$Pre and the attribute HoldAll

Question

Please, consider

$Pre = If[go, #, Null] &;

Now, this

go = True;
a = 1;
b = 1;
go = TrueQ @ (b != 0);
a/b

returns 1, and this

go = True;
a = 1;
b = 0;
go = TrueQ @ (b != 0);
a/b

throws a Power::infy message. The reason is that: Unless $Pre is assigned to be a function which holds its arguments unevaluated, input expressions will be evaluated before $Pre is applied (from Documentation Center)

Let's consider, instead

    $Pre = If[go, #, Null] &;
    go = True;
    SetAttributes[$Pre, HoldAll];

This still give 1

go = True;
a = 1;
b = 1;
go = TrueQ @ (b != 0);
a/b

but this

go = True;
a = 1;
b = 0;
go = TrueQ @ (b != 0);
a/b

should give Null. On the contrary, the error message appears again. Can you, please, explain why ?

addendum

With

$Pre = Function[{arg}, If[go, arg, Null], HoldAll];

by

go = True;
a = 1;
b = 1;
go = TrueQ@(b != 0);
a/b

I get

If[go,go=True;,Null]
If[go,a=1;,Null]
If[go,b=1;,Null]
If[go,go=TrueQ[b!=0];,Null]
If[go,a/b,Null]

and

go = True;
a = 1;
b = 0;
go = TrueQ@(b != 0);
a/b

returns

If[go,go=True;,Null]
If[go,a=1;,Null]
If[go,b=0;,Null]
If[go,go=TrueQ[b!=0];,Null]
If[go,a/b,Null]

Answer 1

First of all start a new kernel. The following evaluates without a problem:

go = True;
a = 1;
b = 1;
go = TrueQ@(b != 0);

So far so good, i.e. go == True. Now

$Pre = Function[{arg}, If[go, arg, Null], HoldAll];
a/b

evaluates to 1 which is correct. Now restart the kernel again, and type

go = True;
a = 1;
b = 0;
go = TrueQ@(b != 0);

Again everything is working, i.e. go == False. Now evaluate

$Pre = Function[{arg}, If[go, arg, Null], HoldAll];
a/b

and you will get Null. Again, this is the correct response. However you are doing something quite differently. You start by assigning a value to $Pre like this:

$Pre = Function[{arg}, If[go, arg, Null], HoldAll];

and then you do this:

go = True;
a = 1;
b = 1;
go = TrueQ@(b != 0);
a/b

and it's not working, because the function you assigned to $Pre is preventing the assignment of a value to go, and any subsequent assignment, from happening.

Function[{arg}, If[go, arg, Null], HoldAll][go = True]
-> (arguments are held) Function[{arg}, If[go, go = True, Null]]
-> If[go, go = True, Null] (cannot be evaluated, no assignment is happening)

If[go, ...] does not evaluate because go is neither true nor false.

What is happening in your first example, $Pre = If[go, #, Null] &;, is slightly different. Again reset your kernel and run the following:

$Pre = If[go, #, Null] &;
go = True;

What happened here? Mathematica did something like this:

If[go, #, Null]&[go = True] 
-> (the argument is not held) If[go, #, Null]&[True]

when the argument was evaluated, go was assigned a value as a side effect because the argument is not held. This actually solves the problem with the previous approach, but it has the new problem that you noticed: because arguments are not held, a/b is evaluated. The flow is like this:

If[go, #, Null]&[a/b]
-> If[go, #, Null]&[ComplexInfinity]
-> If[go, ComplexInfinity, Null]
-> Null

So the result is right, but because a/b was evaluated it will still print the error message. You can avoid the error message by typing Quiet[a/b].