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I often come to think about the limits of logic, and the idea of thinking outside logic - but that seem like a very odd thought, as logic is suppose to represent our way of thinking, so to say "think outside of logic" is paradoxical, if not even nonsensical. But then I come to think about meta-logic (not the mathematical study, but rather the idea of "meta"-system), or "thinking about logic", which seem very close to the concept of Gödel's sentence - meaning that as a system, logic can produce propositions about itself, making meta-logic apparently possible.

Now if we conclude that meta-logic is indeed possible, we can proceed ask if "thinking outside of logic" is possible - even though it's at least paradoxical, according to Gödel it can be suggested that it's not only possible, but even necessarily true (as the system of Logic cannot be complete). But now I'd like to ask, if it is possible to think outside of logic, how can we produce a proposition in order to do so? And if we cannot do so, how can we discuss logic?

  • Have you Googled the phrase 'metalogic'? Completeness, soundness, consistency, etc. are metalogical notions, metalogic is what mathematical logic has been about since the '20s. – Not_Here Jun 30 '18 at 16:47
  • Systems like Coq implement the calculus of constructions, in which proof systems can be formalised. – Keelan Jun 30 '18 at 17:13
  • @Not_Here I know that metalogic is a field of study, the question is how is that possible, on the epistemological and ontological level. The more problematic emphasis in the question is about "thinking outside of logic", where logic stands for the only language we can use to make sense of the world. – Yechiam Weiss Jun 30 '18 at 18:11
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    You are using "logic" with two different meaning: the first one is a synonym of "reason/rationality". The second one is that more familiar to mathematical logic: a specified formal system/theory. In this second case, the mathematical study of the properties of the first system/theory (considered as a mathematical object) like e.g propositional calculus and Peano arithmetic is legitimate as any other disciplie. This one is meta-logic. – Mauro ALLEGRANZA Jun 30 '18 at 20:40
  • @MauroALLEGRANZA I hope the edit makes it a bit clearer what I'm talking about. Tell me if another edit is needed. – Yechiam Weiss Jul 1 '18 at 4:12
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There are two directions one might take to this. The first is to say that Logic is, in generality, the study of inference, rather than thought per se. One studies the relationship of premises to conclusions, and while this relationship might be usefully applied in considering the nature of thought, not all logics need be thought. For example, Godel's model of formal logic works over a language coded entirely in the syntax of natural numbers, which does not seem to be a reasonable representation of human thinking practice.

So one might think your problem simply is a question of using the mathematics of logic as a model of human inferential reasoning in practice. Meta-logic is just Meta-mathematics, and that seems to be not particularly bizarre (than mathematics is by itself, at any rate).

For the other direction, this concept of "representation" draws out something interesting about the notion of human thinking, which is that in as much as we seek to model the "thought processes" of other humans, it has a very strong semantic component. The idea of receiving a visual stimulus and perceiving the world seems to be of a different character to what we would want to attribute to them as holding this particular image of the world in thought.

One might say that you've got things the wrong way around. Logic, or more generally our Language, patterns of speech and word association, and Grammar, partly constitute "thinking". We don't actively represent the world in logic, but rather our linguistic/semantic limits determine the boundaries of how we "think".

That's because when we try to work out what we mean by "thinking", we've historically had to do so with reference to asking other minds questions and listening to their responses. Human cognitive capacities may well (in fact they almost certainly do) outstrip this perhaps limited notion of thought, but that's not a fault of logic, but rather how our capability to infer the possibilities of the inner minds of other people has been filtered through the channel of linguistic expression.

Now, with the advent of modern experimentation, this boundary between minds seems less linguistic, and the concept of "thought" seems like an artifact of folk psychology. Yet at the same time, we want to try to make sense of the cognitive individuation of areas of the human brain with reference to behaviours and reports of behaviours in order to demonstrate "what the bits of the brain do". This line of investigation goes into the burgeoning field of Cognitive science, and your question of "how is Cognitive Science possible?" might well result in some interesting philosophical discussions of its own.

In short then, have a mull over exactly what aspect of "logic" you're really interested in, and delve into one of the two avenues of discussion this draws out. One line's metatheory is very different from the other, though both are almost certainly going to be fascinating.

(Also, thanks for this question. I was halfway through an initial answer when I was led to Wittgenstein's Philosophical Investigations to sense-check something I was about to attribute to him. I normally hesitate to recommend it, but it does seem entirely appropriate here - Wittgenstein's musings very much stand in for this kind of consideration about the limits of language and logic, and his theraputic remedies are particularly useful.)

  • Thanks for flag. See my question to MU above. Best - Geoffrey – Geoffrey Thomas Oct 29 '18 at 20:54
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The idea of thinking outside of logic seems like a very odd thought as logic is supposed to represent our way of thinking.

This is a common mistake and one could ask why this is such a common mistake and one might point to the high valuation placed on logic in a society that places a great deal of trust in the scientific process.

It's a mistake in that thought itself is logical. The usual confusion here is to think of reason as a synonym for logic but in fact reason is a far wider concept. Deductive Logic - which is what most people think of logic is, formally speaking, a way of moving from true propositions to true propositions; this movement is called deduction and is done by way of inference - for example modus tollens.

The other kind of logic is inductive and this is closer to thinking in that it is far less bound to formal concepts such as rules as in deductive logic. This is the kind of logic that primarily guides progress in science - and Poincare in Science and Hypothesis calls it's the real motion of a science - it adds to knowledge in a real way.

Nevertheless these two logics does not exhaust thinking. It takes thinking for example to plan and plot a story. This is generally considered to be imaginative and creative work and logic actually hinders the process rather than facilitating it.

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