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Recently I was having a discussion with user21820 in this chatroom. There very naively (in the sense that I didn't choose carefully each word of my following statement) I expressed the opinion that,

The very crucial thing to human reasoning process as I see can is that it depends on context, the very thing mathematical logic totally ignores.

That's the point where I am confused. My questions are,

(1.) When I said that mathematical logic totally ignores context, I meant that the (formal) deductions totally ignores context. While I think that this isn't true in case of Natural Deduction systems, I believe that to see that it indeed isn't true we need to know what we mean by "context of statements in Natural Deduction systems", isn't it?

Background of (1.)

The first one asks whether to answer the question I asked in the title one needs to know what we mean by "context of statements in Natrual Deduction systems". This is necessary for answering my next question.

(2.) Once we know we mean by "context of statements in Natural Deduction systems" we need to make sure that this actually corresponds to (or is analogous to) the "context of statements in English Language" because if this doesn't turn out to be the case then how can we say "mathematical logic doesn't ignore context"?

Background of (2.)

I assume that when you go to (2.) you have already answered (1.). So if your answer is "yes" (which I think is the most natural answer and so I am not going to say anything about its other alternatives), then you agree that we indeed need to know what we mean by "context of statements in Natural Deduction systems". Now, assume that you already know we mean by "context of statements in Natural Deduction systems". Then my point is that to claim that the what we mean by "context of statements in Natural Deduction systems" actually corresponds to (or is analogous to) whatever we mean by "context of statements in English Language" (which, I think, is a crucial thing to assert that mathematical logic captures human reasoning process), we need to know what we mean by "context of statements in English Language". So, my point is that if "this doesn't turn out to be the case" then how can we say that "mathematical logic doesn't ignore context"? The thing is, if you assert that you have different definitions for "context of statements in Natural Deduction systems" and "context of statements in English Language" then I can't see how "mathematical logic captures human reasoning process" and if you say that "mathematical logic captures human reasoning process" (which is similar to what was asserted in the chatroom I linked in my post) then I can't see how you can have different definitions for "context of statements in Natural Deduction systems" and "context of statements in English Language" (here I use the word different in the sense that no definition is a 'special case' of the other one). So " how can we say "mathematical logic doesn't ignore context""?

(3.) In the above quote by 'context' I meant "the meaning of statements in different contexts". Note that in my extremely vague articulation of what I mean by context just stated, I emphasized on the meaning itself and not on the context of that meaning but in saying this I am implicitly assuming that "the meaning of a statement may differ in different context". This seems to suggest (although I admit that I may be pathetically wrong in stating the conclusions which follow) that the meaning of a statement isn't a property of the statement only, nor it is the property of context only, which leads me to the conclusion that the meaning of a statement is "different" from its context. If this is so then how is it true that Natural Deduction systems doesn't ignore context and in what sense?

Background of (3.)

The basic point that I wanted to make in (3.) was (in disguise) that my so called definition of context isn't acceptable so long as you accept my reasoning. It is because my definition doesn't seem to make any difference between the context of a statement and its meaning. Put another way, my definition seem to assume that context of a statement and its meaning have same ontological and epistemological status. However this (as I argued) isn't true. Then "how is it true that Natural Deduction systems doesn't ignore context and in what sense"?


N. B. - I agree that the question isn't written in a very clear manner (and honestly, I was a bit hesitant to post it). If there is anything in the question that makes it not suitable for this site, please let me know. I will remove it.

  • The answer to your title question is very simply yes, (classical) mathematical logic is intended to be completely general and doesn't attempt to capture any context. The questions in the body: I'm not sure what you mean to ask there. – Eliran Jan 7 '17 at 17:53
  • @EliranH what do you mean by 'classical'? Do you mean historically classical, such as what Aristotle would have written? I hardly think that should be regarded as mathematical logic, that discipline has been defined for over a hundred years now as something completely different. If you mean classical as opposed to other forms of logic, such as fuzzy logic, then model theory is definitely included in the definition of classical logic and therefore it most certainly does capture context. – Not_Here Jan 7 '17 at 19:50
  • Everything ignores (some) of the context, mechanics ignores all properties of bodies except mechanical ones, for example. Mathematical logic obviously does not ignore it "totally", or it wouldn't be applicable to anything, but it does focus on purely logical/quantitative side. But natural deduction is equivalent to other deduction systems, and so has no special status in this regard, except perhaps for being closer in form to natural reasoning. You may want to look at proof-theoretic semantics for rules as meaning-givers. – Conifold Jan 8 '17 at 0:14
  • @EliranH: You said, "The questions in the body: I'm not sure what you mean to ask there." Can you point out exactly what it is(are) in my questions that isn't clear to you? – user 170039 Jan 8 '17 at 4:05
  • The edit has been made as per the discussion in this chatroom. – user 170039 Jan 8 '17 at 7:50
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After a long discussion with the author of the question I am going to section off this answer into two parts. The first part deals with the answers to the three questions posed in the body of the question. The second part deals with the answer to the title of the question, which is "Does mathematical logic totally ignore context?"

1

In regards to the first question which asks (paraphrasing based off of our conversation) “In order to answer 'does mathematical logic totally ignore context?', do we need to have a definition of 'context in natural deductive systems?'” The answer to this question is yes, we need to have a definition of context before we can ask if mathematical logic ignores context.

In regards to the second question which asks (paraphrasing based off of our conversation) "If the answer to (1) is yes, then what we mean by "context of natural deductive systems" corresponds to "context of statements in the English language? Otherwise logic doesn't capture what human reasoning is." The answer to this question is yes but it is a subtle yes and it only applies to when we are expressing the formulas from our logical system in English. When we make a declarative statement in English we are making some sort of factual claim. However, this may be in reference to many different contexts. This could be in context to the real world, it could be in context to some fictional world from a movie or a book, or it could be in reference to some hypothetical alternate history. So, this clearly shows that sometimes statements in English are true and sometimes they are not, depending on the context we put them in. If we say "he shot the butler" and we are referring to some film wherein a character shoots a butler then that statement is correct. However, if we are referring to the real world and "he" denotes someone who did not shoot a butler, then that statement is false. This is the exact idea of a domain of discourse which I define below. This is where mathematical logic gets its concept of "context" or "meaning." The "context" of a natural deductive system, just like English semantics, changes depending on how we use it. In English it refers to what we are talking about (our planet, some other alternate history, the world in a book, etc.) and in mathematical logic it refers to the model (which part 2. of my answer defines extensively).

In regards to the third question which asks (paraphrasing based off of our conversation) "If we assume that the intended definition of context as has been stated in the beginning of the question isn't acceptable (i.e., the meaning of a statement is "different" from its context) then how is it true that natural deduction systems don't ignore context and in what sense?" First, we'll define what meaning and what context have to do with this situation. The "meaning" of a statement in this context refers to its semantic content. In terms of mathematical logic, this means the model theoretic properties of the particular statement. "Context" refers to what specific domain of discourse we are discussing. The context is the actual domain (the real world, or a book, etc..) and the meaning is whatever factual propositions are being asserted about this world. This leads directly into part 2 of my answer which shows that deductive systems on their own do not deal with meaning. In just the same way, English sentences alone do not deal with meaning. If no world exists, if we have no domain of discourse, then the sentence "The cow is happy" has no truth value, because it doesn't refer to anything. Model theory, in mathematical logic, is what gives natural deduction something to refer to.

To reiterate a very clear and concise answer to (3): By itself natural deduction does not refer to specific context or meaning, because it is only syntactic in nature and refers to just the deductive processes. The "natural" part of "natural deduction" does not mean "referring to the natural world" (nor does it refer to "natural languages" which may be where some of your confusion regarding English semantics is coming from). "Natural" was used because the inventors of the theory wanted the deductive process to feel more like what the human mind experiences when it does deductions. This says nothing of the context the deductions take place in. To understand the difference between purely syntactical (proof theoretic) ideas and semantic (meaning and context, model theory) ideas, keep reading on with section 2. Only looking at proof theory gives you only half the story.

2

Now we can move on to answering the titular question:

The definition of "context" in mathematical logic is given by the domain of discourse that the formulas range over. The definition given on wikipedia is as follows:

In the formal sciences, the domain of discourse, also called the universe of discourse, universal set, or simply universe, is the set of entities over which certain variables of interest in some formal treatment may range.

Given this definition, the variables that appear in a formula are just random variables until we give them context by providing a domain of discourse. In mathematical logic the domain of discourse is given by model theory.

From the content of your question it looks like you are only focused on half of mathematical logic, the syntactic half. Yes, it is completely true that the syntactic half of mathematical logic, proof theory and recursion theory, don't care about context in the sense that you defined it ("the meaning of statements in different contexts"). However, this ignores the other half of mathematical logic, set theory and model theory. Model theory is the discipline that explains how we can assign context, meaning, to syntactic structures. Set theory is the study of those specific objects and how they work.

When we say something like A ⊧ φ in logic, we are asserting that φ is true in the model A, which means in the context of A. It is very common for people who have only a basic introduction to logic to say "oh, logic doesn't care about truth! It just tells you what arguments are valid, not what arguments actually are true given a specific context" and this is because most introductory logic classes don't delve into mathematical logic. As Peter Smith said in this outline of how to learn logic, he refers to it as "baby logic." (I don't mean for this to be dismissive towards you or to even imply that this is what you are talking about. If you know what mathematical logic is then you already know more than the people who get a basic introduction. My point is that it is very common for classes and introductions to logic to be very basic and ignore the concept of semantics and truth because it is a lot more complicated, and unfortunately this gives people the wrong impression of what logic is.)

In the Oxford Handbook Of Philosophy Of Mathematics And Logic Steward Shapiro has a paper about logical consequence, both syntactic and semantic. The syntactic half of mathematical logic, proof theory and recursion theory, deals with no context, it is purely the study of syntactic consequence and deduction. The semantic half, model theory and set theory, are the tools we have to put deduction into context. They are the disciplines that show us how to add meaning to arbitrary logical statements and show in what context those statements and their deduction hold to be true. In your question you made statements referring to the deductive system used in mathematical logic to not include context. This is true, but again, this only acknowledges half of mathematical logic. So if your question is about mathematical logic as a whole the answer is no, it does include context. If the question is just about deduction, which is not all of mathematical logic, then yes, it does ignore context.

Considering if what you proposed were to be true, it would be horrible if mathematical logic had no way to establish context. If it had no sense of context, we could not prove a single thing about numbers, not even the basic notions of arithmetic! When we look at the deductive rules and axioms of arithmetic (PA for this argument) we are looking at the syntactic structure of some axioms and their deductive system. However, we use model theory to discuss these axioms in the context of the natural numbers. This means that we are concerned with the truth of these statements as they pertain to a certain context. If we were unable to discuss models mathematical logic would be pretty useless as a foundation for all of mathematics.

To give an example that is very concrete we can look at the axioms of a group. The model of a group is a set (A,M) where A is a set of elements and M is a function that obeys the group axioms. The group axioms are closure, associativity, identity, and inversion. Now, an example of a group is the set of integers, Z, equipped with the function of multiplication. This is a group because in the context of Z and multiplication all statements in the language that are deducible from these axioms are true. Now, in the context of a different set, say the natural numbers, N, equipped with multiplication, the group axioms are not satisfied. Why? Because, the natural numbers do not obey the axioms of inversion. There is no inversion of "1" if we do not have "-1" as an element. Clearly this demonstrates that the deduction is purely syntactic but mathematical logic as a whole is about applying syntax to specific semantic contexts.

  • What is the notion of "context" in Mathematical Logic and what about the answer of (2.)? – user 170039 Jan 8 '17 at 4:13
  • I'm afraid I don't understand what you mean by "notion." Did my answer not give enough of a definition of context? In mathemetical logic, which is more than just deduction and proofs, context is established by models. Models are sets and model theory shows us how to construct and apply these sets to the axioms and rules of inference. – Not_Here Jan 8 '17 at 4:18
  • As for (2) I'm afraid that it really is not clear what you are asking. I avoided answering it because I did not know what exactly the question was. My answer was trying to establish what context means in mathematical logic which is the title of your question and felt like the main question you were asking. Again, the idea of context is found in model theory. – Not_Here Jan 8 '17 at 4:20
  • Regarding the definition of context, I don't think that your answer gives me a precise definition of "context" (here I am using the word in its most general sense), nevertheless it gives me an idea of what context in Mathematical Logic may 'mean'? As for (2.), can you tell me what exactly isn't clear to you in the question? – user 170039 Jan 8 '17 at 4:36
  • Can we move this conversation over to the chat that you posted, comment sections aren't supposed to be for long conversations – Not_Here Jan 8 '17 at 4:38
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We need to make sure that this actually corresponds to (or is analogous to) the "context of statements in English Language" because if this doesn't turn out to be the case then how can we say "mathematical logic doesn't ignore context"?

The answer to this question is that "context" in logic can be precisely defined syntactically, and the syntax is intended to mimic the idea of subordination that we find in natural reasoning and language. So yes, logic is intended to syntactically represent contexts in the same way as natural language. But no, it cannot actually capture the contexts of natural language in an absolute sense, because there must always be a layer of interpretation that converts the symbolic contexts and statements in any logic system to the semantic meanings that (we believe) we understand.

For a simple example, we can say:

Let L denote the statement that natural language is useful.

The logic system can never perform that mental connection between "natural language is useful" and the actual meaning it conveys to most speakers of English. Moreover, different people can interpret it differently and it is opaque to the logic system. However, we purposely ignore that issue so that we can manipulate statements logically according to inference rules before we go back and interpret our conclusions. If the audience agrees with the soundness of the inference rules according to their interpretations, then it is enough to force them to accept conclusions that are validly deduced from accepted premises. Their interpretations may be totally different from ours, and we may even have no way to tell. That aspect can never captured by logic or mathematics.

How is it true that Natural Deduction systems doesn't ignore context and in what sense?

Natural deduction is merely a paradigm, not a system by itself. Its main defining feature is that it utilizes syntactically defined contexts to allow and facilitate symbolic reasoning inside and outside them. In particular we often have introduction and elimination rules to govern contexts.

For instance, the modal logic S4 has a necessity operator "☐" that can be governed by the following context rules loosely described as follows.

If you have deduced:

☐:

...

P.

You can deduce:

☐P.

And vice versa.

We also permit classical reasoning in any context, which together with the ability to create a new context under a "☐", will give rise to the distribution axiom as defined in this SEP article. If we further add the necessitation inference rule, we automatically get both it and (4), which shows the close relation between the two in the framework of natural deduction.


As an aside, a brief discussion with Not_Here revealed that there are some differences in our perspectives, so I'll point them out. Firstly, we both agree that mathematics (and in particular natural deduction) can easily deal with and analyze the same statement in different contexts. However, he/she also claims that model theory is the answer to semantics, and links to SEP's article on model theory. I consider that it presents an incomplete and misleading picture, from a logician's perspective.

The reason is that ultimately any form of reasoning has to be based on some meta-system. Without a precisely defined meta-system, there is no way to validate or invalidate arguments. But the only known way to precisely define a meta-system, as of today, is through syntax. One common way is to specify inference rules of the form "If you have deduced statements of the form ..., then you can deduce the statement ...", and then specify that the meta-system deduces no other statements besides those generated by the inference rules. However, in the interest of making my claim precise, this type of formal system is too restrictive. To achieve the most generality, we define a formal system S to be a system that has a proof verifier V, which is a program (in some fixed Turing-complete language) that, given as input a pair (P,X) (in some fixed encoding), decides (always outputs "yes" or "no") whether P is a valid proof of the statement X or not. While I use the terms "proof" and "statement", they are merely to help the intuition and are not actually part of the definition. In other words, we actually define that a finite string P is a proof over S of a finite string X iff V(P,X) is true, in which case we say X is a theorem of S.

Now we can get down to the contended claim. Model theory, as the term is used by modern logicians today, refers to a branch of mathematics, and modern mathematics is usually based on a particular formal system called ZFC set theory. But ZFC set theory is just one of many incompatible possible foundations for mathematics. So before one can claim that anything can capture real-world meaning correctly, one would first have to justify that it is based on a formal system that is truth-preserving for statements about the real-world. In the case of modern model theory, one would have to justify that ZFC set theory has real-world meaning in this very sense, and this would require that one can give a real-world interpretation of any sentence over ZFC, and that one can show the axiom schemas of ZFC to be truth-preserving (here I'm already granting that classical logic is sound). This is a very tall order, and no logician has ever done this, and remember that there are many incompatible candidates for a foundation of mathematics, so no two of them can be both compatible with the real-world.

Of course, "model theory" as used by philosophers may mean much less than "model theory" used by logicians. That is perfectly normal. But the first consequence of the issue of having to precisely specify the meta-system is that one should not just refer to model theory as if there is only one such notion. In fact, there is a fine spectrum of meta-systems and corresponding philosophical justifications or lack thereof, and I give a brief account and some links in this post.

It should be clear after looking at the spectrum of possible meta-systems that things are not so easily justified as absolute as one might think. For instance, one cannot ever consider "the natural numbers" to be absolute. All useful meta-systems will have a structure that satisfies the axioms of PA, but it is peculiar to each meta-system. No meta-system is able to refer to the so-called 'real' natural numbers in the real-world, even if they exist. The reason is simple and can be justified as follows (in a suitable meta-system).

Take any (consistent) useful formal system S that can construct any collection of 'natural numbers' that is first-order definable over PA. Then there is an arithmetical sentence Con(S) such that:

  1. S does not prove that S is consistent.

  2. S proves that N satisfies Con(S) iff S is consistent.

Now S cannot prove its own consistency otherwise (1) would make S inconsistent. So S cannot prove that N satisfies Con(S). Also:

  1. ( S + S is inconsistent ) is consistent.

  2. ( S + S is inconsistent ) proves that N satisfies PA.

Of course, we reject ( S + S is inconsistent ) as being actually useful, but why? Simply because (here in the meta-system) we can prove that ( S + S is inconsistent ) must have the wrong notion of N, namely that it is different from the 'real' N (that the meta-system knows). But this shows that PA is utterly insufficient to capture the notion of natural numbers. Historically, Godel also did this to prove the existence of a nonstandard model of PA.

But this has serious consequences. We are unable to pin down the natural numbers, but we need to specify a meta-system that has that very collection (that one might perhaps believe has some platonic existence). No extension of PA will be enough, and so our meta-system may very well be referring to some strange model of PA that is not really the same as the natural numbers of the real world (if such a structure exists). Even the meta-system itself knows this possibility (as shown above)!

This is why ultimately it is a philosophical question whether some given formal system has real world meaning, which then has vast implications for model theory that is carried out in that formal system as the meta-system of choice. Too weak, and one would not be able to deduce basic things about logic, such as described in the linked post. Just one 'wrong' inference rule, and the meta-system would necessarily have a collection of 'natural numbers' that it believes is standard but is actually not. If we believe that there is a collection of representations in physical media that obeys PA, when suitably interpreted using algorithms, (which explains why HTTPS and other known algorithms work,) then we necessarily believe that this model of PA is standard by definition of the physical representations, and hence we cannot accept any meta-system that does not have a standard model of PA. In logic terms, we only accept a meta-system that has an ω-model. The problem is, as you may expect by now, we cannot define "ω-model" except relative to an existing model of PA...

(Actually there is ω-logic, that can actually pin down the natural numbers and the entire first-order theory, but naturally of course ω-logic has no effective deductive system.)

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