Simple is difficult

Would that people asked only for complex things to be explained. Making a living would be very easy if one was in the business of providing the answers. It is the simple things that are a problem and it’s the simplicity that often brings tuts of derision.

The laws of reasoning are supposed to be very simple. Three, in particular, sound almost trivial. Suppose I look out the window and see a fresh molehill in the garden. My sentence is:

There is a fresh molehill in the garden.

It seems like nothing, but it’s a start. What is the meaning of this sentence? The meaning is: ‘true’, because it is a fact, and ‘true’, because first order deductive logic does not recognize other meanings than ‘true’ or ‘false’.

What can I infer from it? I could repeat it, for example, but it would be no use. I would end up with a tautology. I can conjoin to it any sentence I like using the disjunctive word ‘or’, including its own negation: ‘It is not the case that there is a fresh molehill in the garden’, because the meaning of ‘or’ guarantees that, in the first case, at least one of the sentences will be true, and, in the second case, precisely one sentence will be true. Either way, the whole new sentence will be true. The latter represents the Law of the Excluded Middle: either something is of a certain kind or it isn’t. There is no third option.

Suppose I don’t look out of the window, but hold the thought in my head. What is this thought? It is an assumption, or hypothesis. What is its meaning? I do not know. I can repeat my hypothesis at will because, it being a hypothesis, I am not committing myself to anything, only probing and exploring. However, assumptions are not made to be just random thoughts floating freely in my head – that is a stream of consciousness. We make an assumption in order to prove something.

I can, of course, attach to my assumption another sentence via ‘or’:

There is a fresh molehill in the garden or the gate is open.

This in itself suffices as my conclusion – something I derived from the original sentence ‘There is a molehill in the garden’, but I’m not sure I know what to do with it. Far more convenient, if less illuminating, is to repeat the hypothesis and stop at that:

Assumption: There is a molehill in the garden.
Repetition: There is a molehill in the garden.
Conclusion: If there is a molehill in the garden, then there is a molehill in the garden.

This may not be much of a conclusion, but it is in fact an illustration of the Law of Identity, which, like the Law of the Excluded Middle, holds regardless of fact. When I look out and see a molehill, the conditional sentence is true. When I look out and see no molehill, the conditional sentence is also true. If I now choose to paraphrase my two sentences:

There is a fresh molehill in the garden or it is not the case that there is a fresh molehill in the garden.

If there is a fresh molehill in the garden, then there is a fresh molehill in the garden.

then it turns out that I can paraphrase each with:

It is not the case that both there is a fresh molehill in the garden and there isn’t.

The key word in this sentence is ‘and’. The sentence illustrates the Law of Non-contradiction. We can’t say that something both is and isn’t. A closer look reveals that all these laws are in fact one law. At any rate, all three hold regardless of the facts on the ground, so to speak.

These three laws were known to the ancient Greeks, and we use them so automatically that we never give them a second thought. When the Skeptics searched for answers as to what is and each avenue led them to deeper skepticism, they turned to these fundamental laws as something unchangeable throughout time, something that could be relied on. I rely on them alright but the more I try to explain them, including to myself, the more tortuous the explanations get.