posted
I spent like 4 hours trying to research this question today, but to no avail.
Ok, assuming we know that Fact P implies fact Q P>Q, then we also know, and I'm sure of this, that the contrapositive and therefore logical equivalent of that statement is If not Q implies not P. ~Q>~P
now the negation of P>Q and therefore not the case, is P>~Q Right??? and the negation of ~Q>~P which I guess has to be ~Q>P would also be false.
This presents a serious problem(I think) when you look at the fact that the contrapositives of those two new statements are then Q>~P and ~P>Q also have to be false(right?) and so you have a whopper of a logical falacy: If P>Q then ~P(not>)Q So what the Heck's goin on?
Posts: 1103 | Registered: Apr 2002
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posted
Where's the fallacy? Let P="It's a square" and let Q="It's a rectangle." If it's a square, then it's a rectangle -- true. If it's not a square, that doesn't prove it's a rectangle -- also true. Could you give an example where there's a problem?
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posted
um my problem is that if a square is a rectangle, then my 'logic' is saying that if it's not a square, it can't be a rectangle.
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posted
"(Not P) does not imply Q" is not the same as "(not P) implies (not Q)." It's another of many failings of the English language -- it's unclear what's being modified by the word "not." There's no logical problem.
Posts: 6213 | Registered: May 2001
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Actually, it's Old English wifely shorthand for "mind your pints and quarts" said to the menfolk as they strolled out of the house to the local tavern . . . Posts: 5609 | Registered: Jan 2003
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posted
I thought it was said to men in the taverns by the women serving when they felt the men were paying to close of attention to them, though I suppose the wives saying it to them as a sort of joke about not paying to much attention to the girls serving would also make sense, maybe I'm just confused (hey my whole post is one long senteance).
Posts: 733 | Registered: Sep 2003
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posted
Please have patience with me. I dropped symbolic logic (and took eastern philosophy ). Would ~(~p>q) mean doing away with it, or does it act as a negative, and thus you would get p>~q, which doesn't quite make sense. I think it's pretty unnecessary anyway. Since, as Papa Moose said, the existence of p implies q, but that doesn't speak of a dependency of q's existence on p. Unless, it means if and only if p, then q, which I'm guessing it doesn't. Ah, , screw it. I'm gonna go meditate or something. Posts: 4 | Registered: Dec 2003
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posted
Dr.M is teaching logic this semester. I showed him your question and here's what he emailed me:
quote:> now the negation of P>Q and thereforenot the case, is P>~Q Right???
Wrong!!!
The negation of P>Q is P&~Q. If you construct a truth table, you'll find that P>Q and ~(P>~Q) have different truth conditions. In light of that, I guess the paradox unravels.
And Tres, Dr. M says that in order to establish the sentences as truth-functionally equivalent, you need to find equivalent truth values on every line of the truth table, not just the lines where the both sentences are true.
Posts: 3037 | Registered: Jan 2002
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quote: The negation of P>Q is P&~Q. If you construct a truth table, you'll find that P>Q and ~(P>~Q) have different truth conditions. In light of that, I guess the paradox unravels.
Not quite. Just answer me this. What is P&~Q the logical equivalent of? In words? ("if P then possibly or possibly not Q"? No... that doesn't work...) P Q P&~Q P>Q (what I was thinking) T T F T T T F T F T F T F T F F F F T F
What was I thinking! good lord. Just somebody please tell me what P&~Q means in english.
quote:What was I thinking! good lord. Just somebody please tell me what P&~Q means in english.
Sorry, the "&" connective is used to form a conjunction, i.e., "&" means "AND." Many logic texts use a dot or an inverted "V" to signify "AND", but the ampersand (&) is also acceptable.
P&~Q means "P AND NOT Q"
Here is the complete deduction:
1. Start with the negation of P>Q: ~(P>Q) 2. P>Q is equivalent to ~PvQ ("NOT P OR Q") 3. So ~(P>Q) is equivalent to ~(~PvQ) 4. By De Morgan's Law, ~(~PvQ) is equivalent to P&~Q 5. Therefore, ~(P>Q) is equivalent to P&~Q
Here is the Truth table for P&Q :
P Q P&Q T T T T F F F T F F F F
The truth table for P&~Q:
P Q P&~Q T T F T F T F T F F F F
The truth table for ~(P>Q):
P Q ~(P>Q) T T F T F T F T F F F F
Note that the values in the third columns of the two previous tables match up, line for line. That means the sentences P&~Q and ~(P>Q) are truth-functionally equivalent.
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posted
Y'all think you have it rough - I have signed on for a lifetime of this. Dr.M is writing a book of logic puzzles FOR FUN!
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posted
See, I just never see the POINT of symbolic logic. What can it do that regular, everyday logic cannot?
Posts: 37449 | Registered: May 1999
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quote:And Tres, Dr. M says that in order to establish the sentences as truth-functionally equivalent, you need to find equivalent truth values on every line of the truth table, not just the lines where the both sentences are true.
Well, I was going for implication, not equivalence. For A to imply B all that needs to be true is for B to be true whenever A is true.
I did realize, though, that I made the implication backwards, which kinds of ruins the conclusion. Oops!
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quote:See, I just never see the POINT of symbolic logic. What can it do that regular, everyday logic cannot?
"Regular everyday logic" tends to be whatever anyone wants to call logic. People will say stuff like "logic dictates that evolution is true" or "logic says we should invade Iraq" when really it isn't logic at all.
Symbolic logic forces you to actually use logic to prove things, because there is a fixed set of rules, a fixed way of using them, and you can quickly tell when someone is using them wrongly.
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posted
I've never seen any real-world issue answered by symbolic logic that couldn't be answered FASTER without it. Do you have an example?
Posts: 37449 | Registered: May 1999
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posted
I know I should be careful here, but can't help myself.
Symbolic logic is, to me, an extraordinarily effective training program. Developing solution tables quickly can be very helpful in common logic. Building tables quickly, establishing statistical probabilities of occurance of tables not built, managing relevance to the situation and culling solutions to derive a group of 'not wrong' alternatives forms the basis for quick,effective common sense approach to solving the daily problems we face.
Couple that with my strong conviction that we think along the lines of pictorial and spend years perfecting the 'interpreter' for our brain's pictures to the spoken word - this type of thought process fits the internals very well. It is quite similar to the way we train our interpreter function.
If we learn to combine these analysis issues as the common way our thought processes function and then approach life in the Zen like - 'stuff happens', we will be able to respond quickly to the challeges presented without having to have established all the links to possible outcomes.
Whew - sorry about the length and possibly confusing scontent...
Posts: 46 | Registered: Dec 2003
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quote:Symbolic logic forces you to actually use logic to prove things, because there is a fixed set of rules, a fixed way of using them, and you can quickly tell when someone is using them wrongly.
Ah! That's a good point. It's a rather drastic way of achieving this end, though. I'm quite sure that precise use of language does the same thing, and while there are precious few people who use language precisely, the numebr can't be much smaller than the number of people who understand symbolic logic. Posts: 2443 | Registered: Apr 2002
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quote:I've never seen any real-world issue answered by symbolic logic that couldn't be answered FASTER without it. Do you have an example?
Yes - symbolic logic has the added bonus that you can program machines to follow the simple rules and solve problems far faster than a human could. That way you can find the best move in checkers far quicker than you could normally deduce it.
Normally in philosophical argument, though, symbolic logic is used more for ACCURACY than SPEED. If you translate an argument into symbolic logic, usually it's because you're confident it's good reasoning and want to prove it.
quote:I'm quite sure that precise use of language does the same thing, and while there are precious few people who use language precisely, the numebr can't be much smaller than the number of people who understand symbolic logic.
Actually, using English precisely will still be vague. You get things like "Joe's friend waited while he was changing his shirt." Is the "he" Joe or the friend? There's tons of stuff like that. Symbolic logic is designed to be a language that doesn't have that sort of vague stuff.
posted
Its not a question of answering it faster, its a question of proving it. Natural language skips steps -- just take a look at the above thread where slight differences in notation in the rather concise manner of logic resulted in considerable debate -- and this is with a completely precise system!
Of course, I would suggest that computers run on symbolic logic might be a good reason for its usefulness :-) .
And of course, here's a useful challenge: name one natural language document (one that has been the subject of considerable inspection, preferably, and is at least a couple paragraphs long) which meaning is completely agreed upon, including connotation. I can show you scads of symbolic logic derivations that have that property.
And of course, you may have heard of the predicate calculus, which has been used to great effect in mathematics. That is symbolic logic.
Try to solve a problem of that nature with natural language. The mathematically precise nature of symbolic logic enabled a solution to a problem that would have had conventional methods of deriving business practices floundering in details.
That last problem illustrates one of the more commonly useful employments of symbolic logic: equivalence proving. Natural language is usually too inexact to prove equivalence, instead relying on mere similarity.
Then of course there's electric circuit design (closely related to programming). It is absolutely essential that electricians be able to prove a complex circuit works, not merely suggest it. And that proof is usually quite straightforward because of symbolic logic. Not only that, but the derivations of symbolic logic combined with known circuit equivalencies lead to the creation of more efficient and useful circuits.
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quote: -------------------------------------------------------------------------------- What was I thinking! good lord. Just somebody please tell me what P&~Q means in english. --------------------------------------------------------------------------------
Sorry, the "&" connective is used to form a conjunction, i.e., "&" means "AND." Many logic texts use a dot or an inverted "V" to signify "AND", but the ampersand (&) is also acceptable.
P&~Q means "P AND NOT Q"
Here is the complete deduction:
Oops, not neccessary, and I understood all of the above already... sorry to put you through the trouble Mrs. M, I guess I worded my question wrong.
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posted
Really, what I should of asked, I suppose: what would "P and not Q" mean if you translated it out of symbolic logic and in to written english by using some sort of example.
Like for the case of P>Q, you could say, if a dog, then red. That pretty much sums up "if P then Q". Is there an equivalent thing you could do for P¬Q? That's what I'd really like to know. Actually, if you can actually give a singular answer to that question, it would go a long way, in my mind, toward answering the question as to whether Symbolic logic is useful.
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posted
For those of you that don't know, the reason I started asking this question is my statements here which I tried to translate to symbolic logic, but couldn't for the life of me figure out how.
quote:Really, what I should of asked, I suppose: what would "P and not Q" mean if you translated it out of symbolic logic and in to written english by using some sort of example.
First of all, keep in mind that P and Q stand for sentences. So, if you're still looking for an example, let P = "Dean is president", and let Q = "hell is freezing over." Then "P and not Q" = "Dean is president and hell is not freezing over."
So, strictly speaking, the negation of the sentence "If Dean is president, then hell is freezing over" is "Dean is president and hell is not freezing over." When one of these sentences is true, the other must be false.
quote:Like for the case of P>Q, you could say, if a dog, then red. That pretty much sums up "if P then Q".
Come on, man! Is that actually a sentence in our language? "If a dog, then red"? What in the world does that even mean?
P and Q, or whatever letters you want to play with, ALWAYS stand in for well-formed sentences. The logical connectives ('and','or', 'if...then', etc.) combine those sentences to make (usually) more complex ones. So P can't stand for just a noun.
Posts: 3037 | Registered: Jan 2002
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posted
Yes I suppose that I'd have to cede that you are correct, and I suppose I shouldn't really have to be told that because I have studied that enough, its just that symbolic logic does not seem very practical put in that way, and it makes me wanna lean towards agreeing with ae:
quote: Is symbolic logic actually worth spending time on? I've never been convinced
posted
Learning to cope with skimmers is one of the aspects of Net fora I never fully mastered
Nevertheless, I'll come to his defense. "If dog, then red" has the same flavor of many/most operations in control theory.
Posts: 1839 | Registered: May 1999
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quote: Did you perchance read my explanation of the many uses of symbolic logic?
Well, skimmed it before , Read it now. I try to skim pretty thoroughly, though, and I meant no disrespect. I do not know, however, why you think your statements are insightful with regard to my last comments. Posts: 1103 | Registered: Apr 2002
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posted
You ask if symbolic logic is worth spending time on. I pointed out in my earlier response that symbolic logic isnecessarily employed in computer engineering, equivalency proofs, electrical engineering, and ssuch.
If someone didn't spend time on symbolic logic, you wouldn't be visiting hatrack, because there would be no hatrack. And so on, as regards computers and various electrical circuits and such.
Is it worth spending time on for you? You didn't ask. Its an excellent foundation for philosophy, and should be required for many branches of it (such as epistemology). Its rigorous approach is often lacking among those seeking to enter the field, and does them good. It helps to understand things such as computers; if you want to go into CS, particularly if you want a to do postgrad, a course or two on symbolic logic is really useful.
I strongly recommend it for pretty much anyone in the liberal arts, in fact. People are often far too absorbed in their own intuitions of what is true to bother with proving (or even coming close to proving) anything, and symbolic logic is a lightweight way to get past that.
Oh, and a bit of expansion on why it being rigorous is a useful response to Tom's assertion that one can at least come close to doing the same thing in natural language using simple statements.
Consider mathematics. I can say "add two to three, then divide by four" et cetera and describe a mathematical statement. With enough care, I can even be precise. However, there are two side effects of using language instead of notation: its a lot easier to hbe ambiguous and have a disagreement as to meaning (and if one doesn't have notation to fall back on this can be tremendously hard to work around), and it is a more efficiently extensible vocabulary. One can very quickly "glom" statements via known substitutions and equivalencies that are harder to notice in natural language.
The way you write something can be important to the development and advancement of that thing; humans deal with different notations in different ways.
Imagine writing out a matrix in english "you have a four by five matrix, the first row is five, three, eight . . ." et cetera, for quite a while.
If that's an outcome for a rectangular game, for instance, I can see at a glance that the third row dominates the fourth row, which means we can quickly simplify the matrix into one with equivalent optimal strategies and value. That relationship is not so immediately noticeable when the matrix is written out by english.
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