Introduction

Conditional proof and indirect proof will be the 2 last methods we go through until we switch gears from sentential logic to predicate logic. Let’s give sentential logic a proper send-off with these last 2 proof methods. These methods are somewhat different from the inference- and replacement rules we’ve seen so far. But knowing these rules will make your life easier; proofs will become shorter and you will be able to prove theorems from now on (more on this later). 

Both these methods have one thing in common: they require you to start with an assumption, not a premise.

• In an indirect proof, we assume the opposite of what we actually want to prove.

• In a conditional proof, we assume the antecedent in order to derive the consequent. So, if we want to prove p → q, we assume p to derive q. As the name implies, this method is restricted to proving conditionals.

Because we assume our starting point in these proof methods, it makes sense to think of them as hypothetical subproofs within a larger proof (“this would happen were our starting assumption true”). While either method can constitute an entire proof in and of itself (as in the case of a theorem), in this tutorial you will most often encounter either method as being used inside a proof that starts with a bunch of premises. You can use premises inside your indirect/conditional proof, but - because both are hypothetical - what happens inside either method stays inside the method. Once you’ve derived what you want, you have to retire the steps inside the subproof. We call that either retiring or discharging the assumption. If this sounds like gibberish right now, don’t worry. This will make more sense when we see the methods in action.

For now, familiarize yourself with the diagram below, as it reveals the usual structure of these proofs. I’m using a visualization I learned from Virginia Klenk. In this visualization technique, you draw a scope arrow to delineate where the assumption starts and ends. This is because eventually, we need to discharge/retire the assumption; it’s only temporary. The diagram shows the skeleton of an indirect proof being used in the middle of a larger proof (i.e. as a subproof). The black dots on either side of the scope arrow are general steps in the proof: the first 3 black dots are premises and the second 3 black dots are later derivations that make use of the conclusion you derived from the indirect proof. While this structure will often be the case for both conditional- and indirect proofs, either method can be used at any part of a proof (and may even be the entirety of the proof, as in the case of some theorems). 

Indirect proof (I.p.)

As noted, an indirect proof involves first assuming the opposite of what we want to prove. Why? A clue lies in the method’s other name, reductio ad absurdum, which is Latin for “reducing [something] to absurdity”. It’s like arguing with your friend Daniel, and Daniel is taking a really dumb stance (physically and argumentatively). Instead of you just saying “uuuh ur so dum”, you decide to show Daniel how ridiculous his argument is by assuming his position and going through a chain of reasoning that ends up in an absurd conclusion (which Daniel is forced to accept if he is to maintain his stance). Naturally, this destroys Daniel, because he realizes that his biggest opponent is not you but, rather, his own mind (ouch).

In our logics, this works because of the law of the excluded middle (which we mentioned way back in Part One). If something isn’t false, then it must be true, vice versa; and never anything in between true/false. 

So if you want to prove ¬K, you assume the opposite, K. Then, from K, you go through the normal proof motions to the point where you derive a contradiction (the “absurdity”), i.e. you derive some term p and its negation ¬p. Now, because our assumption of K led to a contradiction, then it means that ¬K has to be true, which is what you wanted to prove in the first place. 

Conditional Proof (C.P.)

Conditional proofs (also called direct proofs) are used when you want to prove, you guessed it, a conditional. Basically, what you do is assume the antecedent of the conditional in order to derive the consequent. As with indirect proofs, this assumption is only temporary, so a scope arrow is used to indicate where the subproof starts and ends (and no derivation steps inside of the scope arrow can be used outside of the scope arrow).

Conditional proofs are very handy, as they will in most cases shorten your proofs a lot. Consider the long examples in Part Five. The longest proofs we did had some rather complex formulas that we needed to manipulate with a bunch of rules just to break them up. With a conditional proof: if you have a complex conditional proposition to prove, you’re allowed to just assume its antecedent (i.e. a less complex proposition) and go from there. Pretty great!

Back 2 school: Examples of Proofs

As always, let’s practice. We’ll start with an indirect proof. Similarly to the previous part of the tutorial, I’ll mark important steps in red, and include a purple divider to indicate where we left off.

Example 1

We have two premises, and from those, we’re supposed to derive the disjunction ϕ v C. 

Hot Tip #1: Gain intuition by thinking backwards

We want to prove ϕ v C. Based on the premises, we have ϕ included, but no C. Because the conclusion is a disjunction between something we have and something we don’t, then at least we know that will have to use the Addition rule for C at some later point. No need to worry about C. Also, the conclusion is obviously not a conditional (and there’s no conceivable way to turn it into one, using our rules). So, using a conditional proof is not the way to go here. So it looks like we’re gonna get there through an indirect proof. 

Hot Tip #2.1 - Simplify as much as possible BEFORE starting an indirect/conditional proof

Before we use an indirect proof, we’re going to use a variation of Hot Tip #2 - find hot singles in your area. Let’s call it Hot Tip #2.1. The principle is the same, but it’s adapted especially for indirect and conditional proofs. We’re going to apply it by using Biconditional Exchange on Premise 2 now. Why is it important to do this now? While we could do this inside the upcoming (indirect) subproof, remember that steps derived within the scope arrow cannot be used outside. What if, in our proof, we’re going to use several instances of indirect proof, or some combination of indirect + conditional proof? Then we risk having to do the same derivations over and over again because we did them inside a scope arrow. So, for the sake of efficiency: if you have premises, best to simplify them as much as possible before starting either an indirect or conditional proof. Anyways, let’s apply the rule via Biconditional Exchange on 2:

Okay, now we’ve done all we can with the premises given. We can’t simp anything else. Let’s start the indirect proof. Because we want to prove ϕ v C, we will start the indirect proof with its negation, ¬(ϕ v C). And, here we have an excellent opportunity to use Hot Tip #3 - Derive DeConjuncts. ¬(ϕ v C) turns into ¬ϕ ^ ¬C through DeMorgan’s, which we then can simp. But recall that we don’t need to worry about C, so let’s just simp ¬ϕ. This sets us up nicely to use Modus Tollens on Step 1 to yield its negated antecedent. We’re now up to Step 9. You can probably guess what we’re going to do next.

That’s right, we’re gonna use Hot Tip #3 again. That brings us up to Step 12.

Okay, nice. What now? Know that eventually, we’re going to have to derive a contradiction, i.e. a conjunction between a term (or formula) and its negation. Look at steps 4-5 which we derived via Hot Tip #2.1. We have yet to do anything with these. The nice thing is that our freshly simped ¬F (Step 11) will set us up nicely to do a Modus Tollens on Step 5. That will yield ¬¬G. OOOOOOO, DOUBLE NEGATION - SPICY 🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️What’s more, we already have ¬G. So you know what that means: all we have to do is apply Double Negation to yield G, then conjoin it with ¬G from Step 12, and then we have our contradiction! That means that we can discharge the assumption (Step 6 and onwards) and conclude that (ϕ v C) HAS TO BE TRUE. WELL DONE GOOD LUCK IN THE COMING FISCAL QUARTER! 

*NOTE: instead of deriving (G ^ ¬G), we could have also derived (F ^ ¬F) by Modus Ponens via Steps 12 and 5. This would save us from having to use double negation.

That’s so hot 🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️🌶️

Example 2

Thinking backwards: we need to prove a conditional. Seems like an excellent candidate on which to use conditional proof. And, to me, it looks like we might as well get right to it. So, let’s assume the antecedent of what we want to prove - A. Now that we’ve assumed A, we can get to work on the premises. Eventually, we’re going to have to free up D, but it looks like we’re going to have to put some of our rules to work first. The obvious first step is to apply Modus Ponens on Premise 3. That will give us B. And from there, we can conjoin A and B to create the antecedent of Premise 1. This will set us up for another round of Modus Ponens. We’re now at Step 6.

You can probably see what’s going on: it’s Friday night, we’re carelessly swiping on Tinder. A solar storm of self-hatred and nihilism is eclipsed by our intense desire to get the D. Here’s the best way to get it:

Using our conjunction from Step 6, we apply Modus Ponens on Step 1 to yield the consequent, ¬C. From there we will turn our attention to Premise 2, where D is. The next step should be obvious: using ¬C, apply Disjunctive Syllogism on Premise 2 to yield the conjunction between D and E. Then we just simp it to finally get the D. We did it - we found the D. We’re ready to discharge.

That was great for me, how was it for you?

Part Four Wrap-Up

What we have learned

We have learned 2 new methods of proving arguments in sentential logic - Indirect proof and Conditional proof. These methods are different from the other rules in that they rely on making assumptions. These assumptions are temporary and will eventually need to be discharged. So, the best way to conceive of these two methods is as hypothetical subproofs within a larger proof structure. In an indirect proof, you assume the opposite of what you want to prove; in a conditional proof you assume the antecedent of the conditional you want to prove.

I should note that we worked through fairly simple examples: there are cases where you will have e.g. an indirect proof within a conditional proof. I might add some more examples at a later point. If not, we’ll probably encounter such examples at a later point in the tutorial.

A note on I.P., D.P., theorems and Epistemology

These proof methods are great in the sense that they often expedite the proof process. And, because they are based on assumptions, you actually don’t need premises to use them. As long as you have something you want to prove, you can get to work with either method. Proofs without premises are called theorems. You might wonder how this is acceptable - how can you be sure that it is valid if you have no premises established? You can’t be sure of whether it’s capital T True in the world or whatever; you can just be sure that, assuming your premise is true, the rest follows and is therefore true. This is a general issue of epistemology, and the truth (lol) is: everything has to be based on some fundamental assumptions that cannot be proven true or false; you just have to take your first principles on faith for anything else to make sense. Every way of knowing, every world view, every ontology will be traced back to unfalsifiable assumptions of what exists and what doesn’t. This epistemic issue has emerged in several forms throughout the history of philosophy, but a fairly recent rendition of the issue is as one of the unsatisfactory outcomes in the Münchhausen Trilemma. This thought experiment relates to the nature of formal proofs and how to prove the veracity of a formal system, or a world view in general. As the name suggests the thought experiment involves a forced choice between 3 unsatisfactory outcomes. If you try to prove a system to be Capital T True, you will encounter either of these 3 problems:

• Infinite regress: You try to prove the truth of one proof with another proof, which in turn will have to be proved by yet another; and another, ad infinitum

• Circularity: You try to prove the truth of the same proof/system within itself

• Dogma (this is the one we’ve talked about): the assertion of truth is merely based on an unsupported, unfalsifiable assumption.

Where we are going next

This marks the end of the Sentential Logic portion of this tutorial series. Congratulations! Our next mountain to climb is called predicate logic. It’s a little different from the logic we’ve seen so far; we’re no longer going to treat whole sentences as atoms. We’ll get a bit more specific. Plus we’re going to introduce quantifiers and the set of rules that comes with. If you’ve understood most things discussed up to this point, you’ll be fine! But I think you deserve a little bit of a break before we move on. This next part of the tutorial will be a bit of a palate cleanser. You deserve it.