I am going to lump two different Logical Fallacies together because they both have to do with how we class objects.
We humans like to categorise things. We like to parcel ideas up and put them in labels, and then we like to parcel labels up and put them in categories and so forth. In reality every item in this universe is a discrete entity, which shares some characteristics with at least one other thing, but will always have some kind of difference to those other things.
That is pretty heavy going, philosophically, but try to stick with me here. We get a range of items from a tree which we call apples. Each apple from the tree is different to each other apple, yet we call them apples because basically they are the same. We can look at these objects compared with similar types of items from two other trees. One of these similar items is close to the same, but has red skin instead of green skin. Another has orange, but has different insides. We class the two similar ones as “apples”, but separate them based on their colour (or even their specific species, but still class them as apples). The orange coloured things have a separate name as well, and since we are simple we will just call them Oranges. All three fall into a category called “fruit” due to a similarity in form and location found. Yet when comparing two seemingly equal things, they are still different, even if that difference can only be detected because they are in different places at the same time. It makes sense to call the collection of very similar items a single name such as “apples”, and a looser collection of things a looser name such as “fruit”.
The colours of the rainbow are on a spectrum of frequencies, from red up to blue. In between there are a range of other colours that we humans detect – red, orange, yellow, green, blue. We can see that in this spectrum, which is a continuation from red to blue, or a continuum, are in between frequencies that are also part of the “visible spectrum”. It doesn’t jump from one frequency, miss a few, and then go to the next set. The more we look between each band of light, we find more bands of light. That is, there is no empty bit where there is nothing. So visible light is an excellent example of a continuum.
Fruit, on the other hand, is not. There is a gap between where you class a fruit as an apple and a fruit as an orange. Maybe in the past this was not so, but now there certainly is. When we try to look at a different fruit, such as a pear, and compare that to the apple, we find it hard to define exactly why one is a pear and one is an apple, yet we do.
One of the Logical Fallacies has to do with attempting to class apples as pears, because the edge of where one ends and the other begins is fuzzy and hard to define. Similarly one could try to state that the colour red is actually the colour yellow, because the point where red becomes orange is impossible to distinguish, and the point where orange becomes yellow is also impossible to distinguish. So red is orange, orange is yellow, therefore red is yellow. Clearly that is not the case – it is false. A classic example of this in mundane life is to look at cults and religions. It is hard to define the difference, so one could mistake the two for being the same thing. Yet they aren’t.
The other Logical Fallacy is to insert a gap in the spectrum that doesn’t exist. Going back to our light spectrum, the colour red is at one extreme and blue is at the other. The False Dichotomy implies that there is no colour in between. You can choose a colour – but there is only really red or blue, so pick these. I like to sometimes mess with peoples heads and add a third option in that is equally distant to the other two, changing my one dimensional spectrum into a two dimensional triangular spectrum and give them a false trichotomy. Maybe I’m just mean.
The False Dichotomy asks you to choose between only two alternates when there is a spectrum of choice to make. A subtle side trap of the false dichotomy is that it is very human to draw a direct line between the two options offered and only select on that spectrum, when really there are other options too. Perhaps you want a yellow and violet striped colour selection. Neither of those colours was on offer, nor was combining them in an interesting way, and the violet falls way out of the offered spectrum of choice. The False Dichotomy would resist this selection, trying to push you into the binary choices offered. You either worship god or you fear god. Perhaps neither of these are true for me, yet the Logical Fallacy erroneously pushes me to select one of them.
In summary, the False Continuum and False Dichotomy Logical Fallacies have to do with misusing our classification systems of items. The point of categorisation is to simplify data processing, but that can be a trap if we over simplify or misrepresent the information or the choices as a consequence of that classification.