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Some stuff on p-logic, derived rules, etc (unlikely to be helpful)

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» Some stuff on p-logic, derived rules, etc (unlikely to be helpful)

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Derived rules are where it's at. That's when I realized how fun logic could be...that's when I began to do "Ninja Logic."

The most basic, or primitive, logical operators are negation (~, -) and logical-and (.,?, &, ^). It's important, of course, to point out that these two concepts are similar but not the same as "not" and "and" in natural languages. Supposing that these operators are primitive, all the rest are luxuries. These include: or (v), biconditional (<->) and implication (->). Biconditional is easy enough. It's simply * > # with the symbols switched, thus creating a new formula and adding it as conjunct to the original (by conjunction inclusion).

Any statement in propositional logic that involves the luxuries is logically equivalent to some statement that is expressible in the primitives. We get a whole slew of combinatorial possibilities (most of which are introduced in logic 101 courses as "derived rules").

For instance, "A > B" is logically equivalent to "-(A . -B)," and vice versa.
Also DeMorgan's Law (or Theorem), "A v B" is logically equivalent to "-(-A . -B)."

What logically equivalent means, for practical purposes, is that from one formula as the sole premise, you can deduce (derive) the other, and vice versa.

What I'm getting at, however, is this: there's a pattern.

_ > _ translates to -(_ . -_)
_ v _ translates to -(-_ . -_)

_ . _ translates to -(_ > -_)
_ . _ translates to -(-_ v -_)

_ > _ to -_ v _
_ v _ to -_ > _

-(_ . _) to (-_ v -_)
-(_ v _) to (-_ . -_)

Essentially, translating a formula consists of rearranging or changing its elements and introducing or taking away negation symbols.

1. For implication to logical-and, you always change the right element by negating it (sometimes when you negate, you'll get double negation), then negate the entire formula, and finally change the logical operator, which connects the two elements, to the opposite (between implication and logical-and). Call this "Playing politics."

2. For logical-or to logical-and, you negate every element inside the formula, negate the outside, and then change the logical operator (or connective) which links them. Call this "DeMorgan's Law."

3. For logical-or to implication, you negate the left element of the formula then change the logical operator (or connective) to its opposite (between logical-or or implication). Call this the "Arrow Rule."

4. And vice versa, and so on...

Other fun things I enjoyed doing in p-logic involved leaving a huge space between my conclusion and premises so that I could translate premises into logically equivalent formulae. This way, you might see a derived formula (a logically equivalent one) that might make solving your problem much easier.

For instance, if you see:

1. A v B
2. -A
3. B > C
C. C

Setting up two conditional proofs ("A>C" and "B>C") to get your conclusion (which would take like 10-13 lines) is tedious and boring. An easy and fun way out of monotonous writing would be be to:

1. A v B | Premise
2. -A | Premise
3. B > C | Premise
4. -A > B | Arrow Rule from 1
5. B | Modus Ponens from 4, 1
C. C | Modus Ponens from 3, 5

At 4, you easily buy your way out of a long proof because you can then simply Modus Ponens your way out. This is, of course, very ninja.

Of course, 4 could be translated to -(-A . -B). Pretty tough, yeah? Well, this introduces another tactic I enjoy. Call it "building."

1. A v B
2. -A
3. B > C
4. -A > B
5. -(-A . -B) | negated the right, negated the outside, changed the connective; from 4
X.
C. C

Now what could I possibly do with 5? It's pretty ugly. We could "build" from our preceding resources (our premises and likely our viable provisional assumptions) something to contradict 5.

1. A v B | Premise
2. -A | Premise
3. B > C | Premise
4. -A > B | Arrow Rule (negated the left, changed the connective) from 1
5. -(-A . -B) | Arrow Rule (negated the right, negated the outside, changed the connective) from 4
6. -C | Provisional Assumption
7. -B | Modus Tollens from 3, 6
8. -A . -B | Conjunction inclusion 2, 7 *
9. -(-A . -B) . (-A . -B) | Conjunction inclusion
C. C | Indirect proof 6-9

Or we could've just got 5 from 1 by DeMorgan's Law. But the idea is that I setup 8 for no other reason than my intuitively understanding that I had the resources present to get a contradiction with 5.

A smart mentality to have as one gets into p-logic proofs is that you are playing with Legos or building blocks. With every argument, you are given some building blocks that fit or do not fit with one another. What you make out of them depends on the rules you wish to play, but in most cases, you'll use all of the blocks given. It's always important, though, to look back at your resources to see if you can use anything later on down the proof.

@nerdfiles,

I guess I am more into argumentation and informal logic than formal logic. But thanks for the information. Maybe I may actually think about the construction of my arguments a little more. My "proofs" are generally well reasoned or well supported arguments for some point. I guess you could say that I am not an analytic philosopher at all.

@nerdfiles,

I would not dare suggest that anyone take this stuff into the social world. More or less, this kind of logical wankery ought to be left to private reflections and private puzzle solving. It's practically useless outside of "pure argument" or "on-paper argument."

However, it is useful if you wish to get to the "core" of arguments received or made on one's own behalf. Doing this stuff might help you see clearly what the nature of the argument itself is. However, playing around with derived rules likely won't really help you see that.

I mean, for instance, since logical-and and natural-and have different features, if you take a sentence which contains natural-and and translate it into formal logic, you might thus lose the semantic (or pragmatic) meaning of that sentence. Translating it into logic, thus, would do you a disservice. However, it might be help to translate natural sentences into logical sentences, so that you might use logic as a "skeleton" for analyzing the varied meanings in natural linguistic discourse.

Analytic philosophers usually go no further than simple Modus Ponens, Modus Tollens, Conjunction, Disjunction, Hypothetical Syllogism, Disjunctive Syllogism, Negation, etc.

If p-logic is hardly useful in natural language discourse, then derived rules is much less useful than logic. Another problem is that propositional logic is very pure as an artificial language with respect to, say, English. To hazard an analogy: Think of p-logic as binary language (in computers); it just happens to use our English symbols as its own. It's resemblance to natural language is very, very shallow. Binary language doesn't afford us much use in everyday conversation, and p-logic is like that.

Predicate logic can incorporate some of the richness of natural language, much better than p-logic, but it still has its difficulties.

Perhaps I'm too pessimistic. Essentially what I want to say is that p-logic can help you understand the gist of what you're arguing or that which you're arguing against, but in some cases, it can strip away stuff essential to your argument that is given by the essentials of your natural language.

For instance, "The cat is purple and green." The best p-logic can do is say "The cat is purple" and "the cat is green." But nothing about the statement itself requires that we incorporate the term "mixture of ..." Predicate logic has the same problem.

And sometimes predicate distinctions and gradation is important. In fact, we might get into an argument about whether someone is running quickly or slowly. And p-logic would just be a useless way to solve that debate, supposing it really is one (debates about perceptions are a waste of time anyway, but I hope you get my point).

@nerdfiles,

Thanks for posting this stuff. I pretty much focus on ancient Greek philosophy, empiricism, and applied ethics so it is always nice to be subjected to things outside my realm.

I always laugh when things should be left to the private realm philosophically speaking.

@Theaetetus,

A few questions.

nerdfiles wrote:

Also DeMorgan's Law (or Theorem), "A v B" is logically equivalent to "-(-A . -B)."

Is this your version of DeMorgans or the standard version of DeMorgans? DeMorgans has always been from what I understood ~(PvQ) may replace or be replaced by ~P & ~Q or ~(P&Q) may replace or be replaced by ~Pv~Q. Are you saying there is a better way to prove the DeMorgan Principle? I agree with half of what you say in your #2 below the pattern examples, but how does that correlate with the quoted statement?

In regards to the easier way to solve a proof, I don't see how you can derive ~A -->B (a straight conditional) from line #2. This seems like a broken proof and an illogical assumption.

I disagree with a lot of your bi-equivalence translations for the theory you are positing though. It seems like something too far out to suppose that corners could be cut like this. But it does seem interesting to suppose it could be done. Good stuff!

But I do agree with you about comparing legos to propositional logic. It just seems that we are building that awesome pirate ship they used to make and are left with a large handful of pieces. I think the rules are a matter of choice yes, but to a point, haven't we negated the actual rules themselves which some what standardize formal logic?

@VideCorSpoon,

VideCorSpoon;48506 wrote:

Is this your version of DeMorgans or the standard version of DeMorgans? DeMorgans has always been from what I understood ~(PvQ) may replace or be replaced by ~P & ~Q or ~(P&Q) may replace or be replaced by ~Pv~Q. Are you saying there is a better way to prove the DeMorgan Principle? I agree with half of what you say in your #2 below the pattern examples, but how does that correlate with the quoted statement?

-(A v B) = -A & -B
-(A & B) = -A v -B

These are the standards, yes. I'm not so much introducing a version as I am highlighting a general form, regardless of the actual elements (-A, B, etc) of the formula.

The example I give: "A v B" is logically equivalent to "-(-A . -B)"

is clearly no different, in the sense of form, from "-A v -B" to "-(A & B)" the negation symbols have simply been subtracted from their respective formula. I think this helps highlight the systematic nature of the Law.

If you get "-(-A v B)" as premise, I'd like for up-and-comers in logic to immediately think "DeMorgan's Law." I want to move them away from looking at "symmetry" and get them focused on "systematization."

"A & -B" is much nicer to deal with than that mess. When I first learned DeMorgan's, I'd frequently ignore it because I was caught up in looking for "_ v _" and "-_ . -_" and so forth... symmetrical formulae. I dislike the notion that symmetry in logic is a good thing. My professor thought symmetrical proofs were the best. I don't like symmetrical proofs nor do I like using "symmetrical markers" (formulae) to guide.

I like to use DeMorgan's to get myself out of hairy arguments, even when I don't see the staple, standard DeMorgan's expressions. I consider them "examples," however...and thus not essential to the proof itself. The standard examples DeMorgan's provides help you better see the duality, but I don't think they're the only "valid" expressions of DeMorgan's Law.

Quote:

In regards to the easier way to solve a proof, I don't see how you can derive ~A -->B (a straight conditional) from line #2. This seems like a broken proof and an illogical assumption.

Quote:

I disagree with a lot of your bi-equivalence translations for the theory you are positing though. It seems like something too far out to suppose that corners could be cut like this. But it does seem interesting to suppose it could be done. Good stuff!

I admit, they can be ugly at times, but they're all logically equivalent, and I can show you the proof using no other derived rules. Any particular one you have a problem with?

Quote:

But I do agree with you about comparing legos to propositional logic. It just seems that we are building that awesome pirate ship they used to make and are left with a large handful of pieces. I think the rules are a matter of choice yes, but to a point, haven't we negated the actual rules themselves which some what standardize formal logic?

I don't take myself to be negating the rules, only underscoring their power.

For instance, ----------P

If you saw that as a premise, wouldn't it be easier to just write

----------P | Premise
P | Double Negation

rather than

----------P | Premise
--------P | DN
------P | DN
----P | DN
--P | DN
P | DN

It just gets tiresome! Weird stuff like -(-P & -Q) clearly translate into P v Q by DeMorgan's. But this example isn't DeMorgan's own example. There's no justification, as I see it, that "DeMorgan's Law" is really just "DeMorgan's Two Examples." If we accept this, then we miss, and thus do not act upon it, the awesome fact that it's a "Law" which is supposed to stand the test of even weird premises like -(-P v (Q v R)). Two DeMorgan's and you've got an easy three conjuncts P, -Q and -R.

-(-P v (Q v R)) == (P & (-Q & -R))

It's fun to be able to see that right off the bat.

@nerdfiles,

In regards to DeMorgans, it is interesting to see how what you are positing is helpful to up-and-comers in logic. "It is not the case that it is not Alan and (supposing this is the period symbol you are using) not Barry" seems more epideictic than functional in comparison "it is either Alan or Barry." This makes it less practical in my opinion. But I think your thoughts on an anti-symmetrical approach to logic is fascinating though? I admit I have never encountered that before. Problematic but interesting. But from what I catch from your thoughts on DeMorgans, you are stating that there are other interpretations of syntactical structures open to DeMorgan's law. That is a very interesting position and I am very curious to hear your thoughts on that. You may have a very good point on that note with good examples.

On the second quote of mine, LOL! Sorry about that, I am seeing two's instead of one's. But even in the case of ~A -->B derived from A v B, I don't see the logical equivalence. #4 states something drastically different than what can be inferred from line #1. If "if Alan does not play, then Barry will play" is logically equivalent to "either Alan will play or Barry will play," it is completely ignoring the fact that "either Alan will play or Barry will play" is an open ended, non declarative statement compared to a definite declaration in ~A-->B, where Barry will play and Alan will not. So, no I don't see it. It is plausible in some respects to suppose that #4 can be derived from #1, but that seems more something deserving of a provisional assumption than a derivation. Surely you see this. You may be following the dreaded inductive path of a deductive process.

In the case of ----------P, I thought the prime concern of a double negation was to double negate the entire variable... which of course it is, but in what is this example useful? The example seems rather redundant. Does this redundancy extend to DeMorgans. I don't think so.

As to the "it's fun to be able to see that right off the bat" remark. Well? that's a very "interesting" statement.

@nerdfiles,

Well, for logical equivalence, I steer clear of using English representations. Our professor made us prove some pretty weird stuff at times, stuff I couldn't stand to see as English.

I mean "It is not the case ..." alone is just too awkward.

Anyway, I think I see what you mean.

Though, A v B is logically equivalent to B v A. And that's why it gets off the ground because the same holds for -B > A.

A v B
--(-B & -A) | Provisional Assumption
-B & -A | Double Negation
-A | Conj out
-B | Conj out
-A & -B | Conj in
-(A v B) | DeMorgan's
(A v B) & -(A v B)
-(-B & -A) | Reductio
B v A | DeMorgan's

And vice versa.

So it's odd...

It would see that not only is "-A > B" logically equivalent, but "-B > A" is as well.

Sure, A v B says nothing about a particular state of affairs, but it's also important to keep in mind that in logic, logic-or is typically used in the inclusive. Thus, AvB is really (AvB) & (A&B).

Now is this logically equivalent to both the conditionals thus produced?

(AvB) & (A&B)
-B | Provisional
-A | Provisional
A&B | Conj out
-A&-B |Conj in
(-A & -B) & (A&B) | Conj in * contradiction
A | Reductio
-B > A | Conditional

(AvB) & (A&B)
-A
-B
etc etc etc
B
-A > B

So it really depends on the interpretation of OR.

Basically, A v B is logically equivalent to both -A > B and -B > A.

If you think about it, conditionals aren't really definite either.

For instance, if I say "If you go outside, I will shoot this cat." I am not thereby saying "I will shoot the cat" on no grounds whatever. You must satisfy the antecedent, which has the same feeling of "open-endedness." Arguably, by my saying such a thing at all, you'd likely think I intend to shoot the cat. Obviously the cat's done something that offends me, and you going outside seems hardly relevant. But for all that backstory, logic doesn't and shouldn't care. Still a conditional, still "open ended."

Maybe I'm way off, and it's rather that "either-or-but-not-both" (exclusive-or) that is "open ended," or at least gives the feeling as such.

Effectively, logically equivalence is defined as this: If one formula can be proved to follow from nothing other than the supposed logically equivalent formula, then that proved formula is logically equivalent to the supposed.

Or two formulae are logically equivalent if and only if you can prove one from the other alone as its premise, and vice versa.

So

A v B == -A > B
A v B == -B > A

-B > A
-A & -B
---
Clearly you see the Modus Ponens to Contradiction
---
-(-A & -B)
A v B | Conclusion by DeMorgan's

A v B | Premise
A | Provisional
-B | Provisional
A | Copy / Reiterate
-B > A | Conditional
A > (-B > A) | Conditional
B | Provisional
-B | Provisional
-A | Provisional
B & -B | Conj in * Contradiction
A | Reductio
-B > A | Conditional
B > (-B > A) | Conditional
-B > A | Conclusion Disjunctive Introduction

I hate doing these...which is why I always translate OR-premises to logical-and's or at least implication formulae.

@nerdfiles,

I think that English representations are the basis for this type of logic in the first place. And just because proving things may be weird, it does not negate the fact that it could in fact logically be true. That's the interesting thing about logic? it's a closed system where the sky could be green and unicorns could roam the earth devouring small kittens. But I see what you mean in some respects about it being awkward. I use logic as a practical tool though, which requires a syntactical relation to whatever may need to be done.

In regards to the AvB and BvA, etc, that's the whole thing behind the communtation rule I suppose. Not odd as much as mathematically sensible. The line will tabulate no matter which way the disjunction is formed.

In regards to "Thus, AvB is really (AvB) & (A&B), I don't agree. Syntactically it won't work? more is needed. But I do agree with your statement about the disjunction being typically inclusive. The three variables of possibility implied in a disjunction (one, other, both) is much more broad in usage than the exclusive sense in which a disjunction is just "either, or." That seems a simple conception though.

That conditionals "aren't really definite either," I disagree. A conditional is a solid statement? it is not a "if, maybe," a conditional is a "if, then." I don't follow your statement about a conditional being "open ended." Somehow I think there is some disregarding of the necessary/sufficient condition aspects of a conditional.

As to logical equivalence, I interpret it as simply implying the other? namely that in all situations they have to have a matching truth value. Your interpretation is interesting though. So when you then state following that statement about logical equivalency;

A v B == -A > B
A v B == -B > A

I have to disagree with this because logical equivalency does not function in an amalgamation of statements. The truth values of each of these would be different and not logically equivalent. Now if there was (P-->Q) & (Q-->P) or (P&Q)v(~P&~Q) then there is some possibility of bi-equivalence.

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