In this post, I'm going back to some more specific points from my new language. This can be seen as an application of both the referential transparency ideas and the quantification ideas. I will use Lojban as the “other language” because it's simple and formalised, but most points will also apply to English and other languages.

In Lojban, every concrete sumti looks like a smear across space-time. This immediately looks as if it breaks referential transparency. I'm not who I used to be, for instance. But that sentence gives us a clue as to why referential transparency can still hold. “am” is not the same verb as “used to be”. Verbs essentially choose some piece of space or time, and then act upon a few relevant slices of those irregular hypersmears we call “nouns”.

“I'm not who I used to be” is quite an awkward thing to translate into Lojban. I get {mi na du lo pu du be mi} – “I'm not equal to a thing that was equal to me”. But I'm not sure whether this is actually correct. Following the above logic, the main {du} is acting upon here-and-now slices of its arguments: {mi} and {lo pu du be mi}. But what is the here-and-now slice of {lo pu du be mi}? Surely, {lo pu du be mi} satisfies {da} in {da pu du be mi}. So, we're taking a past slice of {da} and equating it to a past slice of {mi}. But {mi} (uniquely) satisfies this {da}. Hence, that sentence wasn't correct, because it reduces to {mi na du mi}. Incidentally, the English sentence is probably also false by the same logic, unless we arbitrarily fix tenses of nouns that are being used with certain verbs. But that's messy and non-general.

All of this prohibits us from relating nouns from different tenses unless we can give nouns tenses directly. So, why don't we do that? I don't see why not. If our nouns, rather than our verbs, have tense, all of this stuff becomes trivial. But there are a few things to remember. In order to do this, we get rid of the smear concept, and with it formally lose the notion of continuity. In maths, we've rederived such things anyway, so this is not as big a problem as it may sound. It does, though, throw up some counterintuitive interpretations (counterintuitive according to those used to smear languages).

The number of past-dogs that jump is much larger than the number of dogs that have jumped. Why? In non-smear languages, there is no intrinsic connection between a dog and that dog in the past. Every time a dog jumps, it becomes a different dog. Therefore, the number of past-dogs that jump is equal to the number of times any dog has ever jumped.

Of course, some way to get back the meaning of “the number of dogs that have jumped” is necessary. What we have to do is invent a predicate similar to “fa is a successor/predecessor to fe in tense fi by standard fo”. When put through this predicate, some of the past-dogs end up having the same successor in the present tense, and are reduced to a single present-dog. Those present-dogs can be considered “dogs who have jumped”.

The “by standard” place is important. The “fa” and “fe” places need not and do not have a formal connection. We can make the predicate as strict or loose as we like. For example, in translating the claim “Germany have won the World Cup 4 times”, we can specify that we consider West Germany to be a predecessor of present-day Germany, so their titles do in fact count for Germany. A literal translation of an overly-precise statement would go something like “4 predecessors/successors of Germany from the past by standard that the predecessors/successors largely occupy predecessors/successors of the land of Germany win the World Cup”. But the standard is nearly always unnecessary (and I'm not sure whether I've picked a good standard in the example).

This scheme also makes it possible to make truly tenseless claims. In Lojban, every bridi is cryptotyped with tense information about space and time. That applies even to {li re sumji li pa li pa}, in which all of that information is meaningless and weakens the claim, technically. Time and space have no bearing on whether 2 is the sum of 1 and 1, so it makes little sense to mention them. In a non-smear language, 1, 2 and the sum predicate can be taken to be universal constants, referring to things that don't happen to exist in space-time.

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