Is there a formal logic symbol for "why"? For example how would you formulate "Why is 2^4 > 4^2?" Could that be formulated in pure symbols of logic if possible?
Also, the phrase "what is" can it be described with logic symbols? Is it just "=" equals? So for example you walk down the street and hear a bang. And you conclude it's either a car crash or a kid popping a balloon. How would that be expressed logically with symbols?
It is not impossible in principle that there might be a logic of explanation, which is to say, a logic that answers "why?" questions. Different logics, such as intuitionistic, relevance, linear, etc., have different natural semantics. It is not unthinkable that one could have a logic whose natural semantics was that of explanation. Asserting a proposition "A" would then be interpreted as "A is explicable", and a conditional "A → B" would be interpreted as "A explains B", or possibly "an explanation of A can be manipulated into an explanation of B".
The main problem is that I expect it would be fiendishly hard to come up with satisfactory general rules for such a logic. Carl Hempel attempted to describe a logic of scientific explanation, but it is widely regarded as unsuccessful. Different branches of science seem to use different paradigms for explanations of phenomena, so the rules would be difficult to generalize.
In some specialized contexts one might be more successful. In computer programming, for example, running a debugging tool might be interpreted as enquiring why one obtained a particular result from a program. Depending on how the program is structured, it might be possible to produce debugging output that takes the form of a formal logic.
No symbol for 'why ?', I'd say for the following reason.
Isn't logic's concern with the structure of propositions (statements, sentences) and predicates, and the relations of contradiction, implication and independence between them, also the valid and fallacious forms of argument (valid in the case e.g. of modus ponens, fallacious in that of affirming the consequent ? 'Why?' is usually an epistemological question rather than a logical one.
In modus ponens, for example :
If p then q
p__________
q
we do not look to logic to explain e.g. 'why p ?'. We look to it to identify p's role in valid or invalid arguments such as the above schema.
See further Michael Gabbay, 'Logic with Added Reasoning', Ontarioo, 2002, Preface & ch. 1.
My indication of logic's concerns at the start is not offered as a full refinement but only as a broad pointer. Refinement can be found in texts such as Gabbay's or Patrick Shaw's 'Logic and its Limits', London, 1981, where in turn more sophisticated texts are listed in the bibliographies.
You also asked about 'what is'. Your example suggests you are thinking of disjunction : 'either it's X or it's Y'. Symbolically : 'V' or 'v' : 'X v Y'. The name of the symbol is 'vel'.
'Or' is ambiguous in logic : there's the inclusive or the exclusive 'or'. So, for example, 'Either X or Y (but not both)'. This is the exclusive use. But 'Either X or Y (or both)' illustrates the inclusive use : it includes both alternatives as possibly being the case. 'Either it's red or it's antique' - inclusive (it could be both). 'Either it's a circle or it's a square' - exclusive (it is one or the other but not both). Your example seems to employ the exclusive 'or' - either it is a car-crash or it is a kid oopping a balloon but not both.
On the inclusive and exclusive uses of 'or' ('V') see Patrick Shaw, 'Logic and its Limits', London, 1981 : 49-51.
No. Formal logics deal with concrete propositions, not speculative discussions. Speculations have huge levels of nuances and are usually performed with narrative language. In fact, papers or scientific documents use mostly narrative argumentation, except for the concrete formal propositions, which are written using formal logics.
This sort of thing may have some bearing on answering your question (and won't fit in a comment):
The Oxford Handbook of Epistemology p409
An 'explanandum' is what is to be explained
To the extend that 'what is' is '=', in systems of logic that trace deduction literally, such as proof theory, implication (⇒) captures the basic notion of why. What lies to the right of the arrow is true because of something that lies to the left.
But Classical logic simplifies the meaning of implication into something unnatural by insisting all true statements are equal in meaning and all false statements prove anything, regardless of referential content.
People who want to talk about semantics then usually introduce a symbol to recapture the natural meaning of implication. So I would suggest that the semantic implication symbol 'turnstile' (⊨) really at least intends to capture the natural sense of 'why?'
When these are interrogative, what is implying or what is equated is a free variable. "Ǝ X: X ⊨ 2^4 > 4^2" means "There are collections of facts and deductions that entails the fourth power of two exceeds the square of four, we will call some one of those X." (In this case there is no X, because the right-hand-side is simply always false.) Then using X anywhere is, in effect, asking why.
