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To my understanding, we talk about things like propositional, predicate and higher order logic because spoken language is not fully logical. But, how exactly is it not logical? Usually the ambiguity of a sentence in English is caused by context and meaning of words we take.

Secondly, if one were to describe the logical language to someone else, they would require a meta language to communicate in (eg: English). If that is so, due to the lack of illogicality of English, wouldn't that mean the communicated logical language is also to an extent illogical?

tryst with freedom
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  • Logical language is a simplified model of natural language. – Mauro ALLEGRANZA Mar 22 '22 at 17:35
  • Related – tryst with freedom Mar 22 '22 at 18:42
  • Natural language is "circular": there are no primitive (undefined) terms from which all other terms are defined (see a dictionary). In same way, also logic and mathematics is: we have to use "natural logic" the explain what a formal argument is and we have to use "natural ability" to count in order to explain what mathematics is. No way... but it works. – Mauro ALLEGRANZA Mar 23 '22 at 09:16
  • Language is not "logical". Language is the tool that allows communicating information. Such information is what is logical or not. Language is as logical as any rock. Formal language are language subsets, which, in addition to linguistic rules include additional elements to deal with logic. But language itself is not "logical". That's why language can be described by itself: not because it is not logical, but because in some cases (formal subsets) it allows addressing logical issues. – RodolfoAP Mar 24 '22 at 03:18

4 Answers4

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Part 1: Types of Language
Languages cannot be rigorously divided into "spoken" and "logical" categories. Rather, the division you are hinting at is "informal" and "formal". What is extremely important is that a system of logic can be based on either type of language (italics in original, bold added):

Typically, a logic consists of a formal or informal language together with a deductive system and/or a model-theoretic semantics.

When the system of logic is based on informal language, it is called an informal logic. The resulting system is still logical, just of a different type than a formal logic, such as first-order logic.

Part 2: Describing Languages
Spanish is a gendered language. To paraphrase as my Spanish teacher, "I have yet to find genitalia on my table, but mesa [Spanish for table] is still a feminine word". English is not a gendered language (for the most part). The fact that a person who can only speak English can learn to speak Spanish proves that the English language can describe aspects of Spanish that it lacks itself. More generally, a language can describe another language even if the latter has aspects the former does not. Therefor, a informal language can describe a formal language without 'tainting' the formal language with illogicality.

E Tam
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  • All languages are formal. I don't know what you mean by informal language. Can you give an example. What you are describing fall under the categories of an object language & a meta-language not formal & informal. The object language is whatever the communication is between you & other people understand. The meta language is the evaluation of parts of a language you are speaking about semantically. For instance you can describe Spanish language sentences in the English language & how they differ: i.e., gender related words. – Logikal Mar 23 '22 at 11:31
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    A formal language has a definitive form (i.e. all of its rules are set and cannot be changed). Informal languages, which are also called natural languages, are constantly changing. "[Formal languages] are notable for their rigorously defined syntax, semantics, and grammar, and precisely defined proof procedures. In contrast, arguments as they occur in real life discourse are notable for their use of everyday language" (https://plato.stanford.edu/entries/logic-informal/). – E Tam Mar 23 '22 at 14:23
  • Perhaps you misunderstood me. All languages must be formal for other humans to communicate. All languages have a syntax. Syntax describes how to communicate in a given language. In English for instance the noun must come before a verb in a sentence. This is FORMAL in English. If I violate the rules of syntax the communication is likely to be misunderstood. I think you are confusing what I stated with the varies words that can be used to communicate. That is there are multiple ways to Express the same idea. Philosophy uses the term proposition to do that. Propositions are not sentences. – Logikal Mar 23 '22 at 17:14
  • Communication with no formal aspects would serve no purpose. What you are advocating is words can change so that eliminates the form. This is bad because the syntax of the language must still be followed --the words you use are not relevant. – Logikal Mar 23 '22 at 17:16
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Let's take a first pass at the title question. If by "spoken language" you roughly mean "natural language" and by "logical language" you roughly mean formal language, the answer is easy. In this case, the distinctions are multitude, the primary one being that natural languages "evolve naturally" - consult a linguist to see what exactly that means.

Your first paragraph notes that spoken languages are not "fully logical", despite the fact that we might yet be able to construct some "translation" of, say, English into a formula which is itself a member of some formal language. Two comments here: first, we do as you say have tools for resolving ambiguities, eg modal logic for modal operators and supervaluationism for vagueness. This gives rise to the second comment: a language being "fully logical" seems to assume a logic of comparison. As there are many logics, a natural language might be "fully logical" with respect to one logic (although such a logic will almost certainly never be formalized, just think of all the irregularities and constructions that a native speaker might make) and yet not so with respect to another. For example, English is certainly not fully logical with respect to CFOL, we can "refer" to non-existents, say unicorns. So we might need at least a free logic ( or you will have to change what you mean by "refer", but thats another story).

As for your second paragraph: Suppose we have a metalogic M1 and we decide to formalize an object language O1 with it. Suppose that we have fixed a background logic by which we make a comparison for "fully logical", say an intuitionistic logic. Suppose that M1 is not fully logical, say a classic logic. Is O1 necessarily not fully logical? The answer is probably not, since we can easily formalize an intuitionistic logic in a classical metalogic (just open up, say Kleen's text on metamathematics, we should be able to formalize his intuitionistic system in ZFC- but I haven't checked, which is why I say probably). Of course, you may then ask this question about ZFC.

This really opens up some rich questions about the relation of the metalogic to the object logic. Here is where I will point you to Dummett, who considers in great depth what a metalangauge should look like and what relations an object language might inherit from certain types of metalanguage. While I would be happy to point you toward other sources, I am unfortunately a few years removed from the subject material, so perhaps someone else can edit this post as they see fit.

emesupap
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  • Hello. Thanks for the deep answer. I don't think I can comprehend some parts at the moment iwth the knowledge I have. However I'll look into the words and terms you are using – tryst with freedom Mar 22 '22 at 18:40
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Over at Math SE I've said a fair bit on the intrinsic circularity in mathematics, which explains how we can learn English and other things using English, despite circularity in semantics (e.g. you cannot define "if" without using some equivalent concept). This circularity does not prevent everyone from arriving at a precise common understanding of what certain basic constructions (e.g. "if X then Y") mean, which then allows us to bootstrap to define 100% precise formal systems using these basic English constructions.

On the other hand, I want to say that your belief that ambiguity in English is usually caused by context and meaning of words is a bit off. Natural languages have serious issues that cannot be solved by having precise meanings and contexts. For example take a look at this version of Quine's paradox:

" preceded by the quotation of itself is not a true sentence." preceded by the quotation of itself is not a true sentence.

which is a completely grammatical English sentence that appears to be a well-defined assertion Q that asserts something about X preceded by the quotation of X, where X is the string " preceded by the quotation of itself is not a true sentence.". Is Q a true sentence or not?

Think about it carefully. If you don't see a problem, or think that there is an easy resolution, then you're almost surely missing something big. Once you realize the difficult in resolving this paradox satisfactorily, see this Math SE post for one clean resolution (skip to the section "Why Quine's paradox fails" and ignore the rest).

The resolution also tells us that we must be careful in our assumptions about what is meaningful and what is true/false. That is why it is important not to carelessly assume we can just use the full English language for rigorous reasoning.

But as I said above, although natural language has some problems, we can still describe a formal system S to someone else by using only a small part of English. We can first describe the syntax and behaviour of simple programs (it suffices to have string variables, basic string operations, and if-structures and while-loops), and then describe S by writing down a proof verifier program for S, namely a program that given each input (p,x) would output "yes" if p is a valid proof of statement x in the system S, and would output "no" otherwise.

Your only 'hope' of arguing that this is not 100% precise is to argue that we may fail to come to a common understanding of the syntax and behaviour of simple programs. However, the fact that you are reading this post using a browser (which is a compiled program) shows that even complicated programming languages have successfully been described and understood.

It is true that most logic texts do not write down a proof verifier program for FOL, but every logician who knows programming can easily do so based on the description given in any good logic text, so 100% precision is still achievable.

user21820
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  • I can understand why you write what you did above but there are some missing links between Mathematical logic & terminology in Philosophy. True has multiple contexts & math as well as Psychology has the misleading definition. There are at minimum three contexts not one: objective truths, contingencies & scientific truths. What math tends to do is side with science eventhough Math is not itself a science. You say x is true if x fits the criteria you describe with your awareness intact. So any future statements you hesitate on that being TRUE because you are unaware. This is requires senses. – Logikal Mar 23 '22 at 19:44
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    Your definition of truth has too many issues especially when Mathematical logic requires ASSUMPTIONS. Thus you have no 100 percent certainty. If your assumptions are mistaken the rigours of method are lost. Mathematics doesn't teach that arguments must have a specific format or you are playing psychological games. Your language problems go away when premises are written a specific way. Use more nouns & less adverbs & adjectives. You will find less paradoxes with specific details given. The language issues arise because of the over use of adverbs & adjectives. This creates ambiguity. – Logikal Mar 23 '22 at 19:50
  • This is a very good post and links. But can’t simple sentences in English (Quine paradox) and math be imprecise? If so that cancels a lot of the distinction you assert imo. For example, don’t constructivists doubt the “precision” of invoking infinity like in arithmetic laws and successor function? – J Kusin Mar 24 '22 at 00:09
  • For example constructivist N J Wildberger says there isn’t an algorithm to add two real numbers. Even a simple mathematical statement can be “imprecise”. – J Kusin Mar 24 '22 at 00:19
  • @JKusin, the way to resolve ambiguities is to require premises be worded precisely in a certain way. That means all relevant information needs to be directly written out & no vague adverbs or adjectives be included in a argument. What math & other topics do is just accept anything someone offers any kind of way. Philosophers use propositions to represent ideas not what words you use. That means you can mean the same idea with different words & sentences. Propositions need to written a specific way to steer clear of deception. Any kind of way in an argument should not be the norm. – Logikal Mar 24 '22 at 01:42
  • @JKusin: As I said, once you actually understand what a formal system truly is, namely one with a proof verifier program, there is absolutely zero imprecision. Everything comes down to that and nothing else. Constructivists do not doubt PA. Only ultrafinitists do, but most ultrafinitists are inconsistent because even pure FOL fails under ultrafinitist assumptions. – user21820 Mar 24 '22 at 09:18
  • Logikal: Your comments don't make sense, and I repeat that if you don't grasp the precision available via programming then you will never grasp the fact that formal systems can be 100% precise. – user21820 Mar 24 '22 at 09:21
  • By the way, N J Wildberger is a crank, and nothing he says about foundations of mathematics is worth paying attention to. It's not just opiniated but blatantly false! Anyone who actually knows basic logic and set theory can immediately verify this. – user21820 Mar 24 '22 at 13:18
  • @user21820 I hate to press you further because I know you are way more qualified….isn’t any proof verifier going to be limited to number systems computers can compute in finite time? That I get is absolutely precise and those completed proofs are unobjectionable. But if a mathematician working in real numbers says the sum of e+pi+sqrt2 exists, she should prove it right? How besides saying the sum approaches something or just is “e+pi+sqrt2”? I feel like some ambiguity has entered even if you give an infinite limit as the sum. It isn’t “completed” like adding two rationals on a computer is. – J Kusin Mar 24 '22 at 14:09
  • @JKusin: You are absolutely correct that any practical implementation of a proof verifier program will be on a finite computer and hence not allow you to use more memory than it has. However, the idea (which Turing also seems to have stated at some point) is that we do not need a truly infinite memory (or tape). All we need is to guarantee (in principle) that the memory can be extended whenever necessary during the computation. This is sometimes called the distinction between potential and actual infinity. PA (Peano Arithmetic) is compatible with finitistism that accepts potential infinity. – user21820 Mar 24 '22 at 15:56
  • On the other hand, your second half of your comment is simply erroneous. Before we (or anyone else) can prove anything about "e+π+√2", we would have to first define what that means in the first place, and such definitions are easily carried out within any reasonable foundational system! Your error is that you were simply not aware of the rigorous definitions of these constants and operations, and so you believed wrongly that it is ambiguous. In the absence of rigorous definitions of the involved symbols, "e+π+√2" is not ambiguous but simply meaningless. – user21820 Mar 24 '22 at 16:00
  • If you want to see how these are carried out, you might want to start with Spivak's Calculus, which is semi-rigorous and gives you some idea of how these things are defined. In this post I sketch a practical foundational system (which I shall call ST) based on many-sorted FOL and ZFC. Within ST we can construct ℝ (e.g. following Spivak) and prove that ℝ is a Dedekind-complete ordered field, and then construct √2. – user21820 Mar 24 '22 at 16:20
  • @user21820 I do need a rigorous understanding of analysis. But I still think I can wonder about a major aspect of your answer (which I like a lot). Finite math, halting computer programs (like your proof verifier spitting out “yes”), and empirical science share the quality that if you doubt justification of first principles or for axioms , you can reason backwards systematically toward your original axioms. An ultimately abductive method I believe, which they all share. More powerful microscopes empirically justifying the indifference assumption of stat mech.l for example. – J Kusin Mar 24 '22 at 19:17
  • @user21820 I have to think this reverse or abductive process is unavailable to certain portions of mathematics, given that we have terms to distinguish constructive, finite, etc from more “infinite” parts. Without an empirical means of production one day buildable (an infinite tape is not), these first principles rest more softly. We can’t really program these rigorous definitions of e and pi, no empirical process to secure our principles. If the empirical process is a poll of set theorists, then that is another problem too. – J Kusin Mar 24 '22 at 19:24
  • @JKusin: Your very last comment is incorrect, because the rigorous definitions of 2,+,√,e,π can all be done in a million symbols within ST. I've already pointed you to the things you would need to look at to define √2, and you can check out what I wrote about e^iπ (linked from my profile) for rigorous definitions of e,π. So every person who understands (rigorous) basic real analysis and programming would see that it's trivially formalizable with 100% precision. I think you grasp this for simpler mathematics but don't realize that all mathematics that humans can do can be done within ST. – user21820 Mar 25 '22 at 09:47
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    Note that definitorial expansion provided by ST is a key feature that allows humanly writing formal proofs in ST. Every practical formal system has a similar feature. For instance, Mizar is a real formal system that has rigorous definitions for 2,+,√,e,π, with one of the largest machine-checked human-readable libraries for mathematics, and it too has a definition mechanism. You cannot look at systems described in common logic texts because they are not meant for human use! – user21820 Mar 25 '22 at 09:54
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    Anyway, SE is now telling us to stop chatting in comments, so feel free to ask more in this chat-room. – user21820 Mar 25 '22 at 09:56
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Short Answer

What is the difference between spoken language and logical language?

A sophisticated revision of the question is 'what is the relationship between natural language and logic?' To that end, natural language expresses much meaning that is not of logical relationships.

Long Answer

What you are getting at is that when two people speak, not everything that transpires in the conversation is logical. To understand this, it helps to take a look at the philosophy of language for some aspects of language that have nothing to do with logical relationships.

  • Performativity - Language can be used to enact social change by shared, understood rites and rituals.
  • Illocution and perlocution - Language has direct effects on action and behavior.
  • Connotation - Language can be used to arouse emotions or create associations in the mind of the listener.
  • Anaphora - Language can use some meanings that refer to other meanings.

These are the sorts of topics that are studied in the philosophy of language and generally have very little to do with logic per se. Natural language is often understood metaphorically as a system of rules and goals between people where people use the language to achieve the desired outcomes. This is known since Wittgenstein as a language game. Pragmatics is very much concerned with the context of language.

Logic is of course a series of relationships. Formal logic systems are usually built in formal systems where syntax tends to predominate over semantics. Logical connectives are generally viewed as set-theoretic mappings or represented with Boolean tables, especially in regard to truth-conditional theories of semantics. Logic in this sense is seen like a mathematical-like set of problems where what is true and what is false is calculated from rules. In these very abstract forms of logic, known as formal logics, the rules of truth might take a particular form, such as in those of a Boolean algebra, or might be tri-valent, that is having truth, falsity, and unknown as atoms in the system.

Of course, the use of logic in a natural language, such as that in a courtroom will be a constrained form of natural language which attempts to use the logic which is part of the natural language in a practical way in order to achieve a goal, for instance, for a trier of fact to reach a conclusion through argumentation. One excellent conception of informal logic is the Toulmin model, which describes how argumentation in natural language actually works especially as an abductive and defeasible pursuit.

J D
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