Can you have knowledge without belief? I don't know. An admittedly oversimplistic thought exercise.

Dan Kahan and I have been ruminating on this question (as have many of his blog followers and the students of his Science of Science communication course, both virtual and in situ. 
First, some (loose) definitions and a chart:

• Belief: Accepting something as true--independent of whether it is true in the world.
  
• Proposition: a declarative statement
  
• Premise/Conclusion: a proposition that is inferred as true on the basis of other propositions contained in a logical argument.
  
• Logical Argument: (more or less) a series of propositions ending with a concluding proposition (premise/conclusion) that is said to logically flow from the previous prepositions. 

In this chart, belief and disbelief are independent of what is true in the world [although I recognize that there is—or at least should be—a higher likelihood of belief in what is known to be true in the world and disbelief in what is known to be false].

Before I start thinking about tricker topics such as climate change and evolution, I’d like to focus on a simpler model. So, for example, let’s consider whether or not X is a rectangle.

Premise: X is a rectangle:

There are two possible states in the world. One in which X is a rectangle and one in which X is NOT a rectangle (columns). Moreover, I can choose to believe that X is a rectangle or that X is NOT a rectangle (rows).

If X is indeed a rectangle, and I believe that to be the case, then I hold a true belief. If it is a rectangle and I do not believe that to be true, then I have incorrectly rejected that premise. Similarly, if X is not a rectangle (maybe it is a triangle), and I believe it is a rectangle then I am holding a false belief. If it is not really a rectangle and I do not believe it is a rectangle, then I am correctly rejecting that premise.

So, let’s say I am being taught the premise X is a rectangle. I’m given the following argument:

It is possible for me to have knowledge about the argument—I can learn the two propositions that lead to the conclusion. If someone asked me what is the argument for X is a rectangle, I could then give them the above argument. Importantly, I can do this without accepting the premise that X is a rectangle.  That said, Is it possible to have knowledge about rectangles and X without accepting the premise that X is a rectangle? (Open Question)

Now, if I know the argument, then under what circumstances would I choose NOT to believe that X is a rectangle?

I would think that in order to NOT believe that X is a rectangle, I would have to reject one or more of the propositions in the argument. In other words, I’d have to believe that Proposition A is false and/or that Proposition B is false. It would not make sense to accept the two propositions but reject the premise.

Why would I reject one of the propositions?

A. Lacking Knowledge/Holding Misconceptions.

• I may reject Proposition A because I do not have adequate understanding of rectangles. I may have a misconception that planes with four sides and four right angles are called triangles. 
  
• Similarly, I may reject Proposition B because I misperceived X, such as counting 3 sides instead of 4, or thinking the angles were 45 degrees instead of 90, etc. 
  

B. Distrust.
Distrust would occur in cases in which I doubt the information that I received about the propositions (or about information pertinent to the propositions)—whether I received the information though my senses (perception) or others’ testimony.

• For instance, I may reject Proposition B because I’ve only received information about X from a person who has previously been deceptive (see Trust & Learning).
  

Note, that I am particularly biased to see this as the case given that I study trust, but also, this is where I see Dan Kahan’s work on cultural cognition fitting in—we are inclined to trust our in-group members and distrust our outgroup members (this is supported with work in developmental psychology, too).

So far, my thinking is that it is possible to know the argument without believing, but I’m not sure if it is possible to have all relevant knowledge without believing—and people may lack all relevant knowledge because they distrust who its coming from.

Open questions:

1. Does having all relevant knowledge lead to accepting the propositions? If we are to argue that people exhibiting cognitive dualism utilize the theories they disbelieve in instrumental ways, doesn't it follow that they accept those propositions?  Are they simply rejecting some, but not all, of the propositions?
   
2. What other reasons might someone reject a proposition other than lack of knowledge/holding misconceptions or distrust?
   

Admittedly, this rectangle example is much simpler than complex scientific premises like Evolution and Climate Change. If we were to think of those theories as structured like big complex logical arguments, each of the propositions may be subpremises with their own set of propositions, and on and on. Is it this complexity that leaves room for cognitive dualism? If we prodded more deeply would we find that they are simply lacking knowledge or exhibiting distrust?