Simple Logic Examples

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Some basic logic

Elliott

Here is our initial world:

1. If it is Wednesday, then it is raining.
2. If it is raining, then we are wet.
3. If we are wet, then we are unhappy.

So, we know some principles that are true in our world. But, we still cannot say anything very specific. We do not know, for example, if we are happy or unhappy, or whether it is raining, or whether we are wet.

We can add a fact to our world, as follows:

4. It is Wednesday

So, now we know something specific. It is raining. But, using logic we also know some additional facts about the world too:

5. It is raining [(1) and (4)]
6. We are wet [(2) and (5)]
7. We are unhappy [(3) and (6)]

Unfortunately, we also know some other things, so we have to be careful::

8. It is Wednesday [because 4 implies 8]
9. It is raining [(1) and (8)]
10. It is raining [(8) and (1)]
11. It is raining [(10)]
12—infinity. etc.

Another way to consider the statement "If it is Wednesday, then it is raining" is as a restatement, "Either (a) it is not Wednesday, or (b) it is raining."

A short form of this is: Not(Wednesday) OR Raining.

Which means:

Either it is simply TRUE that it is [Not(Wednesday)] , or it is simply TRUE that [it isRaining], or if we have additional information it could also be TRUE that is [Not(Wednesday) AND it is Raining.]

Why? Well the following are all true statements:

We do not know anything about the relationship between days of the week other than Wednesday, and whether it is raining. We have no information at all about those days.
We have two statements Not(Wednesday), and Raining. AT LEAST ONE OF THEM must be TRUE
Thus it is not allowed that both [it is Wednesday] is TRUE, and Not(raining) is TRUE, because then both of our two statements would be FALSE and this is not allowed. In other words, if it is true that it is Wednesday, then, because Not(raining) is not allowed at the same time, then "raining" must be true. So Wednesday ==> rain, is the same as Not(Wednesday) OR rain.
If, on the other hand, Not(Wednesday) is TRUE, then one of our two statements is TRUE, and the other one about rain can be TRUE or FALSE according to our rules. In other words, if the day of the week is something other than Wednesday, then it might be raining, or it might not be raining. We have nothing to go on.
[A OR B] in this king of logic is inclusive which means that A is TRUE or B is TRUE, or both of them are TRUE.

We can now extend our logic by adding variables, using what is called First-order logic:

For this we use the quantifiers ForAall, and ThereExists. We can always write statements using either ForAll, or ThereExists, by restating one in terms of the other.

ForAll (THING) If it is raining, then THING is wet.
[When it rains, everything is wet.]
We could also say this differently:
NOT[ThereExists(THING) such that it is raining AND THING is NOT(wet)]
[It is not true that there is some THING such that it is raining, and that THING is not wet.]

We can also use a techique called resolution to learn new things about our world:

A or B—fact
(not B) or C—fact
A or C [by resolution logic, from 1 and 2]

If:

A is A killed sam.
B is B loved Sam.
C is C killed Sam.

Now consider:

1. It is true that A killed Sam or B loved Sam.
2. It is true that either (B did not love Sam) or (C killed Sam).

Thus, we still don't know who killed Sam, but it was either A or C or both A and C did it together. Think about it: B either loved Sam, or she didn't. If she didn't love Sam, then A killed him because of (1). On the other hand if she did love Sam then from (2) (not B) is FALSE and C must be TRUE with no judgement about A.