The Fallacy of Denying the Inference

Consider the following conversation:

• Timothy: Hi there. I am not married, nor have I ever been married. 
• Bob: Oh, so you're a bachelor, eh?
• Timothy: I never said I was a bachelor. Why are you putting words in my mouth?

I've been encountering this line of reasoning a lot lately, and it drives me nuts. I was hoping to give it a silly or fancy name, but I think Denying the Inference seems pretty straightforward. It happens when people basically accuse you of straw-manning their claims, despite the fact that your description is a perfectly logical entailment from their original premises. It happens so much, in fact, that I am honestly amazed that no one has ever bothered to name this thing before (or maybe there is, and someone can just point it out for me?). 

My personal experience of this came recently when I had a discussion about formalism vs Platonism in mathematics. I tried to explain that formalism is basically the de-facto presumption in a number of modern textbooks, as evidenced by such phrases as "the language of natural numbers" or "the language of logic." We also see a lot of modern pedagogical sources that encourage educators to teach math "as a language," not to mention the existence of "formal language theory" as an actual class you can take in upper-division mathematics. The natural numbers themselves are nothing but a closed, tautological circle, as evidenced by the Peano axioms and the textbook definition of "zero." My detractor, however, rejected all of that entirely. Instead, he demanded that I produce an actual textbook containing the explicit affirmation that "formalism is true."

The obvious problem with this line of reasoning is that you don't have to explicitly affirm certain propositions in order for certain other premises to do it for you. When you tell me that Tim has never been married, it cannot possibly qualify as a straw-man argument for me to infer that Tim is a bachelor. Please don't blame others for your inability to grasp the logical implications of your own claims.

A Simple Challenge for Inspiring Philosophy

 
Here’s a fun little argument given to us by our good friend, Inspiring Philosophy [link]. 

1. Anything that exists has an explanation of its existence, either in the necessity of its own nature, or in an external explanation.
2. The universe has an explanation for its existence, and that explanation is grounded in a necessary being.
3. The universe exists.
4. Therefore, the universe has an explanation of its existence.
5. Therefore, the explanation of the existence of the universe is grounded in a necessary being.
6. Therefore, God (a necessary being) exists.
   

That’s an interesting argument you have there, but I hope you can forgive me for being a bit confused. So let’s take a moment to analyze the basic logical structure by applying the following substitutions:

• x = a thing
• X(x) = x is a thing that exists
  
• Y(x) = x has an explanation
• N(x) = x has an explanation by the necessity of its own nature
• E(x) = x has an external explanation 
• B(x) = x is explained by the grounding in a necessary being
• u = the universe
• Y(u) = the explanation for the universe.
• g = God (a necessary being).

Given these definitions, I am going to do my best to rewrite this argument into something a little more structured. After borrowing heavily from the language of predicate logic, I was able to come up with the following: 

1. For all x, X(x) → Y(x), such that either Y(x) = N(x) or Y(x) = E(x).
2. Y(u), and Y(u) = B(u).
3. X(u).
4. Therefore, Y(u).
5. Therefore, Y(u) = B(u).
6. Therefore, X(g).
   

I apologize in advance if my use of predicate logic isn't perfect, but hopefully you can now see some of the basic problems emerging from this argument. I was able to count three deal-breakers.

1. Premise (2) clearly contradicts premise (1). You cannot lump all possible explanations into two categories, only to then introduce a third category.
   
2. The conclusions (4) and (5) are nothing but verbatim repetitions of what was already asserted in premise (2). At best, this is just needlessly redundant. At worst, it is blatant question-begging.
   
3. The final conclusion (6) is a complete non-sequitur. Nothing in this argument says anything about the existence of an actual necessary being, nor does that existence follow from any established rules of inference. The actual existence of a necessary being is entirely separate from the proposition that some explanation is "grounded" in a necessary being, and there is nothing in this argument to formally connect the two.
   

Notice how I’m not even touching the actual interpretation of any of these premises. All I am examining is the formal structure of the argument itself. Unfortunately, by all objective standards, this is a demonstrably invalid argument. That means anything which comes after this point is entirely moot, and there is simply no point in considering this discussion any further.

But hey, you know what? That’s perfectly okay! Nobody is perfect, and there is nothing wrong with putting out a bad argument by mistake. All that matters is whether or not we have the intellectual courage to admit our mistakes and correct them.

With that in mind, I would like to present a formal challenge to Inspiring Philosophy. Now that you have been duly informed of the mistakes in this argument, would you please kindly take the time to publicly acknowledge your errors and correct them? Nobody is judging you for making an honest mistake. It happens to the best of us all the time. The only way this could ever turn into a problem is if you cling to those mistakes after the fact, thereby perpetuating misinformation. So let’s show the world what an amazing philosopher you are by updating your arguments in light of this new information. 

Thank you for reading.