Questions regarding well formed expressions in the Theory of types I'm dealing with a question in type theory:  Is it possible to assign types to $\alpha$, $\beta$, and $\gamma$ in such a way that both $(\alpha (\beta))(\gamma)$ and $\alpha (\beta (\gamma))$ are well-formed expressions?
Now, I  have a couple of questions:


*

*Since the type syntax I'm looking at is constructed out of types of symbols e and t, where e represents an entity and t represents formulas, such that $<e, t>$ represents a one-place 1st order predicate sentence. Should therefore all sentences have an individual expression; an entity e? Could e.g. a verb function as such an expression and in that case, can the theory account for that?

*How should I tackle the parentheses? Should it be read as in mathematics? Is $\beta$ a function of $\alpha$, and thus $\alpha(\beta)$ then in turn a function of $\gamma$ in the first sentence? I am trying to understand in what order should I construct the meaning. Is otherwise the sentence constructions done quite independently of the brackets which functions rather as "semantic indicators", pointing to semantic ambiguities e.g?

*Do you reckon it is possible?
N.B.
I'm quite new to this so help and guidance would be much appreciated. :)
 A: If both of the expressions in your question are well-formed, then $\beta(\gamma)$ and $\beta$ would have the same type, because they both occur as the argument of $\alpha$.  In the type theories I'm acquainted with, such a thing (a function having the same type as its output) is not possible.  (Indeed, it would seem to defeat the purpose of having types in the first place.)
A: Type $e$ is the type of individual constants or variables : i.e. "names" [I suppose $e$ for $e$ntity].
Type $t$ is the type of sentences, i.e. "boolean" [I suppose $t$ for $t$rue-false].
In general [see Gamut's book, page 79] :

an expression of type $<a,b>$ is an expression which when applied to an expression of type $a$ results in an expression of type $b$. In other words, if $\alpha$ is an expression of type $<a,b>$ and $\beta$ is an expression of type $a$, the $\alpha(\beta)$ is an expression of type $b$. The process of applying an $\alpha$ of type $<a,b>$ to a $\beta$ of type $a$ is called (functional) application of $\alpha$ to $\beta$.

Thus, type $<e,t>$ is the type of (unary) predicate letters : i.e. "properties". You have to think at on "object" of this type as a function that maps a name into a sentence.
Consider the sentence (type $t$) : "Socrates is a philosopher"; we can write it as : 

$philosopher(Socrates)$.

$Socrates$ is of type $e$ and $philospher(x)$ is of type $<e,t>$.
With verbs, we can consider the example : "loves". The (binary) predicate $Loves(x,y)$ (translating : "$x$ loves $y$") is of type $<e, <e,t>>$.
We can think at obtaining the sentence "$Loves(John,Mary)$" with the (functional) application of the predicate $Loves(x,y)$ to the object $John$ of type $e$ getting a new object of type $<e,t>$ : $Loves(John,y)$.
Now we apply it to the objcet of type $e$ $Mary$ to get the final sentence of type $t$.
Thus, $Loves(x,y)$ maps objects of type $e$ into objcets of type $<e,t>$, and so is an object of type $<e, <e,t>>$.
