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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"?

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    Your first reference was a style guide for "scholarly works". Language use in this context is not necessarily the same as the casual uses that the dictionaries describe. The distinctions it makes do not apply to common use. "Thus" and "hence" are probably not used very much in common language, except in some idioms and set phrases (like your "hence his name" example).
    – Barmar
    Sep 9, 2023 at 21:11
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    It's going to be very different in mathematics and sociology. Can you narrow down your science?
    – Stuart F
    Sep 9, 2023 at 21:36
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    Before you go involving logic, which is mathematics, be aware that therefore and hence are both metaphors for motion. Hence means 'from here' (thence means 'from there'), therefore means 'ahead of there' (same fore as in forward). All of these words can be used to indicate a final goal, but thus is the only one that always does so (or at least the speaker or writer is trying to do so). The others can refer to intermediate goals. Sep 9, 2023 at 22:36
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    Question: is this book written in Japanese, or only available in Japan, or not written with an English title? I can't seem to find any references to an English-language version. I can find a similar page by him but it doesn't discuss thus/therefore/hence; it does, however, suggest that he is largely basing this on arbitrary intuitions, without citing any sources.
    – alphabet
    Sep 10, 2023 at 0:01
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    'Although, when used in this way, these words are usually close in meaning and often can be used interchangeably, they do have important differences' looks like a recipe for confusion. Who decides when they're interchangeable and when they bring out important semantic differences? I'd say the registers differ. Sep 10, 2023 at 12:55

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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.

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    I couldn't agree more with this answer (and PaulTanenbaum's): I've countlessly pondered over this issue and decided that the difference in usage among 'so', 'thus', 'therefore' and 'hence'—as far as they are interchangeable—while not arbitrary, is frequently dependent on style, taste and desire for variation rather than down to mere logical inevitability, so Paquette's recommendation sounds somewhat esoteric (just like my personal hierarchy for these 'consequence' words) and more apt as an opinion than an authoritative argument.
    – ryang
    Sep 10, 2023 at 6:41
  • Paquette's book is published in Japan in Japanese as "科学論文の英語用法百科 第1編" and it comes with the English version, which I suspect is the original, I mean, the Japanese version is actually a (self?) translation. The author is a physicist with extensive experience in proofreading physics papers. The reference list in the book is thin. So, his claim might be more of an opinion, hence my question. Sep 10, 2023 at 11:15
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    @MasaSakano Yes, I think this is just his own personal opinion that he has arbitrarily decided to enforce when proofreading.
    – alphabet
    Sep 10, 2023 at 11:18
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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.

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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”.

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