[Haskell-cafe] In what language...?

[Alexander Solla]

On Oct 26, 2010, at 12:43 PM, Andrew Coppin wrote:

>> Propositional logic is quite a simple logic, where the building  
>> blocks
>> are atomic formulae and the usual logical connectives. An example  
>> of a
>> well-formed formula might be "P → Q". It tends to be the first  
>> system
>> taught to undergraduates, while the second is usually the first-order
>> predicate calculus, which introduces predicates and quantifiers.
>
> Already I'm feeling slightly lost. (What does the arrow denote?  
> What's are "the usual logcal connectives"?)

The arrow is notation for "If P, then Q".  The other "usual" logical  
connectives are "not" (denoted by ~, !, the funky little sideways L,  
and probably others); "or" (denoted by \/, v, (both are pronounced  
"or" or "vee" even "meet") |, ||,  and probably others);  
"and" (denoted by /\, or a smaller upside-down v (pronounced "wedge"  
or "and" or even "join"),  &, &&, and probably others).

>
>> Predicates are usually interpreted as properties; we might write
>> "P(x)" or "Px" to indicate that object x has the property P.
>
> Right. So a proposition is a statement which may or may not be true,  
> while a predicate is some property that an object may or may not  
> possess?

Yes.  For any given object a (which is not a "variable" -- we usually  
reserve x, y, z to denote variables, and objects are denoted by  a, b,  
c), P(a) is a proposition "about" a.  Something like "forall x P(x)"  
means that P(x) is true for every object in the domain you are  
considering.

>
>>> I also don't know exactly what "discrete mathematics" actually  
>>> covers.
>> Discrete mathematics is concerned with mathematical structures which
>> are discrete, rather than continuous.
>
> Right... so its domain is simply *everything* that is discrete? From  
> graph theory to cellular automina to finite fields to difference  
> equations to number theory? That would seem to cover approximately  
> 50% of all of mathematics. (The other 50% being the continuous  
> mathematics, presumably...)

Basically, yes.  There are some nuances, in that continuous structures  
might be studied in terms of discrete structures, and vice-versa.