MathematicalPhysicist
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i mean rules such as demorgan's, material conditional, conditional proof and so forth.
This seems to have truth and validity mixed in with it. Here are rules where p and q are formulas and $ is an operator.NickJ said:the rule could give the conditions under which it is appropriate to assert sentences containing the operator, and conditions under which it is appropriate to make certain inferences on the basis of sentences that contain various operators; and I don't see that there's more to the meaning of an operator than what we are permitted to do with it (how we properly use it).
Can you prove that?logicalroy said:There are about 21 formal rules of deduction.
There is a valid, deductive inference form called (mathematical) induction. It is not the same thing as inductive inference.Induction doesn't count because that is just another way of saying scientific method.
Reductio ad absurdum is not included in Conditional Proof, and some of those are equivalencies, so you should count them as 2 inference rules.The rules (or Principles if you want to use that term are all verified by you typing them in google search on the internet) are as follows: modus ponens, modus tollens, disjunctive syllogism, addition, simplification, absorption, hypothetical syllogism, constructive dilemma, destructive dilemma,Conditional Proof (another way of saying 'Assumed premise' which includes "rectio ad absurdum"), DeMorgan's Theorem, double negation, distribution, association, material implication, transposition, commutation, material equvialance, exportation, and tautology. In some logic books the names may differ but the forms are identical (such as conditional elimination is another way of saying Modus Tollens).
Logic certainly can be and is used to model physical systems and make predictions.Again, I repeat sciencfic reasoning is different and ony allows probability from inferences where deduction if done correctly must yield certainty. A deduction that does not lead to certainty is either fallacious or done incorrectly. Logic is not used for predicting the future literally or some magic that can solve all of the world's problems but reveal the world as it is before humans changed things. People refer to the laws of logic because they were already present. All humans did was discover what was "already there"; as opposed to some mad scientist or mind freak making the rules up from his head in his basement back in the day. The same thing goes for astronomy and chemistry. Did the astronomer invent those planent in the solar system? No! The chemist invent all of the elements on the periodic table? No! They just reported what they found to the public. Right?
The completeness of Copi's system of natural deduction..honestrosewater said:Can you prove that?
Okay, there's no point in arguing about it. Who has the burden of proof is something we would have to agree on. As far as I'm concerned, you made the claim, so it's your responsibility to prove that it's true. Your claim doesn't get to be true until someone else can prove otherwise.logicalroy said:I gave the titles of two published books. I also listed the rules. You want proof? I provided some. You have the burden of proof to show that I am mistaken. Is not like I made that stuff up.
because it isn't specific enough -- it's ambiguous -- you need to give more information. As it stands now, it's just a source of confusion for people who don't know better.There are about 21 formal rules of deduction.
Induction is a form of argument. In the most general sense that I can think of, it's an argument in which you conclude something about every member of some collection C from the facts that (i) that something holds for every member of some subcollection of C and (ii) if it holds for every member of that subcollection, then it holds for every member of C. So it's actually a special case of modus ponens, which was the very first rule that you listed. You can formulate induction as a semantic or syntactic argument, and it isn't restricted to math. Inductive proofs are used all over the place. A similar idea is also included in recursive definitions and logical calculi themselves.Mathematical Induction is not "LOGIC", but a system of math found in the matematics department.
Formal logic, the subject of this thread, is one of the foundations of math. What is this distinction I need to make? I doubt you or anyone else could clarify the distinctions between logic, math, and philosophy. What good would those distinctions be anyway? If a logician needs to know something about what you classify as math or philosophy, they still need to know it.Clearly you need to make the distinction between math and logic.
Induction does not involve only numbers. It is a logical argument -- it can apply to anything. I'd like to know what logician or mathematician could do without it.Not one of the rules I listed involve ONLY numbers as your mathematical induction does.
Sure. The University of London. Oh, wait, that's actually several universities. MIT then. So what?Can you name any university on the planet where mathematical induction is taught under the "Philosophy" curriculum?
Again, I think your reasoning is backwards. It's not my responsibility to refute every claim you can dream up, which is assumed to be true in the meantime. Little green aliens visited me last night and told me so.If your reply is "NO" then your mathematical induction is not real "LOGIC".
Math couldn't exist in its current state without deductive logic. What on Earth do you think math is? Anyway, you fail your own test here, as plenty of math departments offer logic courses. This is not surprising given that an area of logic, commonly called 'mathematical logic', is one of the foundations of math.Deductive LOGIC falls under the academic disipline of Philosophy --not math.
Where did I do that?You mistakenly seem to believe all logic falls under the same category, which they DO NOT. Science is science & Deduction is deduction -- two distinct things (that you seem to mix).
First, it doesn't follow that two things are the same because they have two properties in common. In order for two things to be the same, they must not differ in any property.Induction and the scentific method are the same becasue both requires experiment and does not always lead to certainty;
You have this backwards. Logic is not concerned with soundness but with validity. Indeed, you could say that vallid reasoning is exactly what logic is all about.whereas Deductive logic does always lead to certainty without performing a sigle experiment. A sure sign of misunderstanding logic is to focus on "Validity". People who lack the proper information focus on validity. People who know more about logic don't focus on "validity" but they DO focus on "Soundness". Sound arguments reflect reality, whereas just valid arguments can be pure fantasy and have nothing to do with anything. All sound arguments are valid and they refelect reality which is why profesionals and people with better understanding of logic care about "Soundness."
Okay, this could just be a terminology problem. I know of a rule that goes by Conditional Proof (which says that if you can deduce a formula q while assuming a formula p, then you can infer the formula (p imples q)). If you meant a hypothetical rule, a rule that involves assuming some formulas as hypotheses, sure, Reductio is a hypothetical rule. But hypothetical rules are types of rules; it's not a single rule itself. You listed Conditional Proof as a rule.Rectio ad absurdum is a form of conditional proof. Read the books I mentioned. Secondly, tell me how can one do rectio ad absurd without an assumption? All conditional proofs (Conditional introductions) must have assumptions based on the rule.
There are plenty such books because there is more than one type of logic and some of them don't include propositional logic. For example, check out a book on intuitionistic logic.Show me one logic book where the rules do not apply.
This is very wrong. Predicate logic adds variables, operations, relations, and quantifiers, which are significant changes, and there are again infinitely many calculi for it.Predicate logic only added two quantification rules, which I did not list.
Huh? Some logics 'include' or are included in other logics, and some logics don't include or aren't included in other logics. What do you mean by 'progressive'?All of the systems of deductive logic are progressive. Each one picks up where one fails.
If you put all of the systems together, the result would be inconsistent. So if you mean a syntactic completeness, whether you're wrong or right depends on whether your definition allows for inconsistency, but who cares either way? And if you mean a semantic completeness, you're wrong because an inconsistent system can't have a model.All the systems of deduction combine to make a complete system.
Thanks, but that isn't the claim I was questioning. I am looking for proof of exactly what he said: 'There are about 21 formal rules of deduction.'loseyourname said:
This is still correct the way that I meant it, but it doesn't apply if you define completeness so that the sets of premises are non-empty, as he does there. This might seem like a small point, but it makes a big difference, namely, the difference between deriving infinitely many formulas and deriving none. So, yeah, my bad there, which just shows once more the importance of clearly defining what you're talking about.The set of rules in Copi & Cohen isn't even a complete calculus for propositional logic because it has no hypothetical rules. You need at least one axiom in order to get any theorems at all. And if you want all and only the theorems of propositional logic, not just any set of axioms will work. So giving that set of rules without also giving the corresponding set, or sets, of axioms is at best incomplete, if not plainly wrong.
Well, there is certainly a way of defining things so that that is true -- there is way of defining things so that any statement is true. The main problem with this thread is that no one has defined rules or the related concepts well enough to be able to determine these things.Note that this is not a complete system for predicate logic, but rather propositional or sentential logic. Copi's system is also an older system. An alternative can be found in the wikibook on Formal Logic in the Inference Rules section.
I do personally prefer Copi's system because it just feels more elegant, but that is only an aesthetic judgement. What I said earlier about reducability was a misnomer. What I meant was that I believe Copi claims his nine rules (roy is also introducing equivalencies, which are not strictly the same thing, but can certainly be used in a deduction) constitute the fewest number of rules that can be used to construct a complete system of natural deduction. Looking back through the text, however, I'm not sure where this claim came from. It might have been my professor, and not Copi himself, that made the claim, or it could be in here somewhere and I'm just not seeing it.
Ok, now you have to prove that there are “any number of inference rules” since you made that claim! Where’s your proof? Are you empty handed again??Honestrosewater said:“There are many types of logic, and the set of calculi for some of them is infinite and includes calculi with any number of inference rules. I have already explained this.”
Are you kidding? You can’t be serious! HYPOTHETICAL SYLLOGISM is a hypothetical rule as well as conditional proof.Honestrosewater said:“The set of rules in Copi & Cohen isn't even a complete calculus for propositional logic because it has no hypothetical rules.”
Again, where is your proof of this being true? I realistically pointed out that in math, logic is taught differently as opposed to the way philosophy teaches logic. This could be why you are so caught up with “Validity”.Honestrosewater said:“You need at least one axiom in order to get any theorems at all.”
This applies only to math. Propositions don’t come with axioms in reality. If I sate a proposition that “all neo-Nazi skinheads live in the southern United States” where is there an axiom?Honestrosewater said:“So giving that set of rules without also giving the corresponding set, or sets, of axioms is at best incomplete, if not plainly wrong.”
You can’t see that those rules can’t be combined into one? Notice please, they are independent of another and can stand alone just like an independent clause can stand alone in grammar. You can combine all of them? You are the first! You should publish this fascinating theory. Proof anyone? Again, empty handed? You and your imagination! You can combine them, ha! Yes, you can just like I can combine all of the chemicals on the periodic table of elements as taught in chemistry into one element. If I did that on chemistry class test I would surely FAIL.Honestrosewater said:“Also, for any list of rules that you give, there is nothing to -- though something in your answer should -- stop me from just conjuncting those rules together into one, or any other possible number, of longer rules. Indeed, this is routinely done with rules, including some of the ones you listed -- what could be counted as separate rules are written together as what gets counted as one. There isn't one, fixed form for rules or one way of counting them..”
You stated MATHEMATICAL INDUCTION specifically. Now you are substituting that for general Induction. Induction is different from Mathematical Induction which you pointed out:Honestrosewater said:“Induction is a form of argument.”
Do you see the little bait and switch you committed above? Induction is SCIENCE because it does not always lead to certainty. Deduction always leads to certainty when done correctly. Induction does generalize more because the facts are not logically necessary. How does one know inductively that something holds for every member of a group without evidence? Again are you empty handed?Honestrosewater said:“There is a valid, deductive inference form called (mathematical) induction. It is not the same thing as inductive inference.”
Pay attention to the last sentence I quoted you on. “It’s actually a case of modus ponens . . . “ So it should be called MODUS PONENS, DUH! Why do you give it a synonymous name (Induction or Mathematical induction)? You agree that it’s the same rule! Are you Ill? On medication? The rules I mentioned are not limited to math. Mathematical induction (not Induction in general) is limited only to math. General Induction is scientific theory, which requires evidence (and or experiment) –deduction does not. There are only two Roots of LOGIC: deductive types and inductive types. All forms of logic fit under one of the two Roots (deductive and inductive) and they branch out to form different types of logic as you pointed out. Math and deduction are different! Deduction includes a lot of semantics; math does not. Math deals primarily with numbers; deduction does not. Yes logic is the foundation of math, but math branches away because some math formulas are not deductive formulas for propositions or arguments. How does the quadratic formula help one analyze an argument? All logic is not math and all math is not logic point blank!Honestrosewater said:“Induction is a form of argument. In the most general sense that I can think of, it's an argument in which you conclude something about every member of some collection C from the facts that (i) that something holds for every member of some subcollection of C and (ii) if it holds for every member of that subcollection, then it holds for every member of C. So it's actually a special case of modus ponens, which was the very first rule that you listed.”
Show me proof that logic is not concerned with soundness! Are you empty handed again? Yes, you surely lack knowledge because you exaggerate “validity” way too much. You are excessive with it. ALL SOUND ARGUMENTS ARE VALID. So why do you avoid soundness? What is your beef against soundness if you value validitiy so much? Are you unrealistic? Do you think there are millions of hypothetical rules? Where’s the proof? The hypothetical rules are limited and I listed every occurrence in my initial post. Next, your claim there are different types of logic is true but all fall under a deductive or inductive banner. Intuitionistic logic is not a deductive based system, but mathematically based. Read the link you posted. WELL LET ME POINT IT OUT: “Intuitionistic logic encompasses the principles of logical reasoning which were used by L. E. J. Brouwer in developing his intuitionistic mathematics, beginning in [1907]. Because these principles also underly Russian recursive analysis and the constructive analysis of E. Bishop and his followers, intuitionistic logic may be considered the logical basis of constructive mathematics.”Honestrosewater said:“Logic is not concerned with soundness but with validity. Indeed, you could say that vallid reasoning is exactly what logic is all about.
Do you think that my knowledge or understanding of logic is lacking?”
Misleading. Copi and Cohen present a deductive system in which there are 21 formal rules of deduction.logicalroy said:There are about 21 formal rules of deduction. See Irving Copi's book "SYMBOLIC LOGIC" and Irving Copi and Carl Cohen's Book "Introduction to Deductive Logic".
Wrong. Just because something is evidence based does not imply that it conforms to the scientific method.Induction doesn't count because that is just another way of saying scientific method.
Wrong. Deduction only leads to certainty in the case of a tautology, or when you are certain of the premises. Deduction, done correctly, is capable of yielding an uncertain conclusion from uncertain premises, or even a false conclusion from false premises.deduction if done correctly must yield certainty
What? You mean like someone trekked into central Brazil, and found a formal rule of deduction lying on the rainforest floor?People refer to the laws of logic because they were already present.
Mathematical induction is not restricted to numbers. It can be applied to all sorts of things. Propositions, for example.Not one of the rules I listed involve ONLY numbers as your mathematical induction does.
Academic subjects don't have sharp lines between them. For example, deduction is used all the time in science.Science is science & Deduction is deduction -- two distinct things (that you seem to mix).
Funny, I thought we left "reality" to the scientists. :tongue:Sound arguments reflect reality
I could say the same about you. Of course, I wouldn't, because that would be an ad hominem attack, and that's wrong.Once again, you are in your own world.
Wrong. There is a lot of semantics involved in mathematics, at least the way most mathematicians do it. E.g. although one can treat calculus simply as a formal theory, most people instead imagine that they are working with a model of that theory.Deduction includes a lot of semantics; math does not.
Wrong. In addition to numbers, math deals with points, lines, topological spaces, algebras, categories, formal languages, graphs, simplicial complexes, and formal logic.Math deals primarily with numbers
Math only "branches away" in the sense that some things are still merely conjecture, and have not yet been deductively proven.but math branches away because some math formulas are not deductive formulas for propositions or arguments.
Because it explains how one may correctly deduce that x = 1 or x = 2 if it is already known that x² - 3 + 2 = 0.How does the quadratic formula help one analyze an argument?
Since you harp on it, I'll respond a second time.Show me proof that logic is not concerned with soundness! Are you empty handed again? Yes, you surely lack knowledge because you exaggerate “validity” way too much. You are excessive with it. ALL SOUND ARGUMENTS ARE VALID. So why do you avoid soundness? What is your beef against soundness if you value validitiy so much? Are you unrealistic?
Of course it's different -- people use (or at least try to) deductive logic all the time in situations where they can't define their terms, and can't write down a list of premises upon which they're basing their argument, and those clearly aren't mathematical situations.WHAT PROOF ON EARTH MUST I USE TO SHOW MATH AND DEDUCTIVE LOGIC ARE DIFFERENT?
Well like your predecessor, you believe that there are billions of deductive systems. Proof anyone? Another empty handed fellow. You believe there are billions of formal rules of deduction. Proof anyone? Another misinformed person snagged without proof.hurkyl said:It does not follow that all deductive systems must have 21 formal rules of deduction.
Go ahead, prove that the Copi and Cohen deductive system can be presented in a different way. That is just as ridiculous as placing the planets in our solar system in a different way. (If we did that it wouldn’t be our solar system under discussion.)hurkyl said:It doesn't even follow that Copi and Cohen's deductive system cannot be presented in a different way that has a different number of rules.
Duh, yes it does. Deduction does not require evidence nor experiment. For example, take the following proposition: “all dogs are animals.” That proposition is what is called logically necessary and does not require any worldly proof. The words are defined in a certain way that anyone who understands the language would see that it is impossible to find a dog that is not an animal –- it’s part of the context of the word that speakers of the language understand. Induction MUST have EVIDENCE OR PROOF. (There are no semantic considerations in induction as far as the value of propositions.) This is what the quadratic formula or any other math equation can not do. Think you can prove it, go ahead! When I ask for an example of a deductive system you both give me mathematical systems --what’s wrong with you and Honestrosewater? If I were to ask for an even number, you would give me a letter (when I asked for a number)? If I were to ask you for a loaf of bread, you would give me a serpent? Something is definitely wrong with both of your communication skills. NAME A DEDUCTIVE LOGIC SYSTEM THAT HAS NO NUMBERS INCLUDED IN IT AND THAT IS NOT MATH! Understand what SCIENCE is! Science doesn't prove anything semantically. Science REQUIRES worldly EVIDENCE OR EXPERIMENT(for the last time already). MODAL Logic and Intuitionalistic logic are scientific systems. I ask for logic and you give math or science. You even stated that LOGIC and MATH were different in your post here:hurkyl said:Just because something is evidence based does not imply that it conforms to the scientific method.
Your command of logic is already suspect. You are inconsistent. People use induction more than deduction because most people want evidence. Where’s your sense of reality?hurkyl said:Of course it's different -- people use (or at least try to) deductive logic all the time in situations where they can't define their terms, and can't write down a list of premises upon which they're basing their argument, and those clearly aren't mathematical situations.
You must be kidding! Ok, go ahead and show me an example of a valid argument that is sound with true premises and yielding uncertain conclusion. NO MATH! You and Honestrosewater are all talk and no proof. You should definitely use deduction when you are certain of the premises from the start! Conditional proof starts from an assumed premise, but will lead to certainty if true. If it is false it will contradict another known fact. If there are false premises, then that is the user's fault --not a flaw in the system of logic.hurkyl said:Wrong. Deduction only leads to certainty in the case of a tautology, or when you are certain of the premises. Deduction, done correctly, is capable of yielding an uncertain conclusion from uncertain premises, or even a false conclusion from false premises.
Can you demonstrate this? Proof anyone? Another one empty handed.hurkyl said:Mathematical induction is not restricted to numbers. It can be applied to all sorts of things. Propositions, for example.
Dude, can’t you see that is math? Are you ill? On medication?hurkyl said:Because it explains how one may correctly deduce that x = 1 or x = 2 if it is already known that x² - 3 + 2 = 0.
LOOK UP SOUND ARGUMENT! You need help. Sound arguments MUST be true in reality and MUST be valid arguments. P-->Q is only a proposition. Secondly, the is no necessary truth to conditionals; they can be true or false. They are not always true. Thus, if the argument has false conditionals, then the argument CAN NOT be SOUND! For example, If G.W. Bush is president, then I am the Pope. If I am the Pope, then I am a millionare. Therefore, If G.W. Bush is president, then I am a millionare. (An Unsound argument, but VALID.) That can be why those people who exaggerate validitiy then lose sensibility about logic for unsound reasons (and claim they know logic when they really need more knowledge).hurkyl said:Furthermore, every valid argument that Q follows from P is essentially the same as a sound argument that (P --> Q) follows from the empty premise. And, as you've said, every sound argument is valid. So, there isn't even a meaningful distinction between the two.
Math/computer science professional, actually.logicalroy said:Another math or science major, huh.
That's an ad hominem attack. Which, may I remind you, you agreed not to do.You are on the same drugs as Honestrosewater
Here's a (very boring) deductive system.Well like your predecessor, you believe that there are billions of deductive systems. Proof anyone
Allow me to list a few that are sound for boolean logic.You believe there are billions of formal rules of deduction. Proof anyone?
Nope. I was attempting to give you enough credit that you could figure these things out after you were told the basic idea.Another misinformed person snagged without proof.
I don't have their exact list. So, I will demonstrate a trivial modification to Wikipedia's list of rules of inference.Go ahead, prove that the Copi and Cohen deductive system can be presented in a different way.
Like, duh, it totally doesn't, dude.Duh, yes it does.
What is what any math equation can't do? I can't seem to tie this to your previous sentences... though it does look like you are no longer responding to the quoted point.This is what the quadratic formula or any other math equation can not do.
Nothing. It's not my fault if you're a mathophobe.When I ask for an example of a deductive system you both give me mathematical systems --what’s wrong with you and Honestrosewater?
Science, of course, makes the assumption that reality forms a model of whatever scientific theory we're considering, so of course semantics are involved. And, of course, once we have a collection of axioms backed with scientific evidence, science then uses deductive logic to determine what the consequences of those axioms are.Science doesn't prove anything semantically. Science REQUIRES worldly EVIDENCE OR EXPERIMENT(for the last time already).
MODAL Logic and Intuitionalistic logic are scientific systems.
Yes. In the way that dogs and animals are different -- all dogs are animals, but not all animals are dogs.I ask for logic and you give math or science. You even stated that LOGIC and MATH were different in your post here:
Where was I talking about the relative frequency of peoples' usage of induction and deduction?Your command of logic is already suspect. You are inconsistent. People use induction more than deduction because most people want evidence. Where’s your sense of reality?Hurkyl said:Of course it's different -- people use (or at least try to) deductive logic all the time in situations where they can't define their terms, and can't write down a list of premises upon which they're basing their argument, and those clearly aren't mathematical situations.
Let me try this again:You must be kidding! Ok, go ahead and show me an example of a valid argument that is sound with true premises and yielding uncertain conclusion.Hurkyl said:Wrong. Deduction only leads to certainty in the case of a tautology, or when you are certain of the premises. Deduction, done correctly, is capable of yielding an uncertain conclusion from uncertain premises, or even a false conclusion from false premises.
Would you care to prove that?The principles of Logic were already present
See Ebbinghaus, Flum, and Thomas's Mathematical Logic. In the second edition, Section II.4 is titled "Induction in the Calculus of Terms and in the Calculus of Formulas". Induction is performed over formulas, not numbers. To quote the book: (C denotes your calculus -- that is, your rules of inference)Can you demonstrate this? Proof anyone? Another one empty handed.
Yes, I can see this is math. Your point being? ...Dude, can’t you see that is math? Are you ill? On medication?
Any valid argument with the empty premise is automatically sound.LOOK UP SOUND ARGUMENT! You need help.
Notice please that your alleged deductive systems are the inference rules I have already mentioned. Your first deductive system is an inference rule called “tautology”. You state it as a NEW deductive system. That is like me writing “56x+1x=57x is not algebra, but a NEW form of math.” IT IS ALGEBRA! It follows the rules and principles of algebra; thus, it is algebra. No matter how much I disagree that equation I used along with variable x is algebra, it is not a new form of math because I say so. What your post appears to be stating is that whenever you use deductive rule you are using a NEW deductive system. That view is mistaken which is why I used a math counterexample. Did you mean you can create your own logic without using Any of the rules I mentioned? Logic inference rules are not switches where you can turn them on and off as you please or desire. They are universals. If your intention was different please clarify what you meant. Your second example of a new system is the inference rule of “conjunction”. These are not NEW systems or NEW inference rules; they fall under the same deduction system (I already mentioned) even if you personally choose not to use any other rules. The other rules can still be used whether you want them to be used or not. I do not see logic is not a matter of personal choice. You used the rule of conjuction rule above:hurkyl said:Here's a (very boring) deductive system.
It has no axioms.
It has only one derivation rule:
p
-----
(p and p)
Here's another deductive system.
It has no axioms.
It has only one derivation rule:
p
q
------
(p and q)
Here's another (very boring) deductive system.
It has one axiom schema:
p and q
It has only one derivation rule:
p
q
r
------
(p and q) and r
Can you see how there are infinitely many deductive systems? Or do you really need me to spell it out?