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Given that there are different formal logic systems, both classical and non classical (Non classical logic), which one do you think is the valid one? Which system does our mind use to draw inferences from the real world? Since rules like the excluded middle are irrefutable in our experience, is it right to conclude that classical logic is the right system? But we also seem to have a concept of necessity/ possibility qualifiers for the real world like should/may, leading us to modal logic.

So which system is the one corresponding to the actual world?

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    No "right" system; real world and everday experience is full of vague situations, to which excluded middle does not apply. – Mauro ALLEGRANZA Mar 30 '17 at 13:19
  • See e.g. Vagueness and Sorites Paradox. – Mauro ALLEGRANZA Mar 30 '17 at 13:19
  • @Mauro ALLEGRANZA What are the real world situations in which the law of excluded middle does not apply? Can something be and not be something in the real world? – Valandil Mar 30 '17 at 17:43
  • Either a man is tall or short (i.e. not tall). What is the maximum height of a short man? – Mauro ALLEGRANZA Mar 30 '17 at 18:04
  • I don't know that we have an answer to this yet. Philosophers of mind and cognitive scientists are still a ways away from figuring out how the brain works, and the closest we've come to any real answer is scattered commitments to representational theory of mind – commando Mar 30 '17 at 21:27
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These are interesting questions and quite difficult to answer. There are many different logics, and as you say, one's view of logic connects with how one thinks about the world.

When most people think about the "real world" they think of it as existing independently of ourselves, and of facts about it being true whether or not they are known, or even knowable. Other domains, such as mathematics, or ethics, or aesthetics, might be viewed differently. In the history of philosophy, all kinds of debates have arisen about whether certain things are 'real'. Depending on the context, realism can be contrasted with idealism, nominalism, constructivism, phenomenalism, intuitionism, instrumentalism, etc. Michael Dummett introduced the neutral term "anti-realism" to refer to all these. He then made an important observation that connects realism with logic. Our willingness to be 'realist' about a particular domain corresponds to our willingness to accept that a statement about it is either true or false, irrespective of whether it is verifiable, or indeed whether any evidence exists about it. For example, consider the sentence, "it was raining on Manhattan island at noon on 1st May, 10,000 BCE." We have no evidence as to whether this is true, and it seems likely that we never will. But most people are inclined to say that it either rained or it didn't: this is just a fact about the world that is independent of whatever evidence we have.

In other domains, such as mathematics, this is less compelling. Some mathematicians are realists (Gödel was), but it seems plausible to say that mathematics is about what can be proved; there is little reason to think that all mathematical sentences must be either true or false, even when no proof exists.

This suggests that classical logic is the natural logic of realism, while intuitionistic logic, which lacks the law of the excluded middle, is naturally an anti-realist logic. There are logicians who maintain that there is a single 'correct' logic. Michael Dummett defended intuitionistic logic in this way, Stephen Read argues for relevance logic, and Graham Priest for paraconsistent logic. But it is also possible to understand these logics as having a different natural semantics. Classical logic is about truth, and classical validity preserves truth; intuitionistic logic is about verifiability; relevance logic has the semantics of information passing; paraconsistent logic has different flavours, but at least one can be thought of as having the semantics of falsifiability. S5 is about logical necessity; S4 has the natural semantics of a weaker form of necessity that is consistent with branching possibilities.

Understood like this, one could use several different logics, without having to accept one as being the only correct one. It still leaves open the question as to whether there is a natural logic of the metalanguage. What logic am I assuming right now when I talk about logics?

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Which system [classical versus non-classical] does our mind use to draw inferences from the real world?

To draw conclusions in everyday life, I think that most people call upon Abduction, also called Inference to the Best Explanation. The article at the Stanford Encyclopedia of Philosophy is thorough and readable. https://plato.stanford.edu/entries/abduction/

Abduction allows the formation of uncertain, yet useable, conclusions, based on actual observation, relevant personal experience, consultation with knowledgeable sources of information, and anything else that might reduce uncertainty.

I suppose that abduction is a part of classical logic, as it calls upon traditional deduction and induction. A discussion of this question appears at Wikipedia > Abductive reasoning > Formalizations of abduction.

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When Aristotle formulated his dialectic system he thought he was modelling human reason as we use it in practice, and I believe he was right. We do reason our way to conclusions just as he proposed.

But few people use his system as he specified and the result is that his 'laws of thought' and dialectic process have been doubted as a way of arriving at truths. If used properly it is reliable, and I see no evidence that we use any other logic when reasoning.

In philosophy this problem of the misuse of Aristotle's logic is acute, for it leads people to false conclusions and seems to force them to invent paraconsistent logics and other exotic systems.

I would recommend C. W. A. Whittaker's book, Aristotle's De Interpretatione
which explains how to use the dialectic properly.

So my answer would be that Aristotle's system is the one we usually use, or intend to use and should use, but that we don't always use it properly, especially in philosophy. Of course, it depends on having appropriate starting terms. Often his system is inapplicable because we are not able to define a contradiction clearly for logical purposes as in the case of bald men or a pile of sand, which are ambiguous objects requiring definitions and judgements that go beyond logic.

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