'Senses' is a term with two senses.

A=I think
B=I believe

In the sense of a guess or hypothesis, I think and I believe are the same. “I think it is raining.” is interchangeable with “I believe it is raining.” In either instance, we go outside and confirm if it is raining or not—they are both a claim of knowledge.

But “I think you are mistaken” is in the sense of a claim to a judgment, while “I believe in God” is an expression of faith or an attitude or a duty. In any event, there are uses in which they are not interchangeable.

A surprisingly tricky question!

One sense of A is always synonymous with B. Therefore, it will ALWAYS be the case that one sense of A is interchangeable with B. The real question is: can we imagine a context where the alternative sense of A is incompatible with B?

I'm thinking along the lines of A = the square root of 1, which as you know may be 1 or -1, but I'm not sure how to proceed from there...

Spoken language is a vehicle for communication, transactions, instruction and entertainment. It is a great amorphous, ever-changing mass of words and grammar to represent things, actions, persons, states and conditions, relationships, concepts, images and ideas.
In English, many words have two or more accepted meanings. In common parlance, words are often used imprecisely in various constructions. Rise up; hurry up; I give up; hold up; serve up; throw up; step up; break up; cover up...

Verbs replace nouns; nouns are used as verbs, either can become an adjective. Religious and political rhetoric co-opts words and alters their meaning; the military invent new ones from acronyms.
In scientific and technical communication, terminology must be better controlled to prevent misunderstanding. In academic debate, terms need to be defined and agreed-upon at the outset; in internet forums a slight shift in meaning can be employed to misconstrue or misrepresent another poster's comment. Even when every attempt is made to write precisely what one means, the result is still open to some range of interpretation. Even dictionaries are not in total accord.
Language is slippery; difficult to handle effectively. I doubt any hard rule can apply to all the words in one language, let alone across the spectrum of languages, regional variants and dialects spoken in the world.

In the sense of a guess or hypothesis, I think and I believe are the same. “I think it is raining.” is interchangeable with “I believe it is raining.” In either instance, we go outside and confirm if it is raining or not—they are both a claim of knowledge. — Antony Nickles

Not exactly a claim to knowledge: that would be "It's raining."
"I think it's raining" generally means one is responding to sensory input - say, pattering on the roof, diminution of light, moisture on the windowpane. "I believe it's raining" may mean exactly the same as think (depending on habits of speech) or might mean that the speaker has heard a credible weather report.

But “I think you are mistaken” is in the sense of a claim to a judgment, while “I believe in God” is an expression of faith or an attitude or a duty. — Antony Nickles

Now, why did you change the example? A valid comparison would be of "I think you're mistaken" - where the speaker has some knowledge of the subject, but not enough to be certain the other is wrong - and "I believe you're mistaken." - where the speaker is confident of his own superior knowledge, but too polite to say "You're wrong!".

If God comes into it, it should be by way an example such as: "I think there is a god" - uncertainty leaning toward belief - "I believe there is a god" - growing conviction - and "I believe in God" - declaration of faith in a particular deity.

Afterthought: I just noticed I didn't use the word in the thread title to deal with the subject of the thread, because "sense" is itself ambiguous, used differently in too many contexts to have one precise meaning.

Yes. A better word then 'sense' is 'context'. This makes it fairly straight forward as we understand many words meaning change within different contexts.

Language is slippery; difficult to handle effectively. I doubt any hard rule can apply to all the words in one language — Vera Mont

The characterization of language as irrational, unable to be clarified, etc. is only in contrast to the fantasy for certainty (“hard rule”s; mathematical). That words are sometimes interchangeable; that communication involves difficulty, laziness, manipulation, vagary, and the ultimate possibility that we may reach an impasse on the means of understanding, does not mean that language is relative or imprecise. Part of what I take philosophy to do (as does Wittgenstein, Austin etc.) is to make explicit the implications and various ways language works (in order to show intelligibility despite no predictable certainty—reference to something “objective” or being only true/false). Wittgenstein will call these “uses” or “senses”, which in his case is a defined terminology by which he means the various things that language does in various situations (context is important to sense, as @Philosophim points out).

Now, why did you change the example? — Vera Mont

Thus my setting out the various senses of belief by examples (and how “think” is used the same way at times) as: a hypothesis of knowledge, a claim of judgment, or an expression of faith (also, the expression of an opinion). Each has its own separate criteria and contexts in which they apply.

If God comes into it, it should be by way an example such as: "I think there is a god" - uncertainty leaning toward belief - "I believe there is a god" - growing conviction - and "I believe in God" - declaration of faith in a particular deity. — Vera Mont

And so here we are mixing up the criteria and context and way in which belief works differently in each sense. Knowledge is not justified true belief (Plato steered us wrong; exactly because he wanted knowledge to be certain in the face of opinion so it could have power over our interests, to avoid chaos, conflict, limbo). To say “I believe in God” in the sense of an expression is not the conclusion that starts with the other sense of belief as a hypothesis of knowledge (“I think there is a god”). They are two separate uses (senses) with different criteria in different contexts. Mixing them together is what creates the impasse of whether God “exists” (and creates that as a false goal). The criteria for proving a hypothesis of knowledge is not the same as an expression of conviction, and the conviction is not a conclusion or substitute for the claim of knowledge.

Separately, I would offer that “It’s raining” is not a claim to knowledge; it is a report of knowledge (even though you may be shown to be mistaken or lying, etc.). And, yes, part of the context of the sense of belief as a hypothesis is probably some clue or “sensory input”, but that does not change its structure, nor does that imply that belief as an expression of faith needs to include any proof, nor exclude that there are associated empirical “sensory input”, such as the feeling of awe, though this does not operate as evidence or proof.

First time I see God being brought up when someone is asking to be taught about polysemy.

The characterization of language as irrational, unable to be clarified, etc. — Antony Nickles

And where did you find such a characterization of language? I believe this is one of those misconstructions through the substitution of similar but not interchangeable words. The words 'slippery', amorphous' and 'ever-changing' do not mean 'irrational'; nor does 'difficult' mean 'unable to be clarified'.

That words are sometimes interchangeable; that communication involves difficulty, laziness, manipulation, vagary, and the ultimate possibility that we may reach an impasse on the means of understanding, does not mean that language is relative or imprecise. — Antony Nickles

It is subject to imprecise applications and interpretations.

[ 'think and 'believe' ] Each has its own separate criteria and contexts in which they apply. — Antony Nickles

Not really. In many contexts, they can be used interchangeably without causing any misunderstanding, and people do use them interchangeably, due to custom or manners, even when the same application conveys a somewhat imprecise meaning. Using one in a context where both may apply, and than the other - with the addition of a crucial preposition - does show that they can mean very different things also.

And so here we are mixing up the criteria and context and way in which belief works differently in each sense — Antony Nickles

Yes, you did.

To say “I believe in God” in the sense of an expression is not the conclusion that starts with the other sense of belief as a hypothesis of knowledge (“I think there is a god”). They are two separate uses (senses) with different criteria in different contexts. — Antony Nickles

They're three separate uses - very possibly by three different speakers - in the same context: answering the question: "How do you regard God?"

nor does that imply that belief as an expression of faith needs to include any proof, — Antony Nickles

Nobody's implied that. The statement "I believe in God." is about the speaker's internal conviction of the existence of a particular deity, whereas "I believe God" at suggests either a a personal communication from the deity or a conviction that some text was written by the deity. Neither has anything to do with proof. (Which is pretty much the definition of 'faith'.)

“It’s raining” is not a claim to knowledge; it is a report of knowledge — Antony Nickles

Can't report it till you've owned it; can't own it till you claim it. It's very rare for anyone to come in from a downpour, dripping proof all over the carpet say, "I know it's raining since I experienced it."
Knowledge, the accuracy of the speaker's information and the basis of his conviction do not figure into a declaration about one's own state of mind.

The criteria for proving a hypothesis of knowledge is not the same as an expression of conviction, and the conviction is not a conclusion or substitute for the claim of knowledge. — Antony Nickles

Good quote.

The words 'slippery', amorphous' and 'ever-changing' do not mean 'irrational'; nor does 'difficult' mean 'unable to be clarified'. — Vera Mont

Also a good quote.

Same sense of “good” as my last comment.

But language is the only place in this universe where something might truly be fixed, made absolute. This is how we can see two senses to one word, by using other words to define sense one as distinguished from sense two. The line between the two senses is then absolute, or nothing can be said at all.

the term "A" has 2 senses. ("B" always has the same sense)

In one sense, A has the same sense as B.

In another sense, A has a different sense as B. — KantDane21

Those stipulations seem to me to be contradictory. The two senses of "A" could be given the names "A1" and "A2". (Those are meant to be names for the senses of "A", not names for whatever object or property "A" refers to.) The sense of "B" could be given the name "B1". What you are saying, effectively, is that A1 = B, A2 = B, and A1 != A2 (i.e. A1 isn't identical to A2). But this violates the transitivity of identity.

The issue of substituting terms salva veritate usually arises in the context of discussions about Frege's distinction between the (Fregean) sense and reference of a singular term (i.e. a proper name or definite description that refers to an object, although Russell does not regard definite descriptions as singular terms) and the different sorts of sentential context in which those terms figure (e.g. extensional of intensional). But it's not clear from your OP if you mean those terms to be singular terms and if you mean them to have the same reference. In any case, even if you would clarify those issues, your initial stipulations still appear inconsistent.

(On edit: I asked GPT-4 if I might have somehow missed the point and it made some suggestions. Maybe some of them could help you refine your OP.)

I believe this is one of those misconstructions through the substitution of similar but not interchangeable words. The words 'slippery', amorphous' and 'ever-changing' do not mean 'irrational'; nor does 'difficult' mean 'unable to be clarified'. — Vera Mont

I was drawing out the implications of what you said, which was to make language seem sketchy (characterize it as such). Part of language’s “polysemantic” nature (which @Lionino points out) is not only that different words can have the same sense (meaning), but that language can be doing/revealing multiple things. A threat can also be a promise. So what we imagine we are saying, even if we are correct, can be blind to how it is otherwise meaningful. Slippery does imply that something can’t be grasped or nailed down; thus, analogously (which is the sense in which it is used here), with respect to language, unable to be made specific, as if it is imprecise, as if language doesn’t have particular ways in which it works. Further, amorphous implies there are not distinctions (language’s shapes) to distinguish one thing from another. If these do not imply that language lacks reasons for how it works, then I have just made a mistake—though that claim would have to be accounted for—that is not a reflection on the nature of language, as if to say, that it is:

…subject to imprecise applications and interpretations. — Vera Mont

Most of the time we do fine. At times, interpretation is not even a possibility (language is not always subject to imprecision; though of course someone can always erroneously claim anything), and, even when it is, not because of an “amorphous” nature of language, but because of multiple possible contexts (which can be narrowed) or because, as I said, we speak poorly or come to judgment too hastily, etc.

[“I believe in God”, “I think there is a god”]…are …separate uses …in the same context: answering the question: "How do you regard God?" — Vera Mont

This is forcing two statements into the same requirements by dictating a question; that is not there ordinary contexts. When someone says they believe in God, it is in the context of expressing their conviction (thus a claim to community), not a claim of knowledge (believing as a hypothetical answer to a question). Sometimes both are said to be claiming God “exists” but as a conviction: God exists (is “real”) in the sense of importance, impact, centrality.

you might want to consider if there's a charitable interpretation of the original post that could resolve this apparent inconsistency. - GPT-4 — Pierre-Normand

This should be in the forum’s guidebook.

This should be in the forum’s guidebook. — Antony Nickles

For sure. However I now notice that the OP was the last thing this user posted on TPF seven months ago. Unless they come back, we can only guess how they meant their post to be interpreted.

But language is the only place in this universe where something might truly be fixed, made absolute. — Fire Ologist

This why humankind invented specialized language for those subjects in which it's important to communicate precisely: mathematics, musical notation, maritime signal flag code. Scientists and engineers also have standardized terminology in spoken languages.
For day to day communication, most people make do with context-recognition. This leads to many misunderstandings, arguments and even accidents, but on the other hand ('other'? 'hand'?), the malleability of words and grammar also engender poetry, drama, rousing and moving rhetoric. Not to mention the need for dictionaries, thesauruses and discussions of linguistic peculiarities.
Of course, not all spoken languages are as rich in vocabulary or as flexible as English.

I was drawing out the implications — Antony Nickles

You were drawing out the inference you made of what I said. Your interpretation.

This is forcing two statements into the same requirements by dictating a question; that is not there ordinary contexts. — Antony Nickles

What is the ordinary context? People don't normally come up to one another on the street or at a cocktail party and announce: "I think there is a god." or "I believe there is a god." or "I believe in God." (But I have had people come to my door and ask whether I've been saved.) These statements are normally made in a larger context - the discussion of deity and one's relationship to deity. Before every such statement, there is an expressed or implied question. The question doesn't force a response; the statement points back to a requirement for making it.
This is true of most conversations. "Four dollars a dozen." doesn't come out of the blue; it's an answer to "How much are the eggs?" Even "I think it's raining," is an answer to a spoken or implied question, such as "What's the weather like?" or "Do you hear something?" It is not true of proclamations, announcements, preachment or political speeches.

You were drawing out the inference you made of what I said. Your interpretation. — Vera Mont

This makes it sound like it depends on me how language works; as if it depends on you how what you said, says what it does. But the reason we can infer implications (the hidden consequences and acts of our words) is that all of it existed before us in our history. Now, you may chose what words to say in a given situation, but how they do what they do, and the implications of having said it, are not up to you. So to suggest that the implications of what you say are just my opinion, is to overlook the rationality of language, again. Now my claim may be wrong or overblown or too board, but there are ways those can be corrected because what we say is evidence, for which there are requirements or criteria for judging (through inference) what the implications are of what we say (too whom) in a given situation. In fact, I take these as the tools of philosophy.

An ‘ordinary’ context is an example that shows us, that reveals, what makes an expression what it is, why it matters, how we judge it, etc. More to the point, philosophy has a habit of starting at the opposite end from discovering that our everyday world is enough, and creates ahead of time the requirements of a context in order to force a particular solution, such as Plato’s forms being necessary because he’d only allow for certain, generalized, universality; and, imagining everything is in response to one made-up question so they all have to be “answers”.

Before every such statement [“I think there is a god." or "I believe there is a god." or "I believe in God."] there is an expressed or implied question. — Vera Mont

Well, we already established that “I think” and “I believe” are used interchangeably in the role of a hypothesis. The context that comes to mind with “I believe in God” is as an expression of trust, even if that was learned as “…[the word of] God” to tell me what to do; or as an attitude toward the world: that there is reason in what we see as chaos and mystery. Thus why “I do not believe in God” is said by those lost to the particular cause, or hopeless, maybe for anything going the way they hoped someone they trusted would do, but also by those that feel they know everything and have complete control over the world.

“I believe there is a god.” is said in the same senses but also as a response by someone confronted with the dismissal of what they take as important to them; by someone who, through the lense of knowledge, lays down the challenge/attack: “I don’t believe [ I doubt or do not know ] there is a god”. The affirmative is not the answer of a question nor a lesser version of knowledge, only unsubstantiated, but as an exasperated, shocked reaffirmation that, even if you don’t believe you will be judged one day, I conduct myself as if everyone were watching.

the statement points back to a requirement for making it. — Vera Mont

This is very good actually, I agree. And those “requirements” (or factors, conditions) I would call the context, which determines what criteria should be applied to judge what version (sense) of the expression it is. But to say every statement is an answer is to abstract from any context so much to imagine one criteria applies to all of them, like true or false. And that classic example ignores the necessity of the contexts and criteria of other things we say to meet the same values of precision, rationality and identity.

This makes it sound like it depends on me how language works; as if it depends on you how what you said, says what it does. — Antony Nickles

What I say depends on me; what you hear depends on you. If I say 'slippery' and you hear 'irrational' that's only because language can be used in various ways; it's up to people who use it to decide whether they do so co-operatively, so that language conveys the intended messages, or antagonistically, so that a message is diverted. I might have intended to imply something I did not say, or I may not have. You can't know my intention; you can only guess, that is, infer. If you do that with good will, the inadvertent miscommunication can readily be corrected; if you do it with ill will, it can as readily lead to an argument.

So to suggest that the implications of what you say are just my opinion, is to overlook the rationality of language, again. — Antony Nickles

I don't see where the rationality of language requires you to read an entirely different word from the one that was written. When I write 'your interpretation' and you read 'just your opinion', that is not forced upon you by the structure or function of language; that is a choice.

Thus why “I do not believe in God” is said by those lost to the particular cause, or hopeless, maybe for anything going the way they hoped someone they trusted would do, but also by those that feel they know everything and have complete control over the world. — Antony Nickles

That's one hell of a big inference about a whole hell of a lot people you know nothing about.

When you draw the workings of language as: intention—words—interpretation, you insinuate that it is just my trying to read your thoughts, which you take as only known to you, so I can’t possible know what I claim to with any authority other than my skill, generosity, cooperation, etc., which I was equating to the analogous position that unsubstantiated opinion has to certain knowledge. You know your intention; I am only guessing (I have no facts to back it up with, as with my opinion). But the possibilities language has in a certain situation are limited and are proscribed by the history of human practice. Any English speaker is familiar with the version of belief when one says “I believe in God” that is different than “I belief it is raining” and so could agree with me on the senses I outlined or correct any mistakes or provide further necessities of context. Words already have a way they work in the world in certain situations, and you just apologize or promise or make a statement or a proclamation. Language does not work like names for “thought” like for objects. Intention is something you ask about only when something strange happens.

So when you say,

That's one hell of a big inference about a whole hell of a lot people you know nothing about. — Vera Mont

it is not me making a judgment about people; I am just describing how disclaiming belief works in the world. And I’ll consider a competing claim, but dismissing the entire project as impossible claiming that I’m in no position is to remove any rationality from philosophical discourse. If someone is claiming they don’t believe in God, in a certain sense they are saying there is no mystery in the world and nothing outside of (above) our power. Now, they might not want that to be the implication of it, but those are some of the things which are believed, and so some of the things which are refused in the denial.

So when I make a claim about the implications of what you are saying, I am not judging you nor trying to guess your thoughts. When you say a particular thing in a certain situation, there is only so much it can be doing, and not all of it will be what you wanted. Your words can betray you, you can be caught out by them, reveal more than you thought you would, because they work in the world, not as a reflection of your mind. So I await any response other than ‘That’s not what I meant’ or ‘That’s just how you see it’.

it is not me making a judgment about people; I am just describing how disclaiming belief works in the world. — Antony Nickles

... the world of believers. That is not how it works in the world of atheists. We don't think of our ourselves or one another as

lost to the particular cause, or hopeless, maybe for anything going the way they hoped someone they trusted would do, but also by those that feel they know everything and have complete control over the world. — Antony Nickles

The way words work in the world is pretty much any way people choose to use them. Words are helpless in the hands and mouths and minds of manipulators.