Atheism Resource

As promised last time, today we’ll be going over some of the most commonly used replacement rules.  DeMorgan’s Theorem Demorgan’s is handy partially because it helps you get propositions out of parentheses, and also because it lets you change “And” statements into “Or” statements and vice versa. To illustrate this, let’s look at the following example: Suppose I tell you that Kelly either plays Banjo or Ukulele. Letting “B” be “Kelly plays Banjo,” and “U” be “Kelly plays Ukulele,” we can symbolize my statement as: “BvU.” Now suppose that you disagree with me, knowing that Kelly has never picked up an instrument in her whole life. “No,” you say, “It isn’t the case that Kelly plays either the Banjo or the Ukulele.” We can symbolize your statement as “~(BvU)” Now, think about your claim that it isn’t the case that Kelly plays either Banjo or Ukulele “~(BvU).”  Which of the following statements is equivalent to what you said: 1. Either Kelly doesn’t play banjo OR she doesn’t play U...

Welcome back. Today we’ll be picking up where we left off with more replacement rules. Commutation (Comm.) Commutation is a handy step for dealing with “And” or “Or” statements, where the statement you have is close to what you would need to complete another step (such Modus Ponens, for example) but is not quite exactly right. I think the easiest way to explain this would be by example. Consider the premises: If the Sun is very Large but looks very Small, then it must be Far away. The Sun looks very Small, but it is very Large. Our symbolism would be as follows: 1. (L^S)→ F                Premise (P) 2. S^L                            P Now, instinctively, I think we understand that “L^S” and “S^L” mean the same thing, but as was stated in the introduction, formal logic requires you to show your work. Every step must be documented. It’s not enough to assume that th...

I mentioned previously (clear back in Part 3) that one of the benefits of symbolization is that plain English has several different ways of stating the same thing, and this can make it difficult to trace the line of reasoning without doing something to burn off the dead wood. While symbolization certainly helps with this, it does not create a “perfect” system where there is only one correct way to symbolize any given statement. There are still a few instances where one symbolization is logically equivalent to another. Replacement rules help us identify these instances where one phrasing can be replaced with another equivalent phrasing, thereby making proofs much shorter. As we go through these, do keep in mind that replacement rules are reversible. Since they are essentially different ways of phrasing the same thing, you can always swap back and forth. Also, because we are not reaching “new” conclusions, but merely re-stating old ones, replacement rules can be used on parts of lines (t...

At one point or other, you have probably seen some theist say "Well prove that God DOESN'T exist" and seem some atheist respond “You can’t prove a negative”. Don’t worry, I’ve said it too. There’s no shame in being wrong from time to time, the point is to grow, and learn, and do better. I have no idea how that idea first entered circulation, but the thing is IT IS FALSE. And today, we’re going to use an Indirect Proof to prove a negative. An Indirect Proof is somewhat similar to a Conditional Proof, insofar as we will be using (and later discharging) Assumptions. However, in this case, instead of assuming the antecedent and hoping to prove the consequent, we will assume the OPPOSITE of what we intend to prove, and see if it leads to absurdity (specifically, if we can create a contradiction such as A^~A). Contradictions are a big deal for us because, in logic, every proposition has exactly one truth value. Schrodinger’s Cat is either alive or dead, but the cat can’t be bot...

Today we’ll start covering the two more-complex inference rules. We will only get to one today, but I’d rather spread this out over two posts and have it be understandable than try to crunch it into one and have folks not get it. Conditional proof From time to time, it is useful to prove an “If/Then” statement. In order to prove an If/Then statement, we sometimes can use hypothetical syllogism. For instance, Consider: 1. If a nuclear Missile is launched, it will start a Nuclear war. 2. If there’s a Nuclear war, then nuclear Winter will follow. We can use Hypothetical Syllogism to prove “if a nuclear Missile is launched, then nuclear Winter will follow”, as follows: 1. M→N                Premise 2. N→W                P 3. M→W                1,2 HS However some situations are more complex, and simple Hypothetical Syllogism will not do. Consider this more c...

Last time we re-visited two inference rules and introduced three more. This time around we’ll cover three more inference rules, and then we’ll just have two more. First on the list we have: Disjunctive Syllogism (DS) This is our first inference rule dealing with “Or” as an operator. Disjunctive Syllogism states: Given the premises “AvB” and “~A”, conclude “B” 1. AvB                Premise (P) 2. ~A                  P 3. B                     1,2 DS Recall that an Or statement states “At least one of the two propositions is true.” So if we know that either “A” or “B” must be true, and we know it’s not “A”, then simple process of elimination leaves us with only one possible answer: B. Addition (Add) Addition is something of the opposite of a Disjunctive Syllogism. I will say up front that – while this inference rule is not difficult – it is ...