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Logic turns up in the sylabus later, one usually starts with linear algebra and analysis where one assumes the underlying logic to be given.

I think the reason is because the study of logic already needs some intuitive application of logic, i.e. logic is applied to describe what actualy logic is.

Once you have defined what a language is and what a logic is you automaticaly also proved the meta-logic that you used to prove all the statements you made about language and logic in general.

A concrete example: Structural induction - You apply structural induction to prove something about the nature of statements, i.e. if you have proved that a property holds for all minimal structures (variables,True,False) and you know that if a property holds for all imediate substructures of a structure then it also hlds for the structure (If A,B have property then A&B). Then you can say the property holds for all structures (statements). You don't know why the principle of structural induction holds but you are still applying it. Later you will know more about the framework and the objects which it comprises.

My Question is it true that logic is applied to describe what actualy logic is ? or is there something I haven't taken into account?

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    $\begingroup$ It is hard to explain one's thoughts to another person without using a language. So if I want to explain logic to you, I will have to use some language such as English (if you only speak Chinese, it will make my job of explaining my reasoning to you much harder.) So in some sense logic/language is used to describe logic itself. As a side remark, I don't think that you could conclude from this that logic relies on logic though (only the description of logic does). $\endgroup$ – Maximilian Janisch Aug 6 at 21:06
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    $\begingroup$ Possibly related: Is mathematics one big tautology? $\endgroup$ – Arnaud Mortier Aug 6 at 21:10
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    $\begingroup$ @MaximilianJanisch I have to object because that is what I basically wanted to say. Can you please elaborate why logic relies on logic is not implied by logic is described to what logic is? Because you seem to agree that logic is described to what logic is, but at the same time you say this is not the same as logic relies on logic. What is the difference between the two statements? $\endgroup$ – New2Math Aug 6 at 22:59
  • $\begingroup$ @New2Math Part 1: Disclaimer: I think that we are leaving the realms of math here. Now to your question: We both agree that it is necessary for someone to have prior understanding of logic/language in order to explain one's understanding of logic to them. It wouldn't be possible to explain logic to a fruit fly, because the fly doesn't have any prior knowledge of a language/logic. So logic relies on logic in a "social sense". A different and much more philosophical question though is the one about logic itself (it can be disputed if this term even makes sense). $\endgroup$ – Maximilian Janisch Aug 6 at 23:20
  • $\begingroup$ @New2Math Part 2: For instance, one could take a Platonic world view and argue that logic exists completely for itself, independent of language or human thoughts. One might also take a formalist viewpoint and say that logic can be considered as statements about string manipulations that adhere to certain rules. I hope this clarifies why I am making a distinction between logic and "logic for humans". $\endgroup$ – Maximilian Janisch Aug 6 at 23:21

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