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There are some notable persons who criticized formal logic in favor of informal logic for various reasons, like Schiller. So what is the school of thought or tradition that incorporates or adheres to the maxims asserting the much importance of informal logic or practical logic.

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Not a school of philosophy, but on a similar theme of questioning logical foundations: Charles L. Dodgson's What the Tortoise Said to Achilles. –  Niel de Beaudrap Jan 7 '13 at 11:42
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In practice, almost all of them. This includes authors of formal logic texts in general. Almost all authors of formal logic texts will spend a little time defining a formula, or well-formed formula, or statement form, etc. and then precede to use strings which are not well-formed formulas.

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Is that tantamount to a criticism of the importance of formal logic, however? It seems to me that the problem is that the existing systems of formal logic have the unhappy feature of being slightly cumbersome in minor syntactical details (eg. nested parentheses), and that if a formal system were available in which self-expression were easier (eg. using a mathematical style order of operations) without sacrificing ease-of-inference, they'd happily use only well-formed formulae. –  Niel de Beaudrap Jan 7 '13 at 11:30
    
@NieldeBeaudrap Polish notation has existed since the 1920s. Reverse Polish notation since the 1950s. One could argue that they make ease-of-inference easier, since something like C==AN (which abbreviates (p->q)==(~pVq)) or A==CN comes as simpler and more mechanical to use. I do not think syntactical details cumbersome in either PN or RPN. But, unfortunately, very few logicians, or mathematicians for that matter, seem to want to use them. –  Doug Spoonwood Jan 7 '13 at 22:21
    
The problem with PN or RPN, at least from my point of view, is that they don't have the same type of grammar (linguisically speaking) as my preferred mode of expression. Polish notation is perhaps VSO or VOS, while reverse polish is SOV or OSV, while I am most accustomed to expressing myself with a SVO grammar. This is a nontrivial distinction from the point of view of self-expression, however trivial it is logically. –  Niel de Beaudrap Jan 8 '13 at 1:35
    
@NieldeBeaudrap Most natural languages of the world actually fall into the SOV pattern: en.wikipedia.org/wiki/Subject%E2%80%93object%E2%80%93verb I do not think that the SVO pattern lies behind the preponderance of infix notation over other notational schemes. –  Doug Spoonwood Jan 8 '13 at 2:32
    
Fair enough; perhaps written mathematics tends to be SVO due to a historical accident, i.e. because almost all European languages are SVO for simple verbs at least. (Function application would count as an instance where mathematics is either SOV or VSO depending on whether one uses the convention for composition from category theory or from analysis, respectively.) –  Niel de Beaudrap Jan 8 '13 at 12:01
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Brouwer was famous for thinking that the principles of reasoning should not be formalized because language is an inadequate tool for representing mathematical thought.

For this reason, he never formalized intuitionistic logic and it wasn't until his student Heyting formalized it that we got a formal system codifying the principles of intuitionistic logic.

You can find good discussion of this aspect of Brouwer's philosophy of mathematics, here. Brouwer's (somewhat obscure) presentation of these views can be read in his Cambridge Lectures on Intuitionism.

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