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Suppose we are to learn a subject like mathematics, then for it's precision and rigor, the topics discussed are described in terms of theories of logic. Suppose a student wishes to learn mathematics, then he must first see some explanation of the logical language in a way he can understand. This explanation usually is done using a natural language.

If natural language is not fully logical, how could it be possible to explain something purely logical through it?

On the surface level , after the explanation, it may seem we can check the student's understanding by probing them with questions... but one can never know if the concept of the logic they have in their mind is the same as that of the explainer. It maybe that if we have a mind reading technology, and the instructor saw their understanding, the conceptual picture of the student and the teacher are entirely different.

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    Indeed you've touched a common, subtle and perhaps deep issue during teaching and transmission of any rigorous logical stuffs including maths, as it's not hard to notice so many students struggling with it almost everywhere. It certainly depends on the instructor's ability to organize their natural language explanation in a paradox free and lucid way as the first stepping stone if the student cannot truly understand by themselves... Jun 12 at 18:29
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    How is math different from other subjects in this sense? How do you ever know you are communicating successfully? Jun 13 at 10:23
  • "How is it possible to explain a logical language through a natural one?" Every "artificial" language: science, mathematics, must be explained through the natural one: we have only it. Jun 14 at 8:43
  • Finally there's still meta-language behind the natural ones (not unique), since many different natural languages across different populations and regions express nearly same ideas which must be equally influenced by some universal principle (such as mentioned in Hume, see a recent post). If one can directly have the right idea/universal principle with this meta-language, one can bypass the natural ones. It's conceivable one only has FOL as its speaking language... Jun 15 at 4:57
  • A plausible explanation is that a logical language is a subset of natural language, thus by definition an articulation of a logical language is also articulation via natural language.
    – Nikos M.
    Jul 17 at 12:45

6 Answers 6

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Short Answer

If natural language is not fully logical, how could it be possible to explain something purely logical through it?

Well, simply put, "purely logical" is a suspect term. However, let's say what you mean is a formal, that is syntactically abstract, artificial language of logic as used in formal systems. In this case, explanations of the "purely logical" things, like FOL, for instance are ONLY meaningful because they abstract something more concrete; that is, pure logic is "distilled" natural logic, which is called informal logic. It is from the natural language that the ideas of the formalisms are originally drawn, and therefore it is in the natural language and the experiences that they represent they have any meaning at all, at least according to intuitionism.

Long Answer

As someone who programs computers, I don't have the luxury of assuming mathematical and logical expressions are some sort of Platonic entities floating around in a "Heaven of Numbers". Logic operations and computer languages that create and use them, particularly in higher-order languages out of type theories, don't spontaneous exist. They simply have to be built. In the case of high-level languages, they come from the machine codes. In the case of machine codes, they come from the work of computer engineers and the hardware they design. Put simply, they are constructed. The AND is constructed (usually out of NAND gates on a chip). The OR. The XOR. What we know as predication in philosophy is simply a computer procedure.

So how do we describe these operations and primitives of pure logic, like variables, domains of discourse, and identity? With natural language, intuition, and metaphors. But you may protest, how can something impure characterize something pure? And here then is the crux of your dilemma. You are engaged in the fallacy of division in the very question you ask! From WP:

A fallacy of division1 is an informal fallacy that occurs when one reasons that something that is true for a whole must also be true of all or some of its parts.

So, implicit in your question is "how can something that is purely logical be explained by something that is not purely logical"? Simple, the purely logical systems to which you refer have logical parts, but those logical parts don't have to themselves be purely logical! And it's just as simple as that. Thus, we explain AND in terms of choices, and apples, and use our intuitions. The parts of anything which you might consider (like a formal system of logic) to be purely logical can have many properties that are not purely logical. To put it in a different perspective, how can one explain the flight of a plane when none of the parts by themselves are capable of flying?!? Simple. It's the combination, the structure if you will, that creates emerging properties. A formal system might be characterized as purely logical, but that pure logic (a term that is ill-defined) is nothing more than that which comes from a systems whose sum is more than its parts.

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Add new words with strict logical definitions

A "boolean" value is a logical construct that is either true or false. Named after George Boole, who first defined an algebraic system of logic in the mid 19th century. https://en.wikipedia.org/wiki/Boolean_data_type

Repurpose existing words to have strict logical definitions in a certain context.

The word "true" in common English can mean a number of things e.g. "He's no true Englishman!" or "I promise I'll be true to you." But in a mathematical context, the word "true" can be defined as "one of the two possible states of a boolean variable".

Language is context sensitive

Because language is context sensitive, it's possible for me to understand that The value of 𝒙 is true is a strictly logical statement. But "He's no true Englishman" is not a strictly logical statement.

Maybe I don't understand the depth of the question, maybe if you're questioning logic being representable with language, then no sequence of words within a language could even theoretically provide an answer for you. But is a question that can't even theoretically be answered even worth asking?

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Perhaps the "boundary" between natural and constructed languages is vague. The whole idea of precisifying all terms and entangled concepts sounds admirable from the outside, but looked at from within (in the attempt at further and further precision),

[e.g.] Frege’s ideal of precision is itself vague because ‘precise’ is the complement of ‘vague’. Second, the vagueness of ‘vague’ dooms efforts to avoid a sharp line between true and false with a buffer zone that is neither true nor false. If the line is not drawn between the true and the false, then it will be between the true and the intermediate state. Any finite number of intermediates just delays the inevitable.

Since the quoted paragraph qualifies the local problematique as a function from a "finite number" of intermediaries, one is tempted to ask after infinite-valued logics. However, if we have to cycle through infinitely many degrees of truth, one wonders if we have lost track of our desire for precision again.

In the other direction, though, then: syntactic options in so-called formal logics are quite lacking, compared to in the so-called natural case. For example, semiotic offsetting is handled primarily in the former cases by parentheses, whereas in English (say) there are also commas, colons, semicolons, n-place quotation marks, em dashes, and different flavors of parentheses. Now set theory, to be fair, features many of those, too, and yet is often construed as formalization par excellence, yet just the same, by now we recognize how much creativity there is in mathematics, especially on the "foundational" level, so we have found space to reintroduce a lot of syntactic differentiation as such. (C.f. programming languages for software.)

So rather than style natural language as less formal than what is said to be formal language, perhaps it's time to recognize that natural language is (or, better, can be) just as formal on its own terms, too. Someone who can follow the train of a natural-language argument quite strictly is perhaps more adept at reasoning than someone who has to have everything distilled into less intricate strings of symbols with "easy" (convenient) rules for deriving new strings from those just given.

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  • Do you know of any books or authors that talk about formal language as a whole like this?
    – J Kusin
    Jul 17 at 14:54
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    Not quite directly, but Mark Z. Danielewski's House of Leaves, and then his The Familiar books, showcase an intricate blend of "ordinary language," academic formality, programming typography, and visual communication. One might also look into Peirce's logic graphs, the issue of Frege's 2-dimensional notation, and maybe specific geographically/ethnically variant languages. Jul 17 at 18:09
  • Malcolm Gladwell wrote a book in which he assessed the stereotype of "Asian mathematical proficiency" by noting that in many of the stereotyped languages, words for numbers are shorter on the baseline than in, say, English, IIRC opening the door to a 2-second window of extra neurological efficiency in processing. I also think Kant's question about 7 + 5 = 12 being synthetic gets an odd answer if we think of it in terms of Latin numerals (does V + VII = XII seem more or less obviously synthetic than the English counterpart?). Jul 17 at 18:13
  • Also, the closest thing to a book-length treatment of the topic might be the lattermost section of Kant's first Critique. He goes into some detail about why we are not to expect that mathematical formalism will do us much service in reflecting on metaphysical issues. Beyond that, there's probably a treatise or two in older-school "ordinary language philosophy" that touches on the problem. Jul 17 at 20:10
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Wittgenstein claimed language is not rational (“logical”) because there is no way to state all types of sentences.

Searle apposes this point with a comparison to animals; there are stuffed animals, large animals, animals with no beak, mammals, reptiles, and countless other possible types, yet we can taxonomize them. This means there are restrictions to animals. Sentences (language) have this property too according to Searle. So even without “rationality”, we can make sense of countless uses of language.

This can be used to explain how we can understand speech acts and propositions as different primary objects of study. Speech acts and logical propositions are wildly different things. Yet, we don’t need any underlying rational principles as long as we pay careful attention to use.

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In their book How to Read a Book: The Classic Guide to Intelligent Reading (Mortimer J. Adler and Charles Van Doren), they write (pages 277-8):

  1. The Systemization of Philosophy:

In the seventeenth century, a fourth style of of philosophical exposition was developed by two notable philosophers, Descartes and Spinoza. Fascinated by promised success of mathematics in organizing man's knowledge of nature, they attempted to organize philosophy itself in a way akin to the organization of mathematics.

Descartes was a great mathematician and, although perhaps wrong on some points, a redoubtable philosopher. What he tried to do, essentially, was to clothe philosophy in mathematical dress--to give it a certainty and formal structure that Euclid, two thousand years before, had given geometry. Descartes was not wholly unsuccessful in this, and hos demand for clarity, and distinctiveness in thinking was to some extent justified in the chaotic intellectual climate of his time. He also wrote philosophical treatises in a more or less traditional form, including a set of replies to objections to his views.

Spinoza carried the conception even further. His Ethics is written in strict mathematical form, with propositions, proofs, corollaries, lemmas, scholiums, and the like. However, the subject matter of metaphysics and of morals is not very satisfactorily handled in this manner, which is more appropriate for geometry and other mathematical subjects than for philosophical ones. A sign of this is that when reading Spinoza you can skip a great deal, in exactly the same way that you can skip Newton. You cannot skip anything in Kant or Aristotle, because the line of reasoning is continuous; and you cannot skip Plato, and more than you would skip a part of a play or poem.

Probably there is no absolute rules for rhetoric. Nevertheless, it is questionable whether it is possible to write a satisfactory philosophical work in mathematical form, as Spinoza tried to do, or a satisfactory scientific work in dialogue form, as Galileo tried to do. The fact is that both of these men failed to some extent to communicate what they wished to communicate, and it seems likely that the form they chose was a major reason for the faliure.

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  • This is interesting reading, but you could perhaps add a few sentences at the end explaining how it answers the question. I get a general idea, but I don't get an answer to the question from reading this.
    – wizzwizz4
    Jul 17 at 11:31
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Here are a couple of thoughts :

(a): Let the metalogic (and corresponding metalanguage) be M, let the object language + logic be O. If we accept a strong correspondence between metaphysical grounding and epistemic explanation, then the dilemma is roughly the question of what grounds the purported logical properties of O, given that O is itself grounded by M, which by stipulation is a natural language. In general, natural languages are equipped with a metalogic so powerful that we can prove false, or state the liar, etc.

(b) Stated as such we can pick out at least one necessary condition: a shared understanding for the semantics of certain operators or concepts in the metalanguage. Take for example soundness. Clearly, a shared understanding of the semantics of the natural language operator “if” are necessary for shared agreement wrt soundness. Further we will also need a shared understanding of the natural numbers. The situation is analogous to a chess game. We can determine a winning move if we can agree on the rules of chess.

(c) the last paragraph can thus be phrased as a claim that we cannot know whether some agent A shares the same concepts with respect to the metalanguage. But this is quite a strong claim (although not entirely without merit). Here are some discussions that might help to settle the matter: (i) the possibility of private languages - if not possible, then evidence for langauge use is purely public. (ii) conditions on knowledge: mind-reading technology doesn't seem to be the only way that we can know what someone means. Otherwise, how do we successfully communicate ordinarily? Theres probably more, but that should be good enough for now.

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