# How many rules are there in formal logic?

1. Mar 16, 2006

### MathematicalPhysicist

i mean rules such as demorgan's, material conditional, conditional proof and so forth.

2. Mar 16, 2006

### honestrosewater

There is more than one system that could be called a formal logic (propositional logic, first-order logic, second-order logic, modal logic, etc.). I think what you're interested in is a calculus for such-and-such logic. Each calculus has a set of axioms, or axiom schemas, and a set of inference rules. Either of those sets can be empty. If you have a recursive rule, you can have an infinite number of (redundant) rules. If you don't like redundancy, I've seen a propositional calculus with one axiom schema and one inference rule (Modus Ponens). Hm, I'm not sure I'm answering your question. If your logic is nice enough that you can turn theorems into rules and rules into theorems, there are as many rules as there are theorems, of which there are usually infinitely many. If you state your rules as schemas, as is most convenient, there are as many rules as theorem schemas. I don't know how many theorem schemas there are in any logic. Define what it means for two theorems to 'have the same form', and start counting. It sounds like an interesting question.

Actually, let me clarify something. There are two 'systems' here: the object system, which is what you're studying, e.g., some formalization of a propositional logic, and the metasystem, which you use to set up and study the object system, e.g., English and whatever logic you're using. The rules are part of the metasystem. The theorems are part of the object system. So in asking how many rules there are, you're turning your metasystem into your object system and creating a new metasystem with which to study the old metasystem. You're moving up a level. But you might be able to stay back on the original level and ask about theorems instead of rules, given that there is some equivalence between them (and it makes sense that there would be since you use the rules to generate theorems).

If you're really just wondering 'Why this set of rules?', I would guess that the main reason that you are familiar with the rules that you know is that humans have found those rules to be the most useful for whatever they're doing at the moment.

Last edited: Mar 16, 2006
3. Mar 16, 2006

### loseyourname

Staff Emeritus
The simple answer to your question is "indefinitely many." Any valid argument form can be called a rule. The number I was personally taught as the rules of inference was nine. There was some reason for it being limited to this, something along the lines of all other valid inferences are reducable to these nine, but they are not themselves reducable.

4. Mar 16, 2006

### honestrosewater

Well, I think I get what you're saying, but the thing is that I'm not sure there's one clear meaning of reducable. There is what seems to me to be a related concept that given a set of axioms and a set of inference rules, that set of rules is, say, minimally complete, iff you can use it and your set of axioms to generate all and only the tautologies of your logic AND if you take away any one of those rules, then you can no longer generate all and only the tautologies of your logic.

However, given a minimally complete set of rules R, for each rule x in R, there is a set of rules X, possibly empty, such that you can replace x with some member of X and still leave R a minimally complete set of rules. This set of rules X might contain a lot of 'reducable' rules, but that loose concept of reducability could probably be formalized in different ways. For one thing, you'd have to define what it means for two rules to have the same form. Then, you'd have to figure out how to figure out which one is simpler. I'm not aware that either concept, having the same form or being simpler, is a formal concept yet. You would have to define that in terms of other things like string length, number of connectives, or such. I'm not saying it would be difficult, just that, to my knowledge, it isn't given (and might even end up being tricky or impossible).

Last edited: Mar 16, 2006
5. Mar 21, 2006

### NickJ

A different way of thinking about how many rules there are in formal logic is this: how many rules do we need to give in order to completely specify the meanings of all the logical operators? The answer to this question, for a predicate logic that does not involve operators like "It is possibly the case that" or "It is permissible that", is TWELVE (depending upon how you want to count some of them -- see below).

Here are the logical operators we want to define: NOT, AND, OR, ONLY IF, FOR ALL, THERE EXISTS. These are the standard logical operators of a predicate logic. (Take out FOR ALL and THERE EXISTS and you get the standard operators for a propositional logic.)

For each logical operator, there are two rules that specify its meaning completely: an introduction rule, which tells you the conditions under which it is appropriate to ADD that operator; and an elimination rule, which tells you the conditions under which it is appropriate to REMOVE that operator. I'll give some introduction and elimination rules for propositional logic, so you get the idea.

Introduction Rule for AND: From p and q, infer p AND q
Elimination Rule for AND: From p AND q, infer p; FROM p AND q, infer q.
(I'm counting this elimination rule as one rule; you might want to count it as two rules.)

I-Rule for OR: From p, infer p OR q.
E-Rule for OR: From p OR q, a proof of r from p and a proof of r from q, infer R.

I-Rule for ONLY IF (material conditional): From a proof of q that contains p as it's only undischarged premise, infer p ONLY IF q.
E-Rule for ONLY IF: From p, p ONLY IF q, infer q.

And so on.

When one does this, one can then treat other rules, like DeMorgan's, etc, as derived rules -- more concise ways of prescribing what the introduction and elimination rules already prescribe. But these other rules aren't needed in order to give the meanings of the logical operators; so, in this sense, they aren't really necessary -- one would have one's logic without ever bothering to write them down.

6. Mar 21, 2006

### honestrosewater

Welcome to PF, NickJ!

One quick point: quantifiers (FOR ALL, etc.) don't work the same way that connectives (NOT, AND, etc.) work. For what it's worth, calling quantifers operators confuses me.

The question that jumps out first at me is why you chose that set of operators. You can get away with only one quantifier (e.g., EXISTS) and one connective (e.g., NAND) -- you can define all of the others in terms of those two. And there are other operators that you didn't include.

It's also not clear to me what you mean by specifying or defining operators. If operators have any meaning, it is by virtue of how they are used to assign truth-values to formulas. But rules are syntactic devices -- as far as a rule is concerned, formulas don't have truth-values -- they are just meaningless strings of symbols. So how rules can define or specify, or, for that matter, say anything at all about, an operator is a bit beyond me.

And the problem still remains that you are dealing with ill-defined concepts. Don't you think it makes sense to say what a rule is before you ask how many of them exist? If you insist on trying to deal with the (yet to be formalized) metalanguage, I think you should consider what your count is actually going to be worth if you don't bother to formalize your system.

7. Mar 21, 2006

### NickJ

Thanks for the welcome.

There are (at least) two ways to think about the meaning of logical operators like AND, OR, etc.

(What do I mean by "meaning of a logical operator"? I mean "what the sentences in which that operator occurs, mean. So, for instance, if someone said that "Jack goes to the bank AND Jill goes to the well" means "Jack goes to the bank only if Jill goes to the well", they wouldn't know what "AND" means.)

One way is truth-functional: the meaning of an operator is given by its truth-table. So far as I'm aware, there aren't truth-tables for quantifiers. But no matter.

The second way is rule-based: the meaning of an operator is given by its introduction and elimination rules. Here is a reason for preferring the rule-based method: Some people think that it is illegitimate to infer the truth of a sentence p from a proof that "not-p" is contradictory. For instance, they think it would be illegitimate to infer that "There is a prime number between 3 and 11" by proving that the denial of this sentence is absurd. Instead, they want to say that proving "There is a prime number between 3 and 11" requires a constructive proof -- for example, giving a number and showing that it is prime (e.g., 7). But there is no way to give a truth-table for "NOT" so that it functions in this way; and taking "NAND" as primitive also does not allow one to do this. But one can give introduction and elim-rules for "NOT" that allow this. (What they are escapes me at the moment; look up any resource on intuitionistic logic. An intuitionistic logic is a logic in which the rules of excluded middle and double-negation elimination are not valid.)

(Another reason for preferring my method to your suggestion: my method allows one to define the operators all by themselves; if you use NAND as a primitive operator (and define the others in terms of what "NAND" means), then you don't have independent specifications of the meaning of "AND" and "NOT" -- these are both defined by reference to "NAND". If you use intro- and elim-rules, you give one pair of rules for each primitive operator. And you take "AND" and "NOT" to be primitive, rather than "NAND", because we just don't say "NAND" in ordinary language -- unless we're into computer programming and such.)

How can rules define meanings? That's my sloppy way of saying that we can take the rule to give the constraints on what the meaning of an operator is. In any case, I think I don't understand your question. Here's an analog of the one you asked: //But <definitions> are syntactic devices -- as far as a <definition> is concerned, <words> ... are just meaningless strings of symbols. So how <definitions> can define or specify, or, for that matter, say anything at all about, an <the meaning of a word> is a bit beyond me.//

I don't get whay you would be puzzled over this analog question (if you would be). And that's why I don't get why you are puzzled when I say that a rule could give the meaning of an operator: 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).

What's a rule? A rule (in first-order logic, anyway) is a schema that specifies the conclusion one is to infer from a set of premises. Here is a rule (where the '-----' separates premises from conclusion and the letters represent sentences):

A
---------- (OR-Introduction)
A OR B

Here's another:

P P --> Q
---------------- (ONLY IF-Elimination)
Q

Here's one for THERE IS-Introduction:

P [t/x]
-----------------------------
THERE IS an x such that P,

where P is a formula that contains at least one occurrence of the term 't' and P[t/x] is the formula one obtains by replacing every occurrence of
't' with the variable 'x'.

(Actually, I can't quite exactly remember this rule -- I might have gotten it wrong.)

I hereby refer those who are interested to the logic textbooks. You'll have to find a text that covers non-classical logics, though -- specifically, one that covers intuitionistic logic. The texts will explain things much better than I can.

Here's a good link to start with (although it gives the logic in the old-school way, by giving a set of axioms):

http://plato.stanford.edu/entries/logic-intuitionistic/

Here's a better link, that gives the natural deduction rules (intro- and elim- rules) that I've been giving (towards the bottom):

http://cs.wwc.edu/KU/Logic/Intuitionistic.html [Broken]

Last edited by a moderator: May 2, 2017
8. Mar 21, 2006

### honestrosewater

What does the set of 6 operators that you have listed have to do with the number of rules for a first-order logic? 2 is enough. Using them, you can define as many operators as you want.

Here is why I'm confused: A definition itself must be assigned a truth-value. (When you define something, you are asserting that the definition is true.) Rules are not statements; they are instructions and cannot be assigned truth-values.

Furthermore, rules do not assign truth-values to the strings that they generate. They just generate strings. There are no truth-values involved.
This seems to have truth and validity mixed in with it. Here are rules where p and q are formulas and $is an operator. p$q
------
p

p, q
-----
(p$q)$q

What does \$ mean?

I don't understand the point of the rest of what you said. If your method fails to give the correct number of rules, that is enough reason to reject it. My only real suggestion is that people at least set things up so that they have a chance of being able to prove that there are x many rules for such-and-such logic.

Last edited: Mar 21, 2006
9. Jul 13, 2006

### logicalroy

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". Induction doesn't count because that is just another way of saying scientific method. Deductive logic is a logic of certainty, whereas Inductive logic is only probable (such as DNA test, CSI type stuff,etc).
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). 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?

Last edited: Jul 13, 2006
10. Jul 13, 2006

### honestrosewater

Can you prove that?
There is a valid, deductive inference form called (mathematical) induction. It is not the same thing as inductive inference.

Inductive inference is not the scientific method. Inductive inference is a type of logical inference, whereas the scientific method is not.
Reductio ad absurdum is not included in Conditional Proof, and some of those are equivalencies, so you should count them as 2 inference rules.
Logic certainly can be and is used to model physical systems and make predictions.

11. Jul 13, 2006

### loseyourname

Staff Emeritus
The completeness of Copi's system of natural deduction..

Toward the bottom of the page.

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.

12. Jul 16, 2006

### logicalroy

Hi honestrosewater,

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.
Mathematical Induction is not "LOGIC", but a system of math found in the matematics department. Clearly you need to make the distinction between math and logic. Not one of the rules I listed involve ONLY numbers as your mathematical induction does. Can you name any university on the planet where mathematical induction is taught under the "Philosophy" curriculum? If your reply is "NO" then your mathematical induction is not real "LOGIC". Deductive LOGIC falls under the academic disipline of Philosophy --not math.
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). Induction and the scentific method are the same becasue both requires experiment and does not always lead to certainty; 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."
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.
Loseyourname,
You are correct that the rules I mentioned applies to sentenial and propositional Logic AS WELL AS predicate logic as well. Show me one logic book where the rules do not apply. Predicate logic only added two quantification rules, which I did not list. All of the systems of deductive logic are progressive. Each one picks up where one fails. All the systems of deduction combine to make a complete system.

Last edited: Jul 16, 2006
13. Jul 16, 2006

### honestrosewater

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.

But even if I wanted to, I couldn't prove anything about your claim,
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.

Asking how many rules there are for formal logic is like asking how much a piece of fruit weighs. What kind of fruit? What measurement system? What's the gravitational acceleration? Your answer of 'about 21' is just as good for both of them.

I have read Copi & Cohen and know their Natural Deduction rules. That is one set of rules for one type of logic. 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.

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.

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. I've already pointed this out too.

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.
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.
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.
Sure. The University of London. Oh, wait, that's actually several universities. MIT then. So what?
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.
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.
Where did I do that?
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.

Inductive inference is a type of logical inference. Science and the scientific method are not types of logical inference. That is a property in which they differ, so they are not the same.

Also, inductive inference does not require experiment.
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.

Do you think that my knowledge or understanding of logic is lacking?
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.

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.
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.
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'?
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.

14. Jul 16, 2006

### honestrosewater

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.'

That proof did make me realize that I need to clarify something I just 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.

And it's still of course true that your set of axioms is restricted. For one thing, it can't be inconsistent, which, with those rules, would allow you to derive all but not only the formulas of propositional logic.
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.

Incidentally, there's a concept similar to what we were talking about before, superfluous rules, called independence. It applies to sets of axioms, but I imagine you can alter it a little to make it apply to sets of rules, perhaps allowing you to transfer over some or all of knowledge already gained about independence.

15. Jul 17, 2006

### logicalroy

Hi again Honestrosewater,

You must be a science major or math major, huh? Once again, you are in your own world. I had the decency of listing two sources and listing the rules themselves. I had the decency to come to the table with something other than JUST WORDS like you have done. You ask me for proof and I provided two sources. That is SOME proof. Then you have the audacity to outright reject the sources listed without the decency of explaining why you reject the sources. You think you are rational? Think again. All you did was shoot down everything I stated with your own words and no proof. It’s not like you listed any sources. You’ve came to the table EMPTYHANDED and you have the audacity to talk? I provided some proof which my not float your boat, but I came to the table with SOMETHING as opposed to your NOTHING. Let rational people here decide who should be more believable –the guy with some evidence or the empty handed guy. Being realistic is what I am, you are not. Truth is not up to me or you. My initial answer was not specific enough because I did not want people like you to take the rules I listed as absolutes and then later read about you whining if I accidentally left some rules out. So I listed what I know for sure.
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??
Are you kidding? You can’t be serious! HYPOTHETICAL SYLLOGISM is a hypothetical rule as well as conditional proof.
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”.
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?
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.
You stated MATHEMATICAL INDUCTION specifically. Now you are substituting that for general Induction. Induction is different from Mathematical Induction which you pointed out:
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?
Deduction, on the other hand, deals a lot with semantics and has nothing to do with the ways of the world. Thus, no scientific inquiry is needed. Reread what you wrote here:
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!
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.”
WHAT PROOF ON EARTH MUST I USE TO SHOW MATH AND DEDUCTIVE LOGIC ARE DIFFERENT? IS THERE ANY SUCH PROOF POSSIBLE TO YOU? If not, then how can you ask for proof?

Last edited: Jul 17, 2006
16. Jul 17, 2006

### Hurkyl

Staff Emeritus
Misleading. Copi and Cohen present a deductive system in which there are 21 formal rules of deduction.

It does not follow that all deductive systems must have 21 formal rules of deduction.

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.

Wrong. Just because something is evidence based does not imply that it conforms to the 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.

(And, of course, this all assumes you have complete faith in your system of deduction)

What? You mean like someone trekked into central Brazil, and found a formal rule of deduction lying on the rainforest floor?

Mathematical induction is not restricted to numbers. It can be applied to all sorts of things. Propositions, for example.

Academic subjects don't have sharp lines between them. For example, deduction is used all the time in science.

Funny, I thought we left "reality" to the scientists. :tongue:

Sound arguments reflect reality only so far as you have confidence that the premises "reflect reality". Since professionals and people with a better understanding of logic realize the difficulties in establishing a premise is "true", and realize that doing so lies outside the purview of deductive logic, they focus on validity.

I could say the same about you. Of course, I wouldn't, because that would be an ad hominem attack, and that's wrong.

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.

Wrong. In addition to numbers, math deals with points, lines, topological spaces, algebras, categories, formal languages, graphs, simplicial complexes, and formal logic.

Math only "branches away" in the sense that some things are still merely conjecture, and have not yet been deductively proven.

Because it explains how one may correctly deduce that x = 1 or x = 2 if it is already known that x² - 3 + 2 = 0.

More precisely, the quadratic formula is a pair of functions (which, remember, are just special kinds of logical relations), and there is a theorem which says that given a quadratic equation, one may deduce that the roots of the equation are the two numbers given by the quadratic formulas.

Since you harp on it, I'll respond a second time.

(1) You need to have access to "absolute truth" in order to classify an argument as sound.
(2) Except for pure tautologies, acquisition of the initial "absolute truth" lies outside the purview of deductive logic.

Therefore, it is rather silly to, when studying deductive logic, be overly concerned with soundness.

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.

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.

P.S. You've been using intuitionist logic all your life -- boolean logic is a special case of intuitionist logic that assumes the law of the excluded middle.

Let me restate that:

(Boolean Logic) = (Intuitionist Logic) + (Law of Excluded Middle)

Last edited: Jul 17, 2006
17. Jul 18, 2006

### logicalroy

Hi Hurkyl,

Another math or science major, huh. You are on the same drugs as Honestrosewater about some of your points and like Honestrosewater you are empty handed. Let me address your mistakes now.

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.

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.)

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?

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.
[Note: truth and awareness are two different things; a proposition can still be true even if you are unaware of its truth. For example, "there are green aliens on pluto 2 inches tall" is still either true or false regardless of our awareness. Your so called "absolute truth" requires you to be aware. Logic has nothing to do with your sensual awareness, but analyzing arguments and meaningful propositions.]

The principles of Logic were already present and mankind simply discovered what was already there. The principles of physics were already present and mankind named what was there. Astronomy followed the same path. Not all subjects are man made.

Can you demonstrate this? Proof anyone? Another one empty handed.

Dude, can’t you see that is math? Are you ill? On medication?

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).

Last edited: Jul 18, 2006
18. Jul 18, 2006

### Hurkyl

Staff Emeritus
Math/computer science professional, actually.

That's an ad hominem attack. Which, may I remind you, you agreed not to do.

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?

If we're simply talking about interesting deductive systems, Boolean logic is still not the only choice. Off the top of my head I know of intuitionist logic, and the various modal logics.

Allow me to list a few that are sound for boolean logic.

p
----
not not p

p
----
not not not not p

p
----
not not not not not not p

can you see how I can write down infinitely many rules of deduction that are sound for boolean logic? Or do you need me to spell it out?

Nope. I was attempting to give you enough credit that you could figure these things out after you were told the basic idea.

I don't have their exact list. So, I will demonstrate a trivial modification to Wikipedia's list of rules of inference.

My trivial revision replaces Wikipedia's addition rule:

p
-----
p or q

with this rule

p
-----
not not (p or q)

and uses all of the other listed rules. The result is the same logic, but different presentation.

Unfortunately, off hand, I don't know where I can find a more interesting alternative list of inference rules.

Like, duh, it totally doesn't, dude.

Go read up on the scientific method. http://en.wikipedia.org/wiki/Scientific_method]Wikipedia [Broken] is a reasonable place to start. To be scientific, you have constraints like reproducibility, attempts to control the experiment to eliminate other factors, et cetera.

Inductive inferences are not automatically scientific.

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.

Nothing. It's not my fault if you're a mathophobe.

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.

I'm not really sure what you're trying to say in the quoted passage, so I hope the above hits upon it. I had wanted to say it anyways, so there it is.

The tone of your whole post suggests that you are trying to distance science from deduction -- and then you say this? Okay, what do you mean by a "scientific system"?

Yes. In the way that dogs and animals are different -- all dogs are animals, but not all animals are dogs.

Where was I talking about the relative frequency of peoples' usage of induction and deduction?

Let me try this again:

A valid deductive argument with UNCERTAIN premises can yield an UNCERTAIN conclusion.

Did you understand me this time?

Would you care to prove that?

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)

The principle of proof is evident: in order to show that all strings derivable in C have the property P, we show that everything derivable by means of a "premise-free" rule has the property P, and that P is preserved under the application of the remaining rules".​

Yes, I can see this is math. Your point being? ...

And I'm in good health. Thank you for asking!

Any valid argument with the empty premise is automatically sound.

Any unsound, yet valid argument that Q follows from P can be easily modified to yield a sound and valid argument that (P-->Q) follows from the empty premise.

P.S. you can edit your posts in this forum. Even if you don't want to, you shouldn't duplicate your previous post in your next one.

Last edited by a moderator: May 2, 2017
19. Jul 19, 2006

### logicalroy

Hi Hurkyl,

I will take the time to take a step back and apologize about what you feel to be an ad hominem attack. I will change my tone and mood to something more appropriate. So I ask for forgiveness if I made a mistake.
An ad hominem attack is when a person dismisses an argument without justification because (the person has no justification) an attack on the person is made in place of reasons. I DID NOT dismiss your argument because I wrote something negative about you. I dismiss your argument because you are misinformed about deductive logic. My negative statements (which I now apologize for) had nothing to do with you being misinformed about logic. Thus, what I did was not an ad hominem attack. I will now start fresh and restate my justifications of you and honestrosewater being mistaken about logic.

P
q
------
p and q

I can STILL apply the inference of simplification and deduce “p” alone. I can STILL use the “Commutation” inference and derive “q and p”. I don’t have to have your permission (or anyone else’s permission) to use other inference rules validly. Logic is NOT about personally choosing the rules of inference that can be used. The deduction inference rules are universal and don’t belong to any person or group as in “His (or their) personal logic”. Your post and the post by Honestrosewater give the vibe that logic can be personalized and is subjective: for example, Roy can have his logic; Bob can have his logic; John can have his logic, et cetera. The view that logic is a subjective is mistaken. A subjective or relative matter would be one like Roy can have a wife, which Roy’s wife will be DIFFERENT from Bob’s wife and John’s wife. All people can use the SAME inference rules of deduction and the inference rules DO NOT change depending upon the person. Deductive logic is universal and not personalized. Your posts and some others give the vibe that everything including deductive logic is subjective and depends . . . ! If I am mistaken about your true intentions then I apologize in advance, but can you clarify what you meant so we are on the same page?
Your other version of a Boolean system with the multiple “not” included are instances of “double negation”; two negations make positive assertion. So no matter how many even number of “not” you include, they reduce to double negation which is already in the deductive inference list and NOT a separate system. The presentation of deductive logic certainly does not matter to an objective person. You can put a pig in a custom made suit; does that mean the species is no longer a pig because you present it differently? So far you have only recited the rules I have already mentioned when you gave logic examples. Your other examples refer to mathematics, which I also mentioned differs on many things and are not the same. Sure math can use some deductive logic but there are times when math just deals with numbers. That makes math branch away from philosophy.
Here is a definition of induction according to Wikipedia:
“Induction or inductive reasoning, sometimes called inductive logic, is the process of reasoning in which the premises of an argument support the conclusion but do not ensure it. It is used to ascribe properties or relations to types based on tokens (i.e., on one or a small number of observations or experiences); or to formulate laws based on limited observations of recurring phenomenal patterns”

DEDUCTION assures the truth of the conclusion where induction does not. I am not sure why you can not comprehend this fact. Here is information AGAIN from the link you suggested from “Wikipedia” under Induction (and scroll down to validity):

“In contrast to deductive reasoning, conclusions arrived at by inductive reasoning do not necessarily have the same degree of certainty as the initial premises. For example, a conclusion that all swans are white is false, but may have been thought true in Europe until the settlement of Australia. Inductive arguments are never binding but they may be cogent. Inductive reasoning is deductively invalid. (An argument in formal logic is valid if and only if it is not possible for the premises of the argument to be true whilst the conclusion is false.) In induction there are always many conclusions that can reasonably be related to certain premises. Inductions are open; deductions are closed. It is however possible to derive a true statement using inductive reasoning if you know the conclusion.”

You seem to reject what I say no matter what evidence I bring and you can’t justify your rejection. You shoot down all that I bring as evidence without stating what is wrong with the evidence I provided. That my friend puts you in the “irrelevant conclusion” fallacy (or better yet a real AD HOMINEM -- you don’t personally like me; thus you reject all I say). INDUCTIVE INFERENCES NEED SCIENCE TO PROOVE THEM; whereas deductive inferences DO NOT REQUIRE SCIENCE FOR PROOF. Therefore, deduction and science are different. Science can in fact use deduction and that seems why you can’t distinguish between the two. All induction is scientific as the wikipedia definition defines above. I never stated that all sciencfic methods are inductive. I stated all general induction is scienctific. Math and deduction are also different and you seem not able to distinguish those writings from people with a “Philosophy only background” and those with a “math and possibly some philosophy”. Those people who only deal with philosophy will most likely agree with me. Your type of people have a philosophy background plus something else. That “something else” is where you get your views and branches off from “Philosophy”.

Last edited: Jul 19, 2006
20. Jul 20, 2006

### loseyourname

Staff Emeritus
Thread's done guys. I think LQG got a good enough answer for his purposes. Please don't fight.