Philosophy Radio! drlabasham777's Podcast
Society & Culture:Philosophy
Here's a sheet on the Logic problems discussed in the cast immediately prior to this one. Use it to follow along as you listen to the Logician out here explain it all; logically, of course!
Two new rules:
Assumption for conditional proof (ACP)
Works like this. Suppose you want to prove and IF this THEN that claim: If p then q, for instance. Any IF this THEN that. If r then p, for instance
Here’s what you can do! Let’s imagine an example
We did it. Walla! If we assume r is true (ACP) then we can prove p is true, so: if r then p!
You can look at the premises and see what’s up: if you get r, you get s so then you get p. Cut to the chase: r get’s you p. If r the p! So we assume r, and get p, and say: Hey, if r then p. Nice!
As it happens there are very few cats out here in the ranchlands. Why? The hawks fly away with them and eat the poor things; killing them in the process. The cats die. So, here we go:
Let r be “hawks dive”
Let s be “cats fly”
Let p be “cats often die”
So,
If hawks dive then cats fly away (if r then s)
If cats fly away then cats often die (if s then p)
Let’s assume a hawk dives (r) ACP
Oh look, there’s a doomed cat! He’s dead. (if r then p)
Very simple when you think about it. What ACP let’s do is short cut to a new IF this THEN that statement.
Here’s a second new rule. One of my favs! Assumption for indirect proof: AIP
Here’s the thinking. Let assume some thing (p, for instance), show if that is true then something impossible must be true (p is true and not true at the same time, for instance). Then we realize p can’t be true. It leads us to an impossible statement. P is true and p is not true at the same time. It’s like a circle that is at the very same time a square. Cannot happen. Impossible. Not even God can do that.
Impossible statements are called: Contradictions.
So with AIP we derive a contradiction. Easy and fun! Lawyers love this stuff.
So let’s do a simple AIP,
Let p be “my client was out of town”.
Let q be “my client we to buy a used car” (here, irrelevant)
Let s be “my client killed the victim: We want to show the client did not kill the victim.
So it goes like this,
My client was out of town: p (. E, And Elimination)
If my client killed the victim then my client could not be out of town: if s then ~p (line 2)
Let’s assume my client killed the victim: s (line 4)
So,
My client was both out of town and in town when the victim was killed. (p . ~p .I, And Introduction) Impossible! A contradiction!
So, my client did not kill the victim. (~s)
Proved innocent!
So if s was true, we’d get a contradiction, an impossible statement. We make a contradiction with a “!” mark. Here we get p is true and p is false (~p) at the same time! So s must be false! Which means ~s is true! Walla! Innocent!
Let’s try an ACP now:
This is going to mirror our example above. Give it a try!
Tautologies are the mirror image of contradictions: All contradictions are false (impossible), all tautologiesare true (must-be true statements; also known as necessarily true), simply because of their logical structure.
Here’s a simple a tautology as can be imagined. If something is true then that very same something is true. Use ACP:
Use ACP to prove it. Insanely instantly: Proven! Go for it.
Now try to do a basic AIP
See how perfect this is? You might say it’s logical.
Now use AIP to prove our most excellent and basic tautology above:
Peace!
Dr. Lee Basham
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