 

 

 

 

  Connective Name Resulting Sentence Type Component Names Typical English Versions English Statement Symbolization in SL
& Ampersand Conjunction Conjuncts "and", "both ... and ... " Agnes and Bob will attend law school. A&B
> Horseshoe Conditional Antecedent, Consequent "if ... then ... " If Agnes attends, then Bob will. A>B
~ Tilde Negation Negate "it's not the case that", "not" Agnes will not attend law school. ~A
v Wedge Disjunction Disjuncts "or", "either... or... " Either Agnes or Bob will attend law school. AvB
= Triple Bar Biconditional Bicomponets1 "if and only if", "just in case" Agnes will attend law school just in case Bob will. A=B

 

 

 

We can also give a single truth table in one place:

  P Q P&Q PvQ P>Q P=Q ~P
row one: T T T T T T F
row two: T F F T F F F
row three: F T F T T F T
row four: F F F F T T T

How will we memorize these?

 

 

 

Symbolizations (2.1)

You try some:
a. Arthur and Barb are both students.
b. Arthur is a student but so is Barb.
c. Arthur is a student if and only if Barb is.
d. Arthur is a student if Barb is.
e. Either Arthur or Barb is a student.

 

 

ALL our connectives are truth functional. That just means that we can do a truth table to define them. Or, in other words:

 

 

 

 

III. Syntax

We may now more precisely define just what counts as a sentence of SL. We give what is sometimes called an "inductive" or "recursive" definition. But all this means is that we define what counts as a sentence by showing how to construct one from basic parts.

Here's the definition. First,

 

 

 

Second we say how we can build more complex sentences from any sentences which have already been built.

 

 

 

 

Now, let's see how to apply these rules to build up a sentenced like:

~(JvK)>(L=~S)

 

More definitions:

 

 

 

 

 

 

 

 

 

 

 

The sentential components of a sentence of SL are all components used in the building process in order to construct that sentence.

The atomic components of a sentence of SL are all atomic sentences used to construct that sentence.

The main connective of a sentence of SL is the last occurrence of a connective used to construct it.

The immediate component or components of a sentence of SL is (are) the sentential component(s) used in the final stage of its construction.

 

 

One last matter: Logical Form

When we say, for example that a sentence has the form Pv~Q, this just means that the sentence has main connective 'v' and the second disjunct has main connective '~'.

 

 

IV. More symoblizations

  1. unless
    • Think about "we win unless they score"; this means which of the following? (three are correct):
      • ~S>W
      • S>~W
      • ~W>S
      • WvS
















  2. only if
    • Think about "there's fire only if oxygen is present".
      • If there is fire then oxygen is present. (F>O)
      • If there is oxygen then fire is present. (O>F)
      • If there's no oxygen, then there's no fire. (~O>~F)
    • Which are right?















  3. neither nor
    • Think about "Neither Bob nor Carol is a student"
    • There are two ways to do this symbolization; which are correct?:
    ~Bv~C
    ~(BvC)
    ~B&~C
    ~(B&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 only if 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 are in do.
  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.

Mock Exam