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I'm trying to analyse the statement "The computer has the capacity to perform long division", but I can't decide whether to use the connector because or therefore. Which one is more fitting for analysis?

"The computer has the capacity to perform long division because it has the capacity to load numbers, store quotients, bring down the next digit.etc"

"The computer has the capacity to perform long division therefore it has the capacity to load numbers, store quotients, bring down the next digit.etc"

Both sentences seem to make sense, however, they seem to express different relations. Does the division result from the steps, or does it somehow necessitate the steps? How do I know which relation and which word to use, and why?

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    Maybe the first one: "the capacity to load numbers, store quotients, etc. are more "basic" ones and thus are the pre-requisites for "performing complex arithmetical operations". Nov 30 '21 at 12:41
  • It depends on where you're taking the reader. If you're emphasizing ontological dependence and the process is built on the computational primitives, 'because' or 'since', but if the context is to argue something and you want to show that the primitives are present, perhaps because you are drawing parallels between humans and CPUs, then therefore can be used as a transition word rather than a logical or metaphysical relation.
    – J D
    Nov 30 '21 at 12:56
  • Therefore has at least two functions in English, so context matters.
    – J D
    Nov 30 '21 at 13:02
  • It might be useful to think of it in terms of the direction you want information to flow in your sentence. Does the proposition that your computer can load numbers etc. inform the proposition that it can perform long division? ("because" flows right to left) Or does the proposition that it can perform long division inform the proposition that it can load numbers etc.? ("therefore" flows left to right)
    – Paul Ross
    Nov 30 '21 at 13:35
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    @RichardBamford Ah, I think I understand now - If you're trying to give a conceptual analysis, the convention is to make sure both directions work. We sometimes say that the conditions are both necessary and sufficient conditions, and that the bit on the left is true if and only if the bit on the right is true.
    – Paul Ross
    Nov 30 '21 at 14:00
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Logically, the first formulation is more correct, because the description of a computer's capabilities is a more general statement than the specific fact that it can do long division.

By the way, since you seem to be asking which statement follows from which, note that the first formulation can be written using "therefore" too, by swapping the order of the statements.

"The computer has the capacity to load numbers, store quotients, bring down the next digit.etc, therefore capacity to perform long division."

A more interesting version of the question would be if we imagine that we listed only the computer's capabilities that are targeted at performing long division. Then the two statements would be equivalent and the connection between them would be if and only if. In this case both of your formulations would be correct!

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Short Answer

It depends on where you're taking the reader. If you're emphasizing ontological dependence and the process is built on the computational primitives, 'because' or 'since', but if the context is to argue something and you want to show that the primitives are present, perhaps because you are drawing parallels between humans and CPUs, then 'therefore' can be used as a transition word rather than a logical or metaphysical relation. 'Therefore' has at least two functions in English, so context matters.

Long Answer

It's easy to forget that words have more than one use. Sometimes two subtle ideas can be expressed with the same word. In philosophy, particularly metaphysical discourse, those nuances in meanings can have a substantial impact on the ideas expressed. 'Because' and 'therefore' are two words that have the potential to have multiple meanings, and their usage to some degree requires both clarification and context. Therefore is a good example since it can be used to make claims about material implication (which is a logical relation), the necessity of existence (which an ontological relation (SEP)), or just used as a linguistic transition in the passage to organize claims.

Also, technically, it's not decided in mathematical philosophy whether procedures, which have a goal, are constructed, that is assembled from more primitive procedures, or if a procedure is somehow discovered as an objective assembly from other procedures in the way that trees are made of cells. Some mathematicians reject the objectivity of mathematics entirely. For an example of the tendentious nature of the presumptions built into mathematical philosophy, consider constructivism:

Constructivism is often identified with intuitionism, although intuitionism is only one constructivist program. Intuitionism maintains that the foundations of mathematics lie in the individual mathematician's intuition, thereby making mathematics into an intrinsically subjective activity.2 Other forms of constructivism are not based on this viewpoint of intuition, and are compatible with an objective viewpoint on mathematics.

Whether one is a realist or an instrumentalist has a lot of bearing in matters of ontological dependence.

Generally, 'because' and 'therefore' are understood as moving in the opposite direction or order, such as 'A therefore B' is similar to 'B because A'. In a formal system, logical operations can be defined explicitly as inverses, but in natural language, definitions of necessity and sufficiency, and in causality, things aren't so pat.

So, the question is what are you trying to express? If you are saying that it is necessary that long division exists because of specific processes, this is an ontological claim. You seem to be claiming that it is necessary to use a standard algorithm for division to occur inside of a computer. In fact, that's not a true claim, since the ALU of a computer neither uses base-10 nor uses the same algorithms we do to conduct math. In the instance that a number is divided by two, for instance, in a processor, it is done both in binary and using shifting. Therefore, one has to exercise caution when making ontological claims about certain general operations entailing the use of more primitive ones. Computer algorithms are a complex subject, and what you think a computer is doing and what it is really doing relies on a sophisticated understanding of mathematics, code, microcode, and logical gates so let the claimant beware.

The use of logical 'therefores' and 'becauses' are going to be determined by your the intention of your argument. An argument is generally two premises, one explicit at least, that arrive at a conclusion. Here in your example, you only provide one claim, so it's impossible without context to decide. Are you presenting a formal or informal argument about computers and division? We simply can't tell. Lastly, 'therefore' can be used as a transition word. Are you attempting to guide the user through the passage? Again, without a passage, we cannot tell.

Ultimately, the right use of word is going to depend on your intention and the context the sentence you provide is invoked in.

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