Logic: Deductive Science of All Possible Worlds

[tgt]

How is this definition ( I made up):

Logic is the deductive (as opposed to empirical) science of all possible worlds.

 

[Dragonfall]

That definition is pretty murky.

 

[SW VandeCarr]

How is this definition ( I made up):

Logic is the deductive (as opposed to empirical) science of all possible worlds.

How about checking out some real definitions instead making things up?:

"The analysis, without regard to meaning or context, of the patterns of reasoning by which conclusions are validly derived from sets of premises."

Borowski and Borwein, Harper Collins Dictionary of Mathematics (1991).

There are other more specific definitions citing axioms and rules of inference, all of them considerably better than yours

 

[tgt]

That definition is pretty murky.

How about checking out some real definitions instead making things up?:

"The analysis, without regard to meaning or context, of the patterns of reasoning by which conclusions are validly derived from sets of premises."

Borowski and Borwein, Harper Collins Dictionary of Mathematics (1991).

There are other more specific definitions citing axioms and rules of inference, all of them considerably better than yours

Well, just say logic wasn't invented then I'd probably be interested to investigate (create) a field that has the definition that I gave. It's a first definition which usually is intuitive.

 

[SW VandeCarr]

Well, just say logic wasn't invented then I'd probably be interested to investigate (create) a field that has the definition that I gave. It's a first definition which usually is intuitive.

Well, your definition describes aspects of the current state of theoretical physics. I suggest you investigate the Many Worlds interpretation of quantum mechanics.

 

[HallsofIvy]

What is your definition of "science". In the usual definition, "a study that follows the scientific method", logic is not a science at all because it does not involve experimentation on the real world.

 

[tgt]

Well, your definition describes aspects of the current state of theoretical physics. I suggest you investigate the Many Worlds interpretation of quantum mechanics.

What is your definition of "science". In the usual definition, "a study that follows the scientific method", logic is not a science at all because it does not involve experimentation on the real world.

From a modern view point, the terminologies have all been mixed up. But pretend it's 400BC before anything that resemble logic has been thought up of. Back in those days science and maths were closely linked and there wasn't the scientific method.

 

[SW VandeCarr]

From a modern view point, the terminologies have all been mixed up. But pretend it's 400BC before anything that resemble logic has been thought up of. Back in those days science and maths were closely linked and there wasn't the scientific method.

What you're describing is the "natural philosophy" of classical ancient Greece. This was a mix of mathematics and logic with a kind of purely observational version of what we call science. Before 300 BC Aristotle had laid down a formal logic. Modern science dates from around the 16th century in western Europe. So is the pre-Aristotle world your ideal?

I have a lot of respect for Pythagoras, Euclid, Plato, Aristotle and other outstanding thinkers of that time and place, but we have clearly progressed since then (although it wasn't a steady climb, at least in the West). It seems you crave an earlier simpler time. Better ignore my last suggestion and stay away from quantum mechanics and its various interpretations.

EDIT: By the way, don't you think that mathematics and science are closely linked now?

 

[tgt]

So is the pre-Aristotle world your ideal?

Well, if I was to put myself in that period which is without a lot of any theoretical maths then it would be interesting to work something in the area of the definition in the OP. Just to get some undisputable truth.

EDIT: By the way, don't you think that mathematics and science are closely linked now?

They are closely connected but not as close as the ancients had thought.

 

[SW VandeCarr]

Well, if I was to put myself in that period which is without a lot of any theoretical maths then it would be interesting to work something in the area of the definition in the OP. Just to get some undisputable truth.

Indisputable truth?? You're winging off into deep metaphysical philosophy now. What makes you think these guys were any closer (to whatever your idea is) than we are now?

EDIT: Would you agree that 2+3=5 at least, is a bit of indisputable truth?

 

[tgt]

Indisputable truth?? You're winging off into deep metaphysical philosophy now. What makes you think these guys were any closer (to whatever your idea is) than we are now?

EDIT: Would you agree that 2+3=5 at least, is a bit of indisputable truth?

Probably indisputable truth in not just mathematics but other areas. That is why my definition has an empirical feel to it even though it shouldn't have anything to do with the world. I was trying to define something that includes more then just maths. It is a bit philosophical which isn't surprising as most philosophers enjoy logic more than maths.

 

[SW VandeCarr]

Probably indisputable truth in not just mathematics but other areas. That is why my definition has an empirical feel to it even though it shouldn't have anything to do with the world. I was trying to define something that includes more then just maths. It is a bit philosophical which isn't surprising as most philosophers enjoy logic more than maths.

Your OP offers a definition of logic as a science which is strictly deductive (not empirical) and deals with "all possible worlds". Don't you see all the contradictions here? Halls of Ivy already shot down your view that logic is a science. If it is strictly deductive, how can it have an "empirical" feel? And what in heaven's name does "all possible worlds" mean if you say it shouldn't have anything to do with the (any?) world. What are the possible worlds? Observable worlds? A multiverse? Braneworlds? Parallel universes? Everything the human mind can conceive (EHMCC)? EHMCC + everything else?

I've read some of your other posts and you seem to know something about math and logic. Are you simply rebelling against rationality as we know it?

EDIT: Consider the definition I quoted: "The analysis, without regard to meaning or context,...". Do you interpret this to mean "all possible worlds"? If so, we first replace "all possible worlds" with this fragment, remove the word "science" and any mention of "empirical" from your definition followed by some reference to valid reasoning from premises to a conclusion and "your" definition is spot on!

 

[tgt]

Are you simply rebelling against rationality as we know it?

I've just made a very naive definition with the view point of someone in 400BC would have. Would it be wrong to say that at that time science = maths? But obviously the word science and maths would have had different meanings as well at that time. Certainly the definition of science would be very different to the definition we would have now.

 

[SW VandeCarr]

I've just made a very naive definition with the view point of someone in 400BC would have. Would it be wrong to say that at that time science = maths? But obviously the word science and maths would have had different meanings as well at that time. Certainly the definition of science would be very different to the definition we would have now.

You just contradicted yourself again. The great thinkers of ancient Greece were not naive. They were brilliant. They just didn't have the level of knowledge (or the technology) we have. Earlier you indicated they would be closer to "indisputable truths" than we are.

I'm still interested in your answer to my direct question to you: Do you consider your reference to "all possible worlds" to be equivalent to "..without regard to meaning or context,..." ?

 

[honestrosewater]

What is your definition of "science". In the usual definition, "a study that follows the scientific method", logic is not a science at all because it does not involve experimentation on the real world.

The objection that non-empirical sciences do not exist seemed to be precluded by the OP's qualification "deductive (as opposed to empirical) science". I think a more general definition of a science along the lines of a systematic investigation and description, perhaps with certain requirements concerning, e.g., consistency, precision, or repetition, would be acceptable. Of course, empirical sciences still use deductive reasoning, so the distinction between deductive and empirical sciences could use clarification.

Logic does investigate and describe classes of all possible worlds using deductive methods, though I'm not sure this covers everything it does. A possible world is an abstract object. It doesn't have to be interpreted as or modeled by any physical object. I don't see what the problem is. Logic also involves the creation of theories and models and the study of their relationships. All the same can be said of math, so I don't see how logic could be considered the (one and only) science of all possible worlds. Math seems to include logic more than logic includes math, especially if you only consider the deductive parts of logic.

Still, any definition that short is not likely to be helpful because of a lack of precision or comprehensiveness. The field of logic is the study of logical objects in logical ways. As a human endeavor, what count as logical objects or logical ways are not static or precisely predefined.

What is the purpose of this definition anyway?

 

[tgt]

You just contradicted yourself again. The great thinkers of ancient Greece were not naive. They were brilliant. They just didn't have the level of knowledge (or the technology) we have. Earlier you indicated they would be closer to "indisputable truths" than we are.

They are so not naive that they killed someone because that someone discovered that irrational numbers come up naturally in mathematics!

I didn't indicate at all that they would be closer to "indisputable truths" than we are. I said it comparing to the other knowledge at the time.

I'm still interested in your answer to my direct question to you: Do you consider your reference to "all possible worlds" to be equivalent to "..without regard to meaning or context,..." ?

I'd say all possible worlds with respect to the intuition of this world. One example would be that things either is or is not. No in betweens or anything. Hence it's not as broad as some modern definitions. This would probably be what the ancients thought all possible worlds to mean.

 

[tgt]

Math seems to include logic more than logic includes math, especially if you only consider the deductive parts of logic.

what count as logical objects or logical ways are not static or precisely predefined.

Would you consider mathematical logic as logic or maths or an independent field? It's considered to be meta maths so it's not maths.

What is the purpose of this definition anyway?

For interest sake.

 

[honestrosewater]

Would you consider mathematical logic as logic or maths or an independent field? It's considered to be meta maths so it's not maths.

Depends on how I defined them and what criteria I was using to categorize them. I don't know all of the options, so I'm not sure which, if any, I would prefer. Are you attempting a partition? Do you know whether it is possible?

I don't even know what I would try to categorize. Trying to categorize human activities looks way too complicated. Trying to categorize general objects doesn't seem to get anywhere since there can be more than one way to interpret the 'same' object. Perhaps I would try to interpret the fields as computers.

Mathematical logic is not considered metamathematics by everyone. This thread has already shown that these things are not agreed upon.

 

[Preno]

Mathematical logic is not considered metamathematics by everyone. This thread has already shown that these things are not agreed upon.

Yeah, it really depends on what you're focusing on. If by "mathematical logic" you mean the study of the kinds of structures found in logic by mathematical means for their own sake (like model theory), then that's a branch of mathematics. Otoh, if we study the structures because they model valid inferences, then we're doing logic.

 

[SW VandeCarr]

Logic does investigate and describe classes of all possible worlds using deductive methods, though I'm not sure this covers everything it does. A possible world is an abstract object. It doesn't have to be interpreted as or modeled by any physical object. I don't see what the problem is.

There are a number of logics for a number of 'worlds'. Modal logic, with its possibility and necessity operators has its particular applications. Ditto for quantum logic(s). These logics have a different structure than 'classical logic'. I already described the special structure of modal logic in another thread in response to a post of yours. Quantum logic has a different structure altogether. (I may be wrong, but I believe (P^notP) is true is some versions).

If you mean logic in a very general way, I agree (maybe). But just what is this very general logic? I believe the definition I quoted in post 3 of this thread is as good as any. It seems that this very general logic should be definable, otherwise it's not..well, very logical is it?

 

[honestrosewater]

There are a number of logics for a number of 'worlds'.

Same word, different meaning. I have been speaking of logic as a field of study, with subject matter, methodologies, and practitioners. You are talking about mathematical structures. This is much easier to answer because the objects are already well-defined. The only thing to argue about is what to call them. I think a useful way to distinguish logics is by their inference relations: what they allow to follow from what, by derivation or consequence. Different axiomatic systems might be considered different logics. In a narrower context, I have seen a logic defined as the set of all logically true sentences in a (formal) language.

The possible worlds I was thinking of are interpretations and valuations of a theory (or parts of a theory).

If you mean logic in a very general way, I agree (maybe). But just what is this very general logic? I believe the definition I quoted in post 3 of this thread is as good as any. It seems that this very general logic should be definable, otherwise it's not..well, very logical is it?

What very general logic? It should be definable in what language? If it's not definable, it's not very intelligible.

No, I don't mean logic in a very general way.

I have no major problem with that definition. Logic is concerned with inferences, perhaps only certain types. If something looks like an inference, a logician might be interested in it.

 

[SW VandeCarr]

Same word, different meaning. I have been speaking of logic as a field of study, with subject matter, methodologies, and practitioners.

OK. Your saying that logic is a discipline, not unlike mathematics or perhaps even medicine or law. I see a problem in that its subject matter is logic (ie it's a category of itself). In math you have somewhat distinct but related kinds of subject matter: arithmetic, algebra, analysis, geometry, etc each with their special identifying features. In law you have constitutional law, tort law, criminal law, etc. In medicine there are the broad classes of diagnostics, surgery and therapeutics further broken down into numerous specialties and sub-specialties. Logic as a discipline, as far I know, has only itself as the subject matter and I argue again that the definition I quoted is quite adequate in defining that subject matter. (I have no vested interest in the definition. I didn't create it.)

EDIT: This may seem pedantic, but disciplines are defined in terms of their subject matter. So if logic is the subject matter of logic the discipline, then logic the subject matter should be defined. Mathematics for example is a discipline, but it's difficult to define mathematics as a whole. Russell and Whitehead tried and failed. Mathematics is mostly defined in terms of its major branches of subject matter: arithmetic, algebra, and analysis (geometry can be incorporated into algebra).

 

[honestrosewater]

OK. Your saying that logic is a discipline, not unlike mathematics or perhaps even medicine or law.

Logic _is_ a discipline. I don't have to say so. The real world says so.

I see a problem in that its subject matter is logic (ie it's a category of itself).

Why? Calling a discipline and its subject matter by the same name is precisely the norm. Do you also have a problem with logic being studied by logicians?

In math you have somewhat distinct but related kinds of subject matter: arithmetic, algebra, analysis, geometry, etc each with their special identifying features. In law you have constitutional law, tort law, criminal law, etc. In medicine there are the broad classes of diagnostics, surgery and therapeutics further broken down into numerous specialties and sub-specialties. Logic as a discipline, as far I know, has only itself as the subject matter

First, you can split up logic in the same way that you split up the other disciplines. Second, what does it matter? You have just moved the problem that you have with disciplines to subdisciplines. The subject matter of geometry is geometry -- and it is studied by geometers. The subject matter of Euclidean geometry is Euclidean geometry.

and I argue again that the definition I quoted is quite adequate in defining that subject matter. (I have no vested interest in the definition. I didn't create it.)

What you just did is stating more than arguing. You have given no justifications for why the definition is adequate.

EDIT: This may seem pedantic, but disciplines are defined in terms of their subject matter. So if logic is the subject matter of logic the discipline, then logic the subject matter should be defined.

Says who? Give me one example of a discipline whose subject matter is always well-defined. The whole purpose of studying something is to develop new knowledge about it, to make progress, i.e., to _change_ it. The subject matter of disciplines changes.

it's difficult to define mathematics as a whole. Russell and Whitehead tried and failed.

?

Mathematics is mostly defined in terms of its major branches of subject matter:

Says who?

(geometry can be incorporated into algebra).

How?

 

[Preno]

OK. Your saying that logic is a discipline, not unlike mathematics or perhaps even medicine or law. I see a problem in that its subject matter is logic (ie it's a category of itself).

No, the subject matter of logic is (valid) inference.

 

[SW VandeCarr]

Logic _is_ a discipline. I don't have to say so. The real world says so.

I didn't say logic wasn't a discipline. It clearly is. It has a methodology, practitioners and (I believe), a defined subject matter. That's why I quoted a definition. The problem I saw is that you don't get away from defining logic by describing it as a discipline.

Why? Calling a discipline and its subject matter by the same name is precisely the norm. Do you also have a problem with logic being studied by logicians?

No. (see above)

First, you can split up logic in the same way that you split up the other disciplines.

Good. How do you split it up?

Second, what does it matter? You have just moved the problem that you have with disciplines to subdisciplines. The subject matter of geometry is geometry -- and it is studied by geometers. The subject matter of Euclidean geometry is Euclidean geometry.

Only that disciplines need only share a common methodology and rational (the reason the discipline exists). I believe this is commonly accepted. The subject matter may vary within a discipline. I gave a number of examples.

What you just did is stating more than arguing. You have given no justifications for why the definition is adequate.

You're correct. I gave a citation. The source is edited by EJ Borowski and JM Borwein, two leading British mathematicians. (And please don't give the argument about about appeal to authority. I don't claim to know everything. When giving a definition, it's quite appropriate to give a reference. In fact it's required in PF if the definition is not common knowledge among the posters.)

Says who? Give me one example of a discipline whose subject matter is always well-defined. The whole purpose of studying something is to develop new knowledge about it, to make progress, i.e., to _change_ it. The subject matter of disciplines changes.

Off the top of my head, how about Number Theory? New knowledge need not change the defined area of study.

?

I don't know what you mean by "?". As someone who is interested in logic, I would have thought you might know something about Russell and Whitehead's project in the early 20th century.

Says who?

Well, there's a fundamental theorem (FT) of arithmetic, a FT of algebra and a FT of calculus; but no FT of mathematics as a whole. There are
some subjects that might fall outside these three areas such as set theory (with its own set of axioms and defined relations) and probability theory which is primarily based on definitions of a probability and a random variable. Nevertheless, most people consider mathematics as a single discipline.

How?

(RE the re-stating of geometries in algebraic form) If you don't know this, you're out of your depth.

 

[SW VandeCarr]

No, the subject matter of logic is (valid) inference.

Ok. I'll buy that, but its not a definition. You can make valid inferences that hold in specific contexts, but not others. The definition I quoted uses the words "valid reasoning" but yours is more specific. I still think a definition would need to say something about "regardless of meaning or context" and going from premises to a conclusion. But if you start with describing logic as a discipline, than it's subject matter is logic (which is defined as...) Remember, the subject of this thread is supposed to be the definition of logic. I'm just trying to stay on topic. Besides, "valid inference" does not discriminate between logical inference and statistical inference (unless you want to say statistical inference is never valid).