L1 W7 (L2)
Exam: 1 week!
- Problems 4 and 5...
- Subderivation idea again...by informal argument.
Let's do this one formally.
- Note the quizzes!
II. Subderivation Ideas and Rules
- When we finish a subderivation (and move back to the originating column) this
- When a subderivation is terminated, we say that its assumption
- A discharged assumption, and any other individual line in a terminated subderivation can no longer be _____. What's
that word to fill in the blank?
- When a line can no longer be cited, it is called
- Let's see this in action.
III. Negation Rules
- These rules just use the reductio ad absurdum strategy.
- Example: 1. 2. 3.
- To formalize this, we just make an assumption...but we assume the opposite of what we are trying to prove. Let's do some.
- And, now you do:
- A>L,~L / ~A
- V&~C , V>~(P&~C) / ~P
||(note: I'm cheating a bit to use 'V' for "an argument is valid". The premises can be read as "The argument is valid and has a false conclusion" and "if an argument is valid, then it doesn't have true premises and a false conclusion". Makes sense, yes?)
- L only if R
- L if and only if R
- L if R
- R is a sufficient condition for L
- L unless R
- Both neither L nor R and Q.
- Not both of L and R.
- If neither L nor R then Q if S.
EXAM next time
Office Hours...by appointment (tomorrow? I'll be around in the early afternoon.)
- Subderivations and >I. (repeated from above)
- If you need to prove ~P, you
- Homework (try 5,4,3 in that order)
A. Validity: Two show that an argument is valid, we can just take its premises and
B. Logical Equivalence
- Two sentences are logically equivalent
if and only if it is not possible that...
- So, if we set up a derivation, from one sentence as premise to the other as conclusion, that
- So, if we use derivations to test for logical equivalence, then we will use how many derivations?
- Let's do some...
C. Logically True Sentences
- A sentence is logically true
if and only if WHAT?
- Now, an example of a logical truth is "Rv~R". Thinking about 'R' as standing for "it's raining", what kind of evidence do you need to know that "Rv~R"
- Let's try some.
- Now, try these:
- Show that 'A=(AvA)' is logically true.
- Show that 'A=B' and 'B=A' are logically equivalent.
(On the computer...)
D. Logical falsehood and inconsistency:
- What do these two concepts have
in common?...on their definitions, a sentence or a collection of sentences is not _____ ___?
- Finally, let's do some more...
III. Review for exam: Mock Exam (online) and a mini-mock exam (below).
A. Fill in the following:
B. Do derivations to show that the following arguments are valid.
C. Symbolize (using the symbolization key of the last exam)
- Moriarity is a crook; moreover so is Holmes!
- Neither Holmes nor Watson is a crook.
- Unless Holmes is a crook, Moriarity is a crook.
- If exactly one of the three is a crook, then it's Moriarity.