Logic and Rules: Part 2

So I find myself putting off my readings for my USSR history class. Then I remember that I’ve left readers in the dust with my logic articles. Well, here we take off once more! If you remember, last time I went over the basic logic gates that construct arguments, as well as the basic methods by which to appraise those arguments. In hindsight, I forgot some definitions:

Valid- This means that it is impossible that we have all premises true and the conclusion is false.

Invalid- It is possible to have all premises true and the conclusion false.

Sound- An argument is valid and has every premise true. Please note that these are incredibly hard to find outside of mundane statements.

You probably need to refresh your memory, so here's the link to my last bout of insanity:

Moving on, we’ll start with the S-rules, which deal with basic inferring from the logic gates.

AND S-Rule:

If it is the case that you have two items linked by an AND symbol, then you can infer each part.

Banana and chocolate

(B*C)

B

#C

Please note, however, that if it is negated ~(B*C), then you cannot infer a thing.

OR S-rule:

If you have the negation of an OR statement, then you can infer the opposite of both.

Not-either banana or chocolate

~(BvC)

~B

#~C

In the case of a negation of the negation of an OR statement, you cannot infer anything.

IF-THEN S-rules:

If you have the negation of an IF-THEN statement, then you can infer the first part and deny the second. Please note the original definition of an IF-THEN “it is not the case that we have A true and B false.” This negation means that we do have A true and B false.

Not if banana, then chocolate

~(B>C)

B

#~C

As has been usual so far, if there is a positive IF-THEN, it is the case that we cannot infer anything.

These should be rather obvious in their usage, and you may find yourself using these rules to infer things about what another person is saying. Now, let us move on to the I-Rules, which are just referred to as the Inference rules, and while those last rules were technically inferring, they’re just called the Simplification rules.

AND I-rule:

If we have the negation of an AND gate, we can affirm one and deny the other.

It is NOT the case that we have A true AND B true

~(A*B)

A

#~B

Remember, if there is a denial of the first, we cannot infer anything.

OR I-rule:

If we have an OR statement, and deny one of the two premises, then we can affirm the other.

Chainsaw or shotgun

(CvS)

~S

#C

Remember that you can only infer when you have a denial of one of them. If one is affirmed, the other can be true or false…we don’t know!

IF-THEN I-rule:

So, we’re back to this complicated prick. This gate actually has two I-rules, so ‘ere we go!

If we have an IF-THEN statement, and affirm the first premise, then we can affirm the second.

If chainsaw then shotgun

(C>S)

C

#S

If we have an IF-THEN statement, and deny the second, then we can deny the first.

If chainsaw then shotgun

(C>S)

~S

#~C

So yeah, this one proves to still be quite wretched as a logic gate.

Now, I can certainly understand that all this is rather dry, so I’m going to shoot something out to the community: Give me a rules situation that you are having trouble with, and I will apply these principles to it to find meaning, and it will be in the next segment of this…not necessarily frequent series regarding Mr. Spock.

As always, shoot me an email at dominobi@gmail.com

I will be waiting to see what you guys have questions on!

Regards, Ishamael