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Is

I am sure

and

I feel sure

having absolutely the same meaning?

Thanks!

Not to me. "I am sure" means that I know, from facts/data/experience that it is so. "I feel sure" means that I'm guessing/have an opinion that I think is right, but don't have direct observation. For example, I am sure that my coffee cup is empty, because I looked at it just now. I feel sure that there's more in the pot, because there usually is, but someone inconsiderate person might have taken the last cup without starting a new pot.

No. They are different. The person who states "I am sure" vs. feels sure is less sure.

Here's why:

First, the grammar. In both, the subject and object is the same. That makes them similar. Easily interchangeable to mean the same thing in a host of situations. They are. Both mean the person is sure of something.

However, the speaker used a different verb, so there must be some difference however slight. We won't know until we examine each verb's meaning, then compare them.

Am vs. Feel: One needs to grasp a concept of "explicit" this "stating something outright" explicitly with I "am" sure vs. "implicit" or things not said with something "implied" when based on feeling with I "feel" sure.

A person stating "I am sure" is explicitly telling you they are sure. The sentence is explicit, and emphatic vs. unemphatic (Purdue University Online Writing Lab https://owl.purdue.edu)

The person who "feels" is implicit. Feelings don't talk. The speaker may never tell you why they feel sure. Since feel is used, it conveys a small hint of uncertainty. The speaker's feeling could change at anytime. Some circumstance could arise that make him change how he feels. Maybe the speaker feels sure because their wife or their boss told them to do something. Do it this way or that way. The speaker is just following what they were told to do. They feel sure because someone over them told them to do it. This unsaid implicit feeling it's right to do what your wife asks or your boss tells you to do. You feel sure it's right.

Merriam-Webster gives definition for feel/feeling as: 2a. an emotional state or reaction; and, 2b. susceptibility to impression

With this definition for "feel," a reader can compare the verbs "feel" vs. "am" and the difference in these two statements come to light. The person who states "I feel sure" may only be sure based on an emotional state or reaction that could change. Feel susceptible to a different impression later. For example, a child tells his dad he wants a new bicycle for Christmas. The Father feels sure he's doing right by going to go buy it as his son asked for it. Later on, after discussing this with his wife, she tells him no. Gives him a laundry list of reasons dad shouldn't do it. His feeling sure to buy his son a bicycle for Christmas, based on implicit feeling it's right because his own son asked for it, is now changed. The father's feelings are susceptible to impression that can change his feeling of being sure. Stating, "I feel sure" carries this dynamic or connotation.

For this reason, after comparing the verb meanings, "I feel sure" is weaker than the emphatic, or explicit "I am sure."

This Merriam-Webster definition of "am:" 1. just to be. Just to be something. To just be sure. Far less likely to bend under any weight of an impression. Less likely to change vs. based on feelings.

Same subject (I) same object (sure), but different verb. Different verb gives each a separate dynamic no matter how similar having the same subject or object makes them the same, the meanings are slightly different based on the different verb used. I hope this helps.

That's it.