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So we have gone through complex truth tables and how to operate them, which is a very important logical subsystem. But it is important to keep in mind that though truth tables can show if an argument is valid or not, it still possesses a major flaw. The flaw is basically this… In order to do the truth table, you have to write out all the truth probabilities? We were fine with one and two variables, because that was only two and four lines long. But what if you had to solve an argument with six variables? ( 64 lines in them!!) That would be ridiculous.

So in order to make it simpler, we have systematized natural deductive logic.

All that is entailed in this particular method of logic is that we deduce the conclusion from the set premises through inference and replacement rules which parallel natural deductive techniques we use in our everyday lives… that’s it.

Truth Functional Logic Proof Structuring

Before we go any further, it might be best to discuss how to go about proof structuring. THIS IS ESSENTIAL TO KNOW IF YOU WISH TO DO PROPOSITIONAL LOGIC PROOFS!!! It is not difficult to do at all. Your past experiences with truth tables will help you somewhat, but this is a little bit different form that which we previously have done.

This is an example of what a full completed and valid Proof looks like…



This example of a logical proof looks complex at first, but it really isn’t once you understand what each section of the proof is for and they all fit together.

This is a simplified version of a proof…



And this is the simplified version of a proof imposed upon the example proof…



Basic Structure of a Proof
The Basic structure of a proof is essentially a giant “plus.” So when you first draw out a proof, you can just simply draw a big “plus” to start with. It is ESSENTIAL to keep to this format as it separates the four quadrants you will need to complete the proof. The sections not only denote what section they are, but also the step you would take when you begin to write a proof. We will get more in-depth into this once we start doing actual proofs, but this is just to familiarize you with the proof itself and the way to approach them.

Section 1 – Section 1 contains the number lines of the argument you are evaluating in your proof. The numbers of the number lines continue into the 3rd section, but just remember the this section is specifically for the initial lines of your argument.

Section 2 – Section 2 contains the argument itself in logical form. Each premise has its own line. But remember that the Conclusion does not have its own line!!! The conclusion does not have its own line because what we are attempting to do with the proof is prove that the arguments can be worked in such a way that they end with the conclusion. The reason we include the conclusion at the end [separated by the / (slash)] is so that we are reminded of what the conclusion actually is.

Section 3 – Section 3 is a continuation of section 1. So if section 1 had lines 1,2,3 section 3 would continue with 4,5,6,etc… Section 3 number lines continue for as long as it is need to derive inference rules to get your conclusion.

Section 4 – This is the heart of your proof. This section contains everything you will need to develop your deductive argument to prove the conclusion you have been given. This section has two parts. One part is the derivations from previous lines. The other part is the citations of the inference rules and from what lines you derived the derivation from.

RECAP!!!!

A proof contain four sections which have number lines (sections 1 and 3) which tell us which line we are on, an argument with a conclusion (section 2), and the proof itself (section 4)

HOW TO APPROACH A PROOF
Now that we know what the sections are for and what the basic format of the proof looks like, I’ll explain what do in order to get to the final step, actually solving the proof (which we will need the inference and replacement rules to solve.)




That is it for the introduction to proofs… not that difficult I hope. I’m very sure I didn’t explain things clear enough, so please ask if you are confused with any part of this. Now a brief spiel on inference rules.


INFERENCE RULES

At the beginning, we talking about recognized deductive techniques. Those recognized argumentative methods are embodied in Inference rules, which means that a specific conclusion can be inferred when certain premises are given for each inference.

There are 4 primary inference rules to begin with (i.e. Disjunctive Syllogism, Hypothetical Syllogism, Modus Ponens, and Modus Tollens. You have to be very familiar with these rules to move onto the next four inference rules and the two “short cut” proof methods.

The best way to cover inference rules would be to introduce a typical argument. Then we can reduce that argument to logical elements, and then reveal the inference rule in detail.

For all intensive purposes, I’m going to break the four major rules down into two symposiums.

That’s pretty much it for the introduction to proofs and inference rules. As always, if you have any questions, don’t hesitate to ask because I am happy to respond. Also, if it seems vague, it probably is, so please inform me if I could clarify it a bit more... or if I’m wrong.