Propositional Logic Symposia - [7] – Proof Structures and Inferences

[VideCorSpoon]

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.

[VideCorSpoon]

Conclusion Function
When setting up a proof, we put down the argument in the second section. But remember that we also put the conclusion in that proof, but we did not give it its own line. Why? We did this because the entire proof is meant to prove the conclusion. Simply, the conclusion cannot be part of the solution. We put the conclusion at the end of the last argument line to remind us of what we have to prove. We know we have successfully completed the proof when we can finally derive a conclusion that matches the desired conclusion in the second section.



In the example above, we see where the second section conclusion reminder is, and how the final derivation line has the same sentence (i.e. ~H) and we got that final derivation by going through the inference and replacements in order to get there.

It dawns on me that people may not understand why we do proofs in the first place. This is probably the most useful answer. A proof can be solved in any number of ways to get a conclusion (usually). But the simpler the proof (i.e. the shorter the proof length) the more coherent the argument.) If you have to do all sorts of inferences and what not’s to derive a conclusion from 40 something lines, the argument probably isn’t that coherent. There’s more to it, but that is just one advantage to these proofs.

[Arjen]

This is an interesting post. I am used to very different notations. It give me a chance to see which is more clear, or which I like best.

[VideCorSpoon]

Im glad you find it useful. Actually, if you feel like it, could you post the different notations you are familiar with. I think that is one aspect I have not documented, which is the different notations for connectives and the different proof formations and why you find them easier. I think others will find it especially valuable.

Personally, for the bi conditional, I like the tri-equal sign best... but I cannot symbolize it easily on word.

[Zetetic11235]

What are these names for the inference rules? I know of inference from Quine's Method's of Logic as (i) Any schema implies itself (ii)If one schema implies a second and the second implies a third then the first implies the third (iii) An inconsistent schema implies every schema and is implied by every other schema (iv) A valid schema is implied by every schema and implies only valid schema.

1.AvB
2.~A
3.B--> D
4.D--> ~E
5.H-->E /~H
Are these assumed true? If so can the be represented as equivalent to

AvB.~A.B->D.D->~E.H-> E implies ~H?

AvB.~A.B->D.D->~E.H->E:-> ~H
side 1 is true if all conjunctions are true, and side two is true only if 1 is true, correct? Or am I off? For side 1 to be true, the truth values must be :A=F, B=T,D=T,E=F,H=F, same is true of side 2. For it to be false,Any of these, (A=T&B=TVF,D=TVF,E=TVF). Since truth output for side one only comes out from one set of truth values and every other comes out false and in it H must be true by the structure D->~E.H->E heres a truth table for that

DHE D->~E . H_>E
TTT T F F F T T T
TTF T T T F T F F
TFT T F F F F T T
TFF T T T T F T F
FTT F T F T T T T**
FTF F T F F T F F
*The rest of the TT is not needed since H is false in the last two cases, The important thing is the indexed case ** where the proposition comes out true when H is true. for this to hold D must be flase and E true, for this to hold 3 must be such that it falsifies 1 to hold true thus this case is eliminated by the other cases. I assume by the above reasoning that my questions about the form of the proof are going to be answered yes. I'm am not totally sure however, so any help would be appreciated.

[VideCorSpoon]

The inference rules (modus ponens, modus Tollens, Disjunctive syllogism, hypothetical syllogism, simplification, conjunction, addition, constructive dilemma, indirect and conditional nested proofs) and the replacement rules (Communication, association, double negation, demorgan, distribution, transposition, implication, exportation, tautology, and equivalence) start in symposia 8 (although, you have to look for it at this point in time because it is not “stickied” to the rest of the symposiums.) I’m getting there. It’s been my summer mission to put down a simplified account of propositional logic.

It’s funny how we have different accounts for the same things in logic. It’s not a very stream lined system. I had just gone though a previous thread today talking with protoman about the different labels of the same inference.

But to tell the truth, I’m not quite sure what you are trying to state in your post. Are you referring to the general consensus of truth functional logic? Are you reviewing a specific inference rule? Or just the given proof? Etc. It seems like you are grasping at a few things here. But it sounds like you have a valid point to make. I'll be able to answer when you clarify.

[Zetetic11235]

Quine's method makes use of a completely different method using EI(existential instantation), UI(universal instantaiton) ect, with little to no appeal to the method you cite.

I don't know if this is because it is theory of quantification. He doesn't cover proof adise from this.

[Arjen]

Quine speaks of Predicate logic. In predicate logic quantifiers are present: the existential and the universal quantifiers.
Perhaps a new topic would be suited for this discussion? It seem quite offtopic.

[VideCorSpoon]

Arjen is right on the money with his comment. Quine indeed speaks in terms of predicate logic, using universal and existential quantifiers. It is sometimes called quantificational logic or whatever have you. It is a more abstract method, like “there exists some x where” or “there exists some x where.” Like propositional but more hypothetical. But still the method here applies to propositional logic.

[Arjen]

If you ask me predicate logic is where the fun begins. I would like to correct Vide by the way. I'm sure he ment to say it, but he made an error in his typing:

Originally Posted by VideCorSpoon

It is a more abstract method, like “there exists some x where” or “there exists some x where.” Like propositional but more hypothetical.

The existential quantifier means “there exists some x where” and the universal quantifier means "for al x-es goes". Logic gets a lot more complicated here, but it can also far more accurately say what you want to. In predicate logic I can see the basis of all langues coming into view.

I was meaning to ask you by the way Vide, are you going to do an introduction into Predicate logic as well?