//Paraconsistent logic in The Office

Paraconsistent logic in The Office

Team leader, health and safety officer, military tactician, survival expert, kung-fu master, corporate detective, and assistant (to the) regional manager, Gareth Keenan is the quintessential Renaissance man. Yet despite the obvious calibre of the Keenan mind, there remain those who continue to trifle with Gareth, doubting, of all things, his ability to reason. According to Keenan dissenters, Gareth is guilty of committing a number of logical blunders. One such alleged error was supposedly committed during an incident known in Keenan scholarship as The Case of the Missing Tim. A closer examination of this case will demonstrate that Gareth was in fact making a sophisticated argument for the legitimacy of paraconsistent logic.

The Case of the Missing Tim begins with Tim erecting a wall of files between himself and Gareth. In addition to the fact that such an erection obviously constitutes a health and safety hazard (blocking out valuable light and constituting a misuse of company files), it also makes it impossible for Gareth to see Tim. Tim uses this obfuscation to “muck about” (something that would not be allowed in the army). Once the wall is in place, Tim lures Gareth into conversation, only to sneak away mid-chat. Tim’s plan was to trick Gareth into committing the irrational act of “talking to no one”. Yet, as we shall see, Gareth’s mind is too highly trained to be taken in by such a schoolyard prank.

Gareth deduces that Tim may no longer be present, and promptly announces that he knows no one is there. Keenan detractors argue that, rather than allowing Gareth to save face, the announcement is itself irrational. If Keenan really believed no one was present, the detractors argue, to whom was he communicating this fact? In Gareth’s defence, I argue that his announcement was in fact the most rational act he could perform.

The argument against Gareth’s rationality is based upon two premises that we generally accept as true. The first involves the idea that if you believe that no one is present to say anything to, it would be irrational to say anything at all. For instance, if you are trapped alone on a desert island it would be completely irrational to say, “Pass me that coconut, would you Daley.” The second premise proceeds from the principle that it is irrational to declare something you do not believe. So in this case, if Gareth did actually believe Tim was present to talk to, then he should not have declared otherwise. These two premises fit together to form the following argument.

1. If you believe there is no one present, then it would be irrational to say anything.
2. If you believe there is someone present, then it would be irrational to say, “No one is present.”
Therefore,
3. It is always irrational to say, “No one is present.”

This argument seems quite straightforward. Keenan apologists such as myself, however, can defeat this trifling reasoning with an appeal to the Redundancy Paradox.

The Redundancy Paradox (also known in some circles as the lottery paradox) is best illustrated as follows. Imagine that David Brent has been forced by head office to make at least one redundancy. In order to avoid actually making a decision (and therefore attracting blame), David elects to draw the name of the hapless employee from a hat. Let’s assume that the hat contains thirty names in total, one for each employee in the office. Given that the name is chosen at random, there will be a one in thirty chance that any single employee will be given the flick. The question on everyone’s mind now is – who will be made redundant?

To see if we can determine who will be packing their bags, let’s establish a rule concerning when it is rational to believe something is true. We shall refer to this rule as the Rule of Acceptance. The Rule of Acceptance states that, at the very least, it is only rational to believe something if it is likely. Let’s define something as likely if it has more than half a chance of being true. So, for example, it would be rational to believe Gareth could catch a monkey only if, at the very least, there were more than half a chance this was true. With such a rule in place, let’s now consider who will be made redundant.

To help us answer this question let’s take a systematic approach and consider each employee in turn. First, consider the fate of Monkey Alan. Is it rational to believe Monkey Alan will be the employee who is made redundant? Well, since it is unlikely his name will be drawn from the hat (there being only a one in thirty chance of this occurring), and as the Rule of Acceptance states that we should only believe those things that are at least likely to be true, it would seem irrational to believe Monkey Alan will be made redundant. Likewise, we shouldn’t believe that Big Keith will be made redundant either, as it is similarly unlikely that his name will be drawn from the hat. The same can be said of each of the thirty employees of the office.

If it is not rational to believe that, for example, Big Keith will be made redundant, and it also isn’t rational to believe that Monkey Alan will be made redundant, then it stands to reason we should believe that neither employee is going to lose his job. This follows from what is known as the Conjunction Principle, which states that if it is rational to believe proposition p and it rational to believe in proposition q, then it is also rational to believe in proposition p & q. However, as the argument below illustrates, if the Rule of Acceptance and the Conjunction principle are both correct then it would be irrational to believe that anyone at all in the office will be made redundant.

1. Since it is unlikely that Monkey Alan will be made redundant, it is irrational to believe he will be made redundant.
2. Since it is unlikely that Big Keith will be made redundant, it is irrational to believe he will be made redundant.
3. And so on for all thirty employees.
Therefore,
4. It is irrational to believe anyone will be made redundant.

Such a conclusion seems inconsistent with our belief that David will make one person redundant.

Normally, we would accuse anyone who holds inconsistent beliefs of irrationality. However, Keenan apologists may claim that in circumstances such as the Redundancy Paradox it does seem perfectly rational to hold inconsistent beliefs. And we are not alone in this assertion. A small band of philosophers, led most notably by Graham Priest, have also championed this cause, the cause of paraconsistency.

Although the Redundancy Paradox itself provides some indication of when it might be considered rational to hold inconsistent beliefs, such a possibilty is highly contentious. This is chiefly because classical logic operates on the assumption that a proposition and its negation cannot both be true. For example, the proposition, “There is a boy who can swim faster than a shark,” and its negation, “There is not a boy who can swim faster than a shark,” cannot both be true according to classical logic. This assumption is referred to as the Law of Non-contradiction.

Without the Law of Non-contradiction in place, chaos seems to ensue. This is largely because it is possible to argue anything from a contradiction. For example, from the premises, “A good idea is a good idea forever” and, “A good idea is not a good idea forever,” we could infer within classical logic that “the price of Opti-Bright Laser copy paper is £2.98 a kg,” which is of course ridiculous (it is £2.40 a kg). This utterly anarchic outcome is based upon what is known as the Principle of Explosion. In order to avoid the consequences of this principle, a new system of logic, known as paraconsistent logic, has been championed.

Paraconsistent logic is a branch of logic that allows people to proceed rationally from inconsistent premises. In other words, paraconsistent logic denies the legitimacy of both the Law of Non-contradiction and the Principle of Explosion. As we’ll see, Gareth’s actions during the Case of the Missing Tim are evidence of the fact that he is working within a paraconsistent framework. Furthermore, given such a conclusion, Gareth’s actions should be interpreted, not as the bumblings of a weasel-faced fool, but rather as the terribly clever actions of a Blockbuster grand finalist.

Although Keenan apologists accept that, during the Case of the Missing Tim, Gareth was indeed expressing inconsistent beliefs, they reject the notion that holding such an inconsistent set of beliefs, and acting on them in the manner in which he did, was actually irrational. Let’s review the case. Since Gareth had no real way of determining whether Tim was behind the wall of files, it seems perfectly reasonable that he place even odds upon each possibility. Therefore:

• Possibility 1 (Tim is present behind the wall) has a 0.5 chance of being true.
• Possibility 2 (Tim is absent behind the wall) has a 0.5 chance of being true.

The Rule of Acceptance suggests that it is rational to believe a possibility is true only if it is likely, where a likely possibility has more than half a chance of being true. So in this case it seems it is rational for Gareth to believe neither that Tim is present (possibility 1), nor that he is absent (possibility 2). Yet since the two possibilities are mutually exclusive (either Tim is present or he is not), one must be true. In which case, because Gareth does not believe Tim to be present, he is committed to belief that Tim is absent. However, as Gareth also has reason to not believe Tim is absent, he must likewise believe Tim is present. In others words, if the Rule of Acceptance and Conjunction Principle are correct, it seems rational for Gareth to believe the proposition, “Tim is present and absent.”

Given such a result, Gareth’s declaration that he knows no one is present behind the wall is a completely rational utterance. Indeed, it is the only rational course of action open to Gareth, given his beliefs. For if Gareth were to say nothing, given his belief that Tim was still present to talk to, this would be irrational. Yet to talk to Tim would also be irrational given his belief that Tim was not present. Consequently, the only reasonable action for Gareth is to announce that Tim is not present, since the act of announcing is consistent with his belief that Tim was present to be announced to, whilst the content of the announcement is coherent with his belief that Tim was absent. So, rather than the Case of the Missing Tim entailing a logical blunder on Gareth’s part, Keenan’s actions are actually a shining example of, what is fast becoming the new buzz within middle management – paraconsistent logic.

Excerpted and edited from The Office and Philosophy: Scenes from the Unexamined Life edited by J. Jeremy Wisnewski (Wiley-Blackwell)