Why do Consumers accept lambdas with statement bodies but not expression bodies?

First, it's worth looking at what a Consumer<String> actually is. From the documentation:

Represents an operation that accepts a single input argument and returns no result. Unlike most other functional interfaces, Consumer is expected to operate via side-effects.

So it's a function that accepts a String and returns nothing.

Consumer<String> p = ""::equals;

Compiles successfully because equals can take a String (and, indeed, any Object). The result of equals is just ignored.*

p = s -> "".equals(s);

This is exactly the same, but with different syntax. The compiler knows not to add an implicit return because a Consumer should not return a value. It would add an implicit return if the lambda was a Function<String, Boolean> though.

p = s -> true;

This takes a String (s) but because true is an expression and not a statement, the result cannot be ignored in the same way. The compiler has to add an implicit return because an expression can't exist on its own. Thus, this does have a return: a boolean. Therefore it's not a Consumer.**

p = s -> ("".equals(s));

Again, this is an expression, not a statement. Ignoring lambdas for a moment, you will see the line System.out.println("Hello"); will similarly fail to compile if you wrap it in parentheses.

*From the spec:

If the body of a lambda is a statement expression (that is, an expression that would be allowed to stand alone as a statement), it is compatible with a void-producing function type; any result is simply discarded.

**From the spec (thanks, Eugene):

A lambda expression is congruent with a [void-producing] function type if ... the lambda body is either a statement expression (§14.8) or a void-compatible block.

I think the other answers complicate the explanation by focusing on lambdas whereas their behavior in this case is similar to the behavior of manually implemented methods. This compiles:

new Consumer<String>() {    @Override    public void accept(final String s) {        "".equals(s);    }}

whereas this does not:

new Consumer<String>() {    @Override    public void accept(final String s) {        true;    }}

because "".equals(s) is a statement but true is not. A lambda expression for a functional interface returning void requires a statement so it follows the same rules as a method's body.

Note that in general lambda bodies don't follow exactly the same rules as method bodies - in particular, if a lambda whose body is an expression implements a method returning a value, it has an implicit return. So for example, x -> true would be a valid implementation of Function<Object, Boolean>, whereas true; is not a valid method body. But in this particular case functional interfaces and method bodies coincide.

s -> "".equals(s)

and

s -> true

don't rely on same function descriptors.

s -> "".equals(s) may refer either String->void or String->boolean function descriptor.
s -> true refers to only String->boolean function descriptor.

Why ?

• when you write s -> "".equals(s), the body of the lambda : "".equals(s) is a statement that produces a value.
  The compiler considers that the function may return either void or boolean.
  

So writing :

Function<String, Boolean> function = s -> "".equals(s);Consumer<String> consumer = s -> "".equals(s);

is valid.

When you assign the lambda body to a Consumer<String> declared variable, the descriptor String->void is used.
Of course, this code doesn't make much sense (you check the equality and you don't use the result) but the compiler doesn't care.
It is the same thing when you write a statement : myObject.getMyProperty() where getMyProperty() returns a boolean value but that you don't store the result of it.

• when you write s -> true, the body of the lambda : true is a single expression .
  The compiler considers that the function returns necessarily boolean.
  So only the descriptor String->boolean may be used.
  

Now, come back to your code that doesn't compile.
What are you trying to do ?

Consumer<String> p = s -> true;

You cannot. You want to assign to a variable that uses the function descriptor Consumer<String> a lambda body with the String->void function descriptor.It doesn't match !