There’s something wrong with language

I’m usually one of those annoying guys that has read a bit too much of Wittgenstein, Russel and postmodernism and so enjoy to point out that most of the sentences we use are meaningless and/or logically wrong. Imagine that someone stops you from talking just to say that a proposition you used for something like 18 years of your life is bullshit: that really is annoying; and make me wandering how I can still have some friends that listen to me, after all. However if my usual targets are constructions that use double negations like “I din’t do anything” (such an expression is usually used to say “I did nothing” but logically analyzing it lead to the conclusion that it means the exact opposite: “I did something”) this time I want to focus on a specific sentence: “A is better than B in every aspect”.

I really like symbolic notations so I’ll expose the argument two times: one without symbols (only via words) and one using symbols (i.e, functions and things like that); I really think that the exposition in plain words is more that sufficient to understand the argument but I love so much symbolic things that I have to do it. Let’s start.

If “A is better than B in every aspect” is true that we should be able to pick a random property and confirm that A “scores better” under that property (i.e, it’s better than B in that particular aspect) than B. Let’s pick the property of “being good at math”, so if “A is better than B in every aspect” A must be better than B in “being good at math”, hence A must be better than B at maths; there’s no problem here. But now let’s pick the property of “being bad at math”, A can’t be better than B in “being bad at math” because we’ve already stated that A is better than B in “being good at math”; being better than someone in something clearly states that that someone is better than you in doing bad that something. The properties of “being good at math” and “being bad at math” are mutually exclusive (i.e, you can have only one) and so “A is better than B in every aspect” is clearly a contradiction and so, a meaningless sentence.

Let’s jump in the symbolic things now. We can then set up a function for every property that give us numbers according to how good a thing is at that particular property. Those functions general form look like this:

\displaystyle f(x):\mathbb{E}\rightarrow [0,1]

Such a function take in an entity x from the set \mathbb{E} of all entities and output a number 0\leq n\leq 1 that tell us how good is x under a particular aspect. To clarify: we can set up a function \phi(x) such that it would give us numbers according to how good the entity x is at math and we can set up another function \psi(x) that would give us numbers according to how old the entity x is and so on; we can theoretically set up a function for every existing property an entity (i.e, an element of \mathbb{E}) can have. Let’s call the set of all of those functions \mathbb{F}. To understand how those functions work let’s use an example and pick w to be my father (that is an entity ans so an element of \mathbb{E}), \phi(x) to be the function for the property “being my father” and \psi(x) to be the function for the property “having at least 100 years”. Applying the functions to w well’get (I think):

\displaystyle \phi(w) = 1 \\ \psi(w) = 0.47

\phi(w) can’t be anything but 1 because the entity w is defined to be my father and that’s also the property \phi(x) tell us how entities perform at. Then there is \psi(w) that give us 0.47 as result: that’s because my father actually is 47 and so we can say that he have at 47% the property “having at least 100 years” or that w perform a score of 47/100 for the task of “having 100 years”. That’s how these functions work.

Now since \mathbb{F} contains all the function related to a property that a member of \mathbb{E} can have we can find pairs of functions that correspond to opposite properties. Let’s pick again the property “having at least 100 years” but this time let’s consider also its opposite “having less than 100 years”, let \psi(x) denote the same function as above and \psi^\neg(x) the function for the property “having less than 100 years”. Considering again w to be my father with his 47 years:

\displaystyle \psi(w)=0.47 \\ \psi^\neg(w)=0.53

This points out that my father is actually better at “having less than 100” rather than at “having at least 100 years”. It’s also important to notice that the output of \psi(w) is exactly 1 - \psi^\neg(w) and vice versa; this happen because \psi(x) and \psi^\neg(x) are function about two properties that are one the opposite of the other1. This result can be generalized to every pair of function in \mathbb{F} about opposite properties no matter what’s the entity x\in\mathbb{E} chosen:

\displaystyle \forall\alpha\in\mathbb{F}:\alpha(x)=1-\alpha^\neg(x)

Now let’s define another function Bet(A, B, \alpha) where \alpha\in\mathbb{F} and A\in\mathbb{E},B\in\mathbb{E} this way:

\displaystyle Bet(A,B,\alpha) = \begin{cases}  1\quad\text{if:}\ \alpha(A)>\alpha(B)\\ 0\quad\text{if:}\ \alpha(A)\leq\alpha(B) \end{cases}

As you can guess this function give us 1 if A is better than B and 0 in any other case (B better than A or B equals A at a particular aspect). This allows us to write our sentence “A is better than B in every aspect” like this:

\displaystyle \forall \alpha\in\mathbb{F}: Bet(A,B,\alpha) = 1

Assuming this to be true, we can easily derive a bunch of things:

\displaystyle \forall \alpha\in\mathbb{F}: Bet(A,B,\alpha) = 1\\\therefore Bet(A,B,\alpha) = 1 \wedge Bet(A,B,\alpha^\neg) = 1\\\therefore \alpha(A)>\alpha(B) \wedge \alpha^\neg(A)>\alpha^\neg(B)\\\text{Remembering that\ }\alpha^\neg(x)=1-\alpha(x)\\\alpha(A)>\alpha(B) \wedge 1 - \alpha(A)> 1 - \alpha(B)\\\therefore \alpha(A)>\alpha(B) \wedge \alpha(A)<\alpha(B)

And we’ve got the contradiction. Then, via modus tollens, we can say that our hypothesis is false and hence no thing can be better than another under every aspect and thus add one more argument in our “language is bullshit” folder.

Notes:

  1. Indeed this follows from how we define negation in fuzzy logic.

P.S. apparently I can write an article without quoting from Puella Magi Madoka Magica

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