{¬Logic, English} ⊢ (English→Logic): Is this equation Logically correct?

Question

Since I am reading Logic in wonderful Book by Derek Goldrei, a question came to my Mind, Can I Teach Logic to someone who just knows English?!

So first I wanted to check whether it is "Logically Possible" ALSO since I am learning to talk like Mathematicians, I wanted to ask this question in the Language of Logic, I am not sure whether that makes any sense?!

I hope it means, is (English→Logic) derivable from a set of Assumptions {¬Logic, English}. I think {¬Logic, English} ⊨ (English→Logic) also means something similar but my Original Question is, is it possible to teach someone Logic who just knows Natural Language?!

Answer 1

Consider a mathematical logic textbook, like that you have used: D. Goldrei, Propositional and Predicate Calculus: A Model of Argument (2005): the Introduction (page 2) starts with the introductory example of the axiomatic system for real numbers.

And see page 6:

Assumed knowledge. The book is written on the basis that you have already had some experience of using sets and functions, and that you are familiar with a variety of mathematical words and notations. Perhaps most surprisingly for a book about logic, we assume that you already know something about logic and reasoning.

We teach arithmetic to children that know only natural language, but formal logic is not as "elementary" as elementary arithmetic.

We may assume that some part of "natural logic" is conveyed to children through natural language and elementary teaching: the concepts of true and false ("do not lie!"), and some sort of argument is learned through basic arithmetical training.

But IMO the title of the textbook above is relevant: "a model of argument". Formal logic is the study of deductive arguments and Mathematical Logic provides a "mathematical model" of deductive argument.

Thus, it is a quite sophisticated mathematical discipline.

Also "propositional calculus" and "predicate calculus" has been adopted to convey the idea of computation, like the rules for sum and product. The goal is that of "mechanizing" deductive arguments through symbolic manipulation: see Leibniz and Boole.

Answer 2

If your original question is if it’s possible to teach someone logic on the basis of only natural language, then surely the answer is yes given that you, who was (probably) taught natural language before logic, understands logic. (Failing that in some circumstance, you can use me or any number as other people as evidence for the possibility of such a thing, since I among others am certain that I was taught natural language before logic.)

The expression you define isn’t very coherent, though. Assuming that English and Logic are booleans (given that you’re using them in the context of “not” and “implies” operators) knowing the truthhood of the expression requires that you know a lot more about how you can derive things (i.e. what axioms are available). This isn’t something I think you should try and express logically - it’s perfectly fine as a piece of natural language, I doubt it could ever be misunderstood - but it would best be written as something like Learned(English) -> Learnable(Logic).

Issue there is just that every operator and object is used once and this expression just means exactly what it says. There isn’t a reason as far as I can tell to write it in logical language.

Answer 3

That kinda depends on what you consider a "natural language" to be and how and where this operates and takes place.

Like you can express the tokens of logic in terms of the tokens of a natural language. That is a feature readily exploited by... text books and lessons.

Though the question is somewhat whether that logic is a feature of the language itself or whether the tokens of the language merely serve as input to idk the mind, which is the logic doing machine. So in that latter sense access to a natural language is not required for you to perform logic as logic is a feature of the person and the interpretation of communication not of the means to communicate itself.

Which prompts another question of what even is language in the first place, like pointing at things and connecting them likely already serves as a language and if it's what you naturally use to communicate with others, it might even serve as a natural language so is any form of standardized information exchange a language? If so you could only avert a connection between language and logic by discovering logic on your own. Problem is that actually might work. Unless you believe in a deity and consider perception a conversation between you and that deity by the language of nature...

So no you probably can discover logic connections without a need to discuss with other people and without developing a formalized method of exchanging information. That being said is the existence of logic perhaps it's own sort of language regardless of whether you share it with other people? So it might actually be the other way around, in that logic is a natural language, in that having something that is seemingly objective and understood by different people could serve as the underlying necessity to formulate a language by which people can communicate.

TL;DR: So depending on how you define language and logic, you're pretty much able to argue for the thing and it's opposite, which is interesting but also useless.

Answer 4

First of all, which natural language someone speaks is totally irrelevant for the question whether or not a logically valid reasoning can be understood. However, abstract reasoning turns out to be difficult for people, even for adults.

There is empirical evidence that about 80% of children of ages 7-8 are able to identify correct conclusions in an AAA type syllogism. In syllogisms of the form EAE and AII this dropped to 65%. If we accept the Piagetian stages of cognitive development, then this suggests that there is a natural, logical competence concerning categorical syllogisms of the form AAA, in concrete operational children. (See: Inferential Intuitions and Logical Reasoning, in: Efraim Fischbein, Intuition in Science and Mathematics, 1987, p.73) When we try to ascertain this kind of evidence, there is an interesting methodological problem: How do we know a child (or an adult) really understands the logic of the presented syllogism? It turns out that if the same (an equivalent) logical puzzle is presented in one case in the form of a more "abstract" puzzle and in one case in the form of a more "concrete" puzzle, even the performance of adults can show amazing differences. (In order to test yourself, see: The Wason Selection Task, in: What is Logic? by Josh Dever). It turns out that children may sometimes reach correct conclusions, but not necessarily by logical reasoning, and not consistently, but by transduction -- by applying particular, concrete, experiential knowledge. (Try with "If this candy has sugar in it, then it is sweet. So, if it is sweet, then ...?" "So, if it has no sugar, then ..." -- No, it actually does not follow that if it's sweet then it has sugar. And it also doesn't follow that if it has no sugar then it is not sweet...

Apart from the difficult question what kind of reasoning is actually being used, there are also semantic problems and problems of verbal framing. Consider saying "All candy is sweet. So, if it's sweet it must be ...?" (But what about apples, then, aren't they sweet too?) The turn of phrase used here, the rhetorical ploy of using a leading question ("it must be") may lead some children to answer "candy". (And of course many adults, even philosophers, are sensitive to linguistic framing too, without being aware when they are 'being framed'.)

Answer 5

Short answer: Yes. It possible to teach someone Logic who just knows Natural Language.

I agree with @haxor789 and @Mauro ALLEGRANZA. They both point to humans essentially/naturally/necessarily being logical in nature.

Premise 1: Humans are Logical Beings. One can make the argument that every action (action, thought, communication, belief, etc.) is a logical or a logical argument. Although we don't normally see such regular actions (like a child playing with a toy) through the lens of logic, it can be viewed when analyzing. This can be supported through analyzing a child's decisions in play therapy. Why did the child place a specific toy being a fence? Was it an indication that the child feels guarded? Prior to clear decision making and preferences, babies cry, and, at least from my experience, it is a sign/consequence of something else: hunger, dirty diaper, boo-boo, etc. This would be in the form of "If A, then B. A exists, therefore B." If hungry, then cry. I'm hungry, therefore I cry.

Premise 2: Logic as a "Form." I would also argue that logic itself can be comparable to Aristotle's (and others) concept of forms. A form is the essence or defining characteristic of a thing, explaining what it is. Although there are surely other and better definitions, the form of logic would be " a tool for understanding and evaluating the world (and beyond) around us." Logic as we speak of and use here is the mode/language we use to communicate and rationalize the world around us. The form of a chair exists, but how we build chair and what it looks like is more physical and personal. Logic is like the chair.

Premise 3: Natural Language (and all Language) is the Embodiment of the "Form" of Logic. Logic exists and we need a way to conceptualize it and communicate it with others. First-order Logic (FOL) is one language that has been created to express the "form" of logic. However, any language can be used to express the "form" of logic. As used above, English can be used as an expression of logic: "If hungry, then cry. I'm hungry, therefore I cry." The natural language the person uses shouldn't matter either (although I would love to study if there is a connection between what language a person utilizes in their life and its effect on their beliefs).

Whether its FOL or English or any other language, I would argue that it embodies the "form" of logic. Natural Language is the chair a person builds, and Logic is the "form" of the chair.

Conclusion: English, or any language, can be used to teach logic because English, or any language, is the expression of the "form" of logic.

Hot take: English is not a good language for expressing logic, and certainly not the best.

My problem with the English language is how many implied premises there must be in order for it to be effective. I could say, "I love __________." The definition of "love" as used here in English would be entirely dependent on the context or the predicate of the sentence. The logic expressed in another language (or translated) would need to add an implied premise, being the context. I love my wife much differently than I love Oreos. There would be many implied premises to the argument to differentiate the kind of love used. English really uses just love, whereas Greek has many. In Greek, there are multiple words for "love," each capturing a distinct aspect of the concept, the main ones I can think of are: Agápe (Αγάπη); Éros (Έρως); Philía (Φιλία); Storgē (Στοργή); Xenía (Ξενία); and Manía (Μανία).

I would conclude that while English can be used to teach and express logic, it is not the best to do so.