Recently I have been thinking a lot about why so many people seem inconvincible of certain things which I hold to be true. And while I could certainly make a list of some of these things, that is not the intention of this blog entry. Instead, I have been reflecting on open-mindedness and wanted to share.

Many people – myself included – often enter into discourse with someone of a differing opinion with the intention of convincing them to change their mind. For example, maybe your friend Paul thinks all trees in your neighborhood that are taller than 12 feet should be pruned to 12 feet or less, so as not to obstruct anyone’s view of the lakefront. You *know* that he’s clearly wrong! You get into a discussion. Only it’s not really a discussion – it’s an argument each of you is trying to win. Maybe out of frustration you start incorporating personal attacks. Maybe you get so angry at Paul’s refusal to capitulate, as well as the horrible things he is saying about you, that it ends your friendship. Maybe in the middle of the night, Paul prunes all of your tall trees. Maybe the next night you erect a 30 foot statue on your lawn directly in Paul’s line of sight to the lake … and so on.

It’s sad, and you don’t even like the statue, but what choice is there? Paul must be taught a lesson.

I wish this was hyperbole. Sadly, it is not. And the conclusion is clearly suboptimal.

Well … let me construct a basis for discussion with some (hopefully) fair assumptions. In doing so I’m going to have to use a little bit of math terminology, and it occurs to me that some people might not know precisely what I mean, or even be put-off by some of my more mathematical references. If you think this might be the case, I ask you to bear with me. The concepts and symbols I use are the best way for me to illustrate my point, and I’ve included here a bit of a math lesson, in case it is not something you’ve encountered in your life – it will clarify some of the words and concepts I use for the rest of this article. Of course, if you feel there’s no need for you to read this section, by all means scroll past it and keep reading (I won’t feel bad).

**Some Math concepts**

** Sets
**Mathematicians like to talk about collections of values that are somehow related, and when they do, they use the word

*set*. We use curly brackets to list the objects (known as

*elements*) of a set. So for example the set is a set I have named , and just so you know, it is the set containing all the fruits I might bring to work with me in my lunch. A

*subset*of a set is another set that only contains elements from . So for example is the set of fruits I brought to work in my lunch on Monday, and is a subset of . On the other hand, is not a subset of .

** A Little Bit of Algebra (Apologies to the Arithmophobic)
**Consider this simple algebra equation:

The and are understood to be symbolic of numbers, but the use of symbols mean that they vary – which is to say, they are

*variable*. The equation is a statement. In this particular statement,

would be a valid solution (i.e., the equation becomes true), since

.

So would

,

since

.

However

would not be a solution (i.e., the equation becomes false), since

,

which is not 7.

**Statements**

In math and philosophy, a *statement* is a sentence that must either be true or false (but not both, and not maybe). Very often the *truth value* (i.e., “true” or “false”) of the statement depends on values for variables contained in the statement. The algebra equation above is a statement. Another example is the statement “I like cheese”, which contains two variables: “I”, and “cheese”. If the “I” refers to “Rich Dlin” (i.e., it is me speaking and not you), and the “cheese” refers to “Havarti”, then the statement is true. If the “I” is “Rich Dlin”, and the “cheese” is “Cambozola”, the statement (I promise you) is false. Notice that if the “cheese” were to refer to “gingerbread cookie” the statement would be nonsense, since “gingerbread cookie” is not a cheese – even though it is true that I like gingerbread cookies, it is irrelevant in the context of this statement. A mathematician would say “gingerbread cookie” is not an element of the set of all cheeses. Going back to the algebra example, {(1,1),(3,-2)} is a subset of the set of solutions to the equation given. The actual set has an infinite number of solutions in it, but that’s more than I need to talk about here. What I will say is that the truth value of the statement *“Three times John’s favorite number plus four times Gail’s favorite number will yield seven”* is:

True if (“*John’s favorite number*“, “*Gail’s favorite number*“) belongs to the set of solutions of 3*x* + 4*y* = 7,

False if (“*John’s favorite number*“, “*Gail’s favorite number*“) does not belong to the set of solutions of 3*x* + 4*y* = 7, and

Nonsense if, for example, John claims his favorite number is “*cinnamon*“. Be on the lookout for nonsense – it is surprisingly common.

The Assumptions

Ok. Welcome back. Here are the assumptions I was talking about:

**All questions have a right answer** …**
**… when the answer is justified properly with a well framed statement.

The truth value of the statement may be subject to variables that change which answer is correct, but with a fixed set of values for the variables, there is a right answer. For example, the question “Should all trees taller than 12 feet in our neighborhood be pruned?” could be answered “Yes”, justified with the statement “It is unacceptable for some trees in our neighborhood to block sight lines to the lakefront”. Note that here the answer to the question is “yes” if the statement is true, and “no” if the statement is false, and may reasonably depend on whether or not the tree is also so wide, or part of a grove, as to make it impossible for a resident to see the lakefront from a different angle standing on the same property. It may also depend on whether 12 feet is a reasonable height with respect to whether or not sight lines get blocked. In this case these variables need to be introduced into the statement, or else agreed upon as not being variable.

**The right answer may well not be knowable** …**
** … even with the variable values fixed – which doesn’t mean there is no right answer!

As an example, consider the question “How many humans are alive on Earth right now?”

- The number changes many times in a short span of time. So the truth value of the answer depends on what time it is indexed to.
- The answer is subject to a definition of “alive”, and the answers to whether or not some organisms are living humans are in dispute.
- “On” Earth needs to be defined. If I am in an airplane, am I on Earth? What if I am in low orbit?
- However there
*is*an answer, if we fix the variables. - There is currently no way, even with the variables fixed, to know the answer.

* Knowing the truth is inherently valuable.
*This is a big one. Many people demonstrate by their behavior that they do not adhere to this assumption. A simple example is the person who refuses to go to the doctor about a problem because they are afraid of what they might find out. In some ways, not wanting to know the truth is a human quality, especially in situations where a false belief has spawned an entire tree of values and beliefs we have been living by. If the root belief is false, what happens to the tree?

**When it Comes to Truth, What We Want Doesn’t Matter**

So with these assumptions, my position is that for any belief I hold, I am either right or wrong, and that I may not be able to tell. So then what am I to make of someone who disagrees? Can I immediately conclude that they are wrong? Clearly not. However I freely admit I want them to be wrong, so that I don’t have to be. After all, being wrong has some negative implications. On a fairly benign end it means I have been somehow deluded, which injures my pride. On an extreme end it may mean I have to discard an entire tree of conclusions that were premised on my error, leaving behind a buzzing hive of uncomfortable questions and observations about my previous behavior. But if the root belief is actually wrong, what choice do I really have? Since it is rooted in falsehood, the whole tree is an illusion anyway.

Here is a hard truth: What we *want* has nothing to do with what is *true*. I *want* there to be peace in the Middle East. But there is *not* peace in the Middle East, and no amount of wishing on my part, no matter how fervent, can alter the truth value of this or any other statement. On the other hand, what is true can and should definitely impact what I want. What we all want.

Ok. Here is another statement that is tautologically true: *In the set of things I hold to be true, some might be false*. And from a probability perspective, I am also comfortable saying that *in the set of things I hold to be true, some are true, and some are false*. I want to say “most are true and some are false”, but I am honestly not sure I have a reasonable argument to claim that, so we’ll leave it there as a desire more than a fact.

**Shades of Gray**

Now I will focus on statements where the truth depends on fixing values for the variables in the statement., which to me is the core of the shades of gray argument: *In cases where there is a continuum of possibilities between true and false, almost everything in the set of things I hold to be true lies somewhere within the boundaries of the continuum, rather than on one of the ends*.

Here a philosopher or mathematician might (and should!) argue that there can be no continuum between true and false, since those are binary options. My response is that I am talking about a sphere of reasonable answers centered on the truth, where anything outside the sphere is easily demonstrated to be false (or worse, nonsense), but things get a little more touchy inside the sphere. This is a consequence of my point about the truth of a statement depending on fixing values for variables the statement depends upon. To elaborate on this, I am going to define something called an *assumption set*.

**Assumption Set**

Suppose a statement depends on a set of variables. For example, consider the statement “Running is good for you.” The truth of this is not absolute. It depends on some variables:

- How much running (the quantity of the running)?
- How intense (the quality of the running)?
- What preconditions do you have that running would exacerbate (e.g, bad knees, asthma, heart problems)?
- Where do you plan to do your running (road, track, beach)?
- and many more.

So before we could discuss whether the statement is true, we would have to fix values for these variables. I call these fixed values the *assumption set*. So for example an assumption set for this statement could be

.

Notice that one of the elements (the preconditions) in this assumption set is itself a set – that’s completely acceptable. On the whole, I would judge this assumption set to be a reasonable one – which is to say, the elements of the set have a probability associated with them that makes them not unexpected in the context of discussing the claim that “Running is good for you.”

Another assumption set could be

.

On the whole, I would judge this assumption set to be very unreasonable – which is to say, it is highly improbable that this would be an assumption set on which the claim “Running is good for you” would be a relevant discussion.

**Reasonable Answers (Approximately True?)**

A *reasonable* answer to a question can be defined as a statement that is true when evaluated with a plausible assumption set. That is to say, the assumption set is comprised of elements that have probabilities high enough that if we observed them we would not be surprised. In situations where the variables are in constant flux, the *approximate* truth value of a statement may be argued as the one that holds given the most likely assumption set. In cases like this, we may generalize a statement as true, while being willing to challenge it in the face of a game-changing assumption set. We maybe won’t talk about who gets to define “plausible”, even though there are times when that becomes the most relevant thing.

**Arguing(?) With an Open Mind
**Here I have chosen to use the word “arguing”, even though in truth I prefer the word “discussing”. That’s because most people seem to think that discussions between people in disagreement need to be arguments. I disagree. Remember the assumption that we are not right about everything? And remember the assumption that knowing the truth is inherently valuable? These two should premise every discussion we enter into. So when discussing the answers to questions, or the truth about statements, we need to do our best to remember that what we are trying to do is get as close to the center of the sphere as possible, because that is a good thing to do, and because we may not be there yet.

Of course, we all think we are closer than an opponent. If not, we wouldn’t be having the discussion in the first place. But keeping in mind that if two people are in disagreement, one of them must be wrong, a productive conversation is one where at the end of it the parties have converged on something they both hold to be as close to true as they can see getting. When this happens, the world gets a win. I’ll list some techniques for true open-mindedness.

**Discussing With an Open Mind**

**Remember that you might be wrong.**

Put another way, be willing to change your mind, or adjust the approximate truth of what you believe.

See, you believe that you are probably right. You may even believe that you are*certainly*right (although for the truly reflective, certainty is a pretty difficult thing to attain). But your opponent has the same thoughts. Both of you probably have many reasons for these. And they probably have a lot to do with assumption sets, and which one of you is applying the most plausible set. Sometimes the discussion is not about the truth of the statement but on the plausibility of the assumption set. Keep that in mind. Yours may be the less plausible. Or maybe both assumption sets are equally plausible, in which case the statement can be split into two (or more) more detailed statements that include some of the differing assumptions explicitly. But keep in mind that emotional attachment to an assumption set can and will blind you to the plausibility of an alternate set, and ultimately cause you to refute a statement with unreasonable (even fanatical) obstinacy.**Have higher expectations for yourself than you do for your opponent.**

This means you need to challenge yourself to inspect the assumptions and claims of yourself and your opponent objectively, even if they are not doing the same thing. When you do this – and do it out loud – they hear that. Look at elements of the assumption sets and objectively evaluate their probability. Also evaluate whether they change the truth value of the statement or not. And be prepared to evaluate whether or not they render the statement as nonsense – this happens surprisingly often but it’s not obvious until it is isolated. Discussing things this way models a behavior that is necessary for the two of you to converge on a conclusion you both agree with. And if you are consistent with it, your opponent will often adopt the same style, if only because they think this is the way to convince you they are right.I don’t mean this as a politeness. I mean this in the most sincere sense. Any opportunity we get to reflect on our set of beliefs is valuable. Sometimes your opponent and you will converge. Sometimes you will not, and they leave the exchange completely unmoved, perhaps even claiming “victory”. This is sad, since the only true victory would be a convergence of opinion, but ultimately it is not relevant to your own experience. Make it so that if you have moved on a topic, it is because you discovered something you were not considering, or were considering incorrectly, and now you are closer to the center of the sphere of truth. If you do not move, make it because you were not presented with any strong evidence that you needed to. In either case your beliefs will have been strengthened in some way, either because you changed to something as a result of new insight, or because you were challenged in some way, and it was unsuccessful. For this you have your opponent to thank.**Thank your opponent, regardless of the outcome.**

**How to Spot Real Open-Mindedness**

Many people claim to be open-minded. It may be true, or it may be a trick (some people say it so that when you fail to convince them of something it will prove they were right). True open-mindedness doesn’t mean you are ready to believe anything. It means you are willing to change your mind when presented with evidence that objectively compels you to do so. If you know of (or are) someone who has changed their mind in the moment, during rational discourse, but who was fairly slow to do so, they are probably the type of person I am describing. This goes back to my point that we are probably not right about everything we believe. Which means mind-changing can occur. Which means if you’ve seen it occur, it occurred in someone with an open mind.

Thanks for reading,

Rich