With or Without You Venn Diagram Meme

16 05 2012

Apparently my new hobby is playing logic games with memes.

There is a meme floating around that seems to have its origin with George Takei. There are two circles, one labeled ‘With You’ and the other ‘Without You’ with their overlap labeled as ‘Place Where Bono Can’t Live.’ It’s cute, but it is also bad logic. The Venn diagram models conjunctive relationships between two or more sets of objects, whereas the lyrics of the song being referenced detail a disjunction formulated in straightforward(“I can’t live with OR without you”).

Because it’s fun, I’ll map it out. A Venn Diagram describes set relations. Each circle represents one type of set and its members, with the overlap defining a subset of objects that share multiple sets. In this case, these are the set of objects that are ‘with you’ and the set of objects that are ‘without you.’ The subset of objects defined as belonging to both sets are ‘places Bono can’t live.’

We have a problem right from the start. ‘Without you’ is simply a common way of saying ‘Not with you.’ Since the set of things ‘With you’ is defined solely by that property, by definition all objects that do not share that property do not belong to that set. In other words, ‘With you’ and ‘Without You’ are non-overlapping sets.

Let’s pretend, though, that this isn’t an issue. How would we translate the Venn Diagram into a more straightforward logical statement? It looks something like this:

There is a set of places that Bono can’t live which are members of the set of things ‘With you’ and the set of things ‘Without you.’ Or, more straightforwardly, Bono can’t live in places with and without you.

Since ‘without you’ is just the negation of ‘with you,’ we can approximate the relationship logically as:

If a thing is Bono, then it is not living in a place that is both with you and not with you.

That’s no big deal because it amounts to saying that Bono can’t live a contradiction, which is pretty much a given.

However, the logic of the song is quite different. The lyrics don’t describe an overlapping set but two sets of disjunctive states, With you and Without You, Living and Not Living. Using a little predicate logic, let’s map it out thusly:

b=the object Bono;L=predicate applying to the set of objects Living; W=predicate applying to the set of things With you

We’ll need to get the lyrics into a roughly logical form. Given the intensity of the ‘can’t,’ it seems best to model it as an if…then statement. We get something like this:

If I am with you or without you, then I can’t live.

This can be straightforwardly given in formal logic:

(Wb v ~Wb) -> ~Lb

There are thus four sets under discussion: the set of things with you; the set of things without you; the set of things living; the set of thing not living. Since we know that two pairs (Wb and ~Wb; Lb and ~Lb) are non-overlapping, we are attempting to determine the relationship between being with you and being living. The object under discussion is whether or not Bono is a member of all or some of these sets.

There is a little bit of implied logic still going on behind the scenes here, so we need to pull that to the fore now. It is a modal issue. The song strongly implies that it is always the case that an object will fall into one member of the non-overlapping pair of with and without you. To be more precise:

 it is always the case that an object will be with you or without you.

Using the operator $ to indicate always the case we get:

$(x) (Wx v ~Wx)

Which gets us to the thrust of the song’s dilemma. Since any object (x) must either be with you or not with you, and Bono (b) is one such object, then it is always the case that Bono is not living (~Lb). If we want to be flippant, we can say that the song’s narrator must be a dead man.

But that’s exactly not what we should do. The song, by virtue of being sung, is demanding something else. We can wave our hands and say that it articulates some existential truth that cannot be categorized in formal logic, but that’s dull and empty.

Maintaining a firm grip on the strictures of formal logic, we can say that the song is an affect-filled reductio ad absurdum. In articulating this opposition between living and not living, being with and not with you, and the attendant logical contradictions that follow, the song demonstrates the inadequacy of these terms and makes a plea for some other way of conceiving the matter that would enable some coherent logical form to be given to them.

Now, the song doesn’t go further than that reductio, but we the listeners can. How do we overcome the failure? To appreciate them, we need to look at the formal logic as something other than straitjacket. Each element that goes into the absurd propositions are points at which we can re-examine our assumptions. The reductio diagrams a conceptual machine that we can then attempt to fix.

This particular map diagrams several points for reconsideration that might be corrected. What needs to be remembered is that each of these points are possible points of correction, not necessary ones.

(1) We can discard and revise the implied modal statement (It is always the case…) that drive the reductio. As soon as it becomes possible to be something other than with or without you, the conceptual force of the If…then statement disappears. To do that, we need to examine more closely what we understand what ‘being with you’ entails.

(2) We can revisit the proposition itself, re-examining what we understand living to entail and then, following after that, examine living’s relationship to being with you. We may find that we are able to re-conceive of living in a way that includes either being with or being without you.

In either case, we have to depart from the strictly logical formulation in order to re-examine the terms that produce the contradiction. That is one of the the real virtues of logical thinking. It highlights our incoherencies and aids us in thinking our way toward coherence, though it does not itself often resolve them for us.

Terms like ‘living’ and ‘being with’ aren’t simple and discrete. We make use of them by drawing upon their relationship to other terms. It is by exploring those relationship, our understanding and application of them, that we are also able to transform them. With that transformation, we can change our application of them, change our lives to a greater or lesser extent.

That means logic frequently demands we move beyond formal logic, to the terms that we brought to it. This should be nothing surprising–it is the basic point of Godel’s Proof that a system of logic depends upon terms external to it for its operation. Yet, strangely to me, the rethinking that logical thinking demands is rarely performed. Instead, when confronted with inconsistency, one of two patterns tend to take hold.

The first is simply the assertion that whatever produced the inconsistency is simply nonsense to be abandoned. This strategy often ignores how visceral the roots of some contradictions are and instead affirms the logician’s own conceptual preferences without examining their broader foundation in other concepts. Take the recent spate of efforts to define gay marriage out of existence for are a good picture of this.

The second pattern asserts the pointlessness of logic, the inability of logic to deal with matters of real substance. While this has the virtue, perhaps, of validating the visceral dimension of many contradictions, it does little to ameliorate it. Since we are almost incapable of abandoning our conceptual understanding, the abandonment of logic leaves the contradiction intact, capable of returning in force. It also interferes with their use of the sort of rational thinking that allows them to broaden and expand their understanding, which is a vehicle for improving upon their lives.

Worse, these two patterns tend to reinforce each other. Rigid conceptualism encourages those struggling with a contradiction to abandon logic’s aid, while willful obscurantism furthers the conceptualist’s sense that the contradiction is just nonsense.

(Yes, it’s true, it does seem to keep coming back to Kant’s First Critique.)

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2 responses

16 05 2012
Randall Baker

My immediate visceral reaction to this was: “Nerd! Nerdy McNerdenstein! Nerdy nerd nerd nerd!” But… this could be a conversation my friends and I had last week, so, I’ll admit I find it interesting.

17 05 2012
Ian

I’ll own the nerd. This did all start just because I wanted to see if I could still properly describe what was going on in logical terms. That said, I’m just stodgy enough to be bothered by the way so many good logical tools have become the basis for easy jokes while they are so rarely applied to the serious problems for which they are well-suited.

I know, it’s the internet. Still.

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