Difference in logical inevitability between therefore/thus/hence

Question

I am trying to figure out the difference in the degrees of logical inevitability that the words therefore, thus, and hence express, when used in academic scientific writing.

Glenn Paquette explains in his textbook "English Composition for Scholarly Works" (2004) (specifically, Chapter 123):

The adverbs thus, therefore and hence can be used to express connections of causation and logical implication of several kinds. Although, when used in this way, these words are usually close in meaning and often can be used interchangeably, they do have important differences.

and describes the differences for 13 pages (very detailed!).

I summarise his argument in the table below:

	Meaning	Unique-ish meaning	Inevitability
therefore	for this reason		high
thus	as a (necessary) result	in this way, to this point	very high
hence	as a deduction	from this time/place	highest

To add, only `hence' has the meaning and use "that is the reason or explanation for" (Cambridge Dictionary)" which takes a noun phrase: e.g., "His mother was Italian, hence his name - Luca".

Here are some examples (and reasoning) illustrating the differences, taken from his book.

1. Example 1: We must demonstrate that X is a compact space, and therefore we attempt to prove that every infinite set has an accumulation point.
   • "therefore" can be replaced with neither "thus" nor "hence".
   • (I guess this is because the logical inevitability is too weak for "thus", let alone "hence", in this case.)
2. Example 2: To this point, the renormalization of α has been ignored. Therefore/Thus the phase diagram in Fig. 2 should not be considered an accurate expression of the predictions of the model.
   • Either of "therefore" and "thus" is OK, but "hence" would be somewhat unnatural (according to the author).
3. Example 3: However, the elements of S cannot be put into one-to-one correspondence with the integers, and hence S is not countably infinite.
   • "hence" cannot be replaced with "thus".
   • However, "and hence" can be replaced with the following: "and it is thus seen that".
4. Example 4: At the next order, however, the effect of the cross term appears, yielding a term proportional to cos²(ωt) in the solution. Thus/Hence a non-zero time average arises.
   • Here, "therefore" would be inappropriate.

I broadly agree with his description/claim (except for example 4). However, I have so far failed to find any literature or work to back up the claim. So, I am unsure how valid the claim is.

For example, Merriam-Webster defines, apart from their respective unique-ish meanings:

• therefore: for that reason; because of that; on that ground
• thus: because of this or that
• hence: because of a preceding fact or premise

These do not seem to differentiate their differences clearly, apart from the fact "hence" sounds more formal and may be preferred to be used only when there is a strong logical connection. An answer to the question "“Hence”, “therefore” and “so” in mathematical proofs" in StackExchange supports the argument. Similarly, an answer to the question "Which one is less formal: hence, therefore, or thus?" in StackExchange lays out "1) therefore, 2) thus, 3) hence (from informal to formal)". This supports the said argument if the formality is directly connected to the logical inevitability (but is it?).

"painintheenglish.com" summarises their differences from an etymology-type point of view, which perhaps highlights the differences (also note a similar claim by a StackExchange answer):

• ’Therefore’: means ‘for this reason’ - it relates to deductive reasoning
• ‘Thus’: means ‘in this/that way’ - it relates to ‘HOW’
• ‘Hence’: means ‘from this/that’ - it relates to WHERE

However, according to this explanation, "therefore" is used for deductive reasoning, suggesting a need for a strong logical connection.

What are the generally perceived degrees, if there are any, of logical inevitability of the three adverbs (in scientific writing, where clarity is paramount) when they are used to express a logical connection, i.e., excluding their respective specific uses like "thus far"?

Answer 1

I'm going to focus on your main question:

I am trying to figure out the difference in the degrees of logical inevitability that the words therefore, thus, and hence express, when used in academic scientific writing.

There are differences in meaning between these three words. But the question of whether there is a difference in degree of logical inevitability (when these words are used to express logical consequence) is purely a matter of opinion, as demonstrated by the inconsistencies you've found between different sources. It is possible that these terms have specific meanings in physics, but I find it highly improbably that readers in that discipline would consistently read them as having some technical meaning specific to the field.

As this earlier question demonstrates, this is particularly true of "thus" and "hence." One could argue that "therefore" is more likely to indicate deductive inference and thus "logical inevitability," given its usage when describing formal logic, but I find it extremely unlikely that anyone would consistently see a difference in the "degree of logical inevitability" between these words.

This supports the said argument if the formality is directly connected to the logical inevitability (but is it?).

It is not. The level of formality of a word has no connection to the word's meaning.

Glenn Paquette explains in his textbook "English Composition for Scholarly Works" (2004) (specifically, Chapter 123):

I can't find a copy of Paquette's "English Composition for Scholarly Works," but the work I can find seems to be more or less a usage manual, but written by a physicist with no obvious expertise on the matter. He seems to just be listing his own intuitions without citing any sources. I would not consider this a particularly reliable source.

Answer 2

In the mathematics literature, and by that I mean in peer-reviewed journal articles, the three are used interchangeably. And math is about nothing if not careful reasoning and fine logical distinctions.

Answer 3

Academic English is a different language from English

Well, not quite, but the distinction you are looking for is lost outside the particular jargon of scholarly writing and, probably, outside the particular field Paquette was thinking about - physics jargon is different from sociology is different from law is different from history. Anyone who has written a thesis for a post-graduate degree or an article for publication knows how painful it is to learn to write “academically”.