Strategies I use for True / False Questions in Logic
When faced with a True False question, I
think of a few different stategies:
Translate the words back towards the definitions: take the words (or symbols) used in the statement and put down more basic ones.
\Gamma is SD-consistent
\Gamma is not SD-inconsistent;
we cannot derive both \alpha and \not \alpha
for the original set \Gamma.
\Gamma |= \beta
If a tva \tau |= \Gamma then \tau |= \beta
Getting down these alternate forms lets you consider the following strategies (and is probably indispensible for the third and fourth strategy, since it makes more 'logical connectives' appear).
Think of an example. Take one or two letters, write down
some formulae or sets which have the properties being described.
This helps me see what is being talked about. Many times
this is enough to let me see what happens in other situations,
and the words applied to the example can be rewritten
as a general argument. Of course, an example is only a complete answer when the 'unviersal' statement is false and we are looking for a counter-example.
Once I have the sentence in a general form, I consider
the same strategies I would use in SD+.
Given a sentence \alpha -> ¬ \beta,
I write down the sentence \beta -> ¬ \alpha
[Using the informal versions of Transposition and Double Negation.]
Basically I use any changes of appearance which make things easier
for me to think with. It is suprising how much of what we do
in logic, in algebra, in calculus etc. is based on how things appear, not on what they mean (in words).
If I was starting to get confused about the words in the previous strategy, I would consider a pseudo-translation: some semiformal version using logically symbols, maybe quantifiers etc. (see chapter 6). I would then 'see' strategies from SD+ (and PD+ when we get there) which would work. For example, if it became clear that I was trying to 'prove' a negative statement, I would consider a proof by contradiction. This is the same strategy I use in SD when I face a conclusion of the appearance ¬ \beta and investigate ¬ E as the final rule.
In general, we are looking for stategies in logic. Our claim has been that the rules of SD, SD+ (and the rules coming up for PD, PD+) capture strategies logical people use all the time. If they still seem strange to you in natural language, then work back and forth between the language and the formal systems. As Ganong said in his advice in preparing for Test 1, it is best if you feel all these methods as propoer and fitting. They are the best way we know to handle 'truth' and not lose it along the way!
It is sometimes strange how the same methods handle 'falseness' - an inconsistent set of premises does validly entail any given wff as a conclusion.
In the same spirit, given an contradiction, we have a little 'four line game'
which will derive any given wff from the contradiction. One goal is that this becomes 'natural' to you, in this course but also in all your other courses.
I understand that for many of you this way of handling 'falseness' is not what you believed at age 7. [That has become my definition of 'common sense' for most people.] Given this fact, it does take substantial effort, playing around with the possibilities, to change you deep reactions. One of the goals of this course is to have you face these issues and be sure that you future reactions reflect this more precise logic.
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