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.

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).

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.

A few questions.

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?

-(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.



You mean this one? It doesn't. It derives from 1.

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

Surely you see this.

Suppose:
A v B | Premise
-A | Premise
Therefore, B
Hence, -A > B

A v B | Premise
-A | Provisional Assumption
-B | Provisional Assumption
-A & -B | Conjunction Introduction
-(A v B) | DeMorgan's Law
(A v B) & -(A v B) | Conjunction Introduction -- Contradiction
B | Reductio
-A > B | Conditional Proof

The other way is just as easy, for they're logically equivalent.



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?



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.

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 batinteresting" statement.

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.

s. 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.