'I Have' Why Not 'I Has'?

In the present tense, the -s belongs to the 3rd person (sing)

As you can tell from the previous answers, it is the 3rd-person-singular form in the present tense that is "out of step", not the 1st-person-singular. For regular verbs, all forms are the same except for the 3rd-person-singular, which adds an 's'.

EDIT: the quoting feature doesn't seem to be working right now. I used "----" to represent quotes.
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I wish to know "Why 'have' is used after I instead of 'has'.
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The verb to "have" comes from the Old English verb "habban".

I have was "Ic haebbe"
Thou hast was "Thu haefst"
He has was "He haefth"

Later on, the standard form from the 3rd person singular became "hath". After that, the Northern dialect form won out, which used -s as the ending, rather than -th. So, the standard form became "has".

"Thou" dropped out of common use, and was completely replaced with the plural (acc) "you".
Therefore its ending died with it (thu haefst -> thou hast -> 0)

So, in the singular, we are left with "I have" and "He has".

So, as you can see, "has" simply comes from the Northern dialect form for "hath". "I have" was originally "haebbe", but became regularized to "have".

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Because otherwise we use "have" after plurals (they, we, etc.) and "has" after singulars (he, she, it). Why then we use "have" after I (singular).
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It has nothing really to do with singular and plurals. The rule is not "it's have for plurals, and has for singulars." The plural forms were all regularized to "have", as was the first person singular. Originally they had different forms. The third person singular was not regularized, and the second person singular dropped out of common use in most modern dialects of English--it would be "thou hast".

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e.g. We say "I have got a pen". Why not "I has got a pen".
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In certain dialects of English, such as British English, "I have got" is considered correct. In my dialect, as well as in several other North American English dialects, "I have got" is considered incorrect, although the contracted form "I've got" is occasionally used in informal speech by people. We use simply "I have". We do use "I've gotten" though, but it does not mean "I have", but rather "I've received."

Illogical question

But why is it "I was", and not "I were" then?