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Would you check these?

If he should come, I will ask him about it. He might come or not, mightn't he? Should implies less possibility, doesn't it? The First Conditional.

If he should come, I would ask him.
He will not come, will he? Should shows less probability, doesn' it? But..It's unreal. It keeps me confused. The second conditional.

We shall be grateful, if you will send us a telegram. "Will" implies politeness and request, doesn't it? The First Conditional

Thank in advance.
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I would interpret should come as "if he just happens to come"
Fandorin If he should come, I will ask him about it. I would interpret this as "if he just happens to come along, then I will ask him." I hardly ever use "should" in this situation. I say "If he comes, I'll ask him about it."

If he should come, I would ask him. I don't use this should...would combination.

We shall be grateful, if you will send us a telegram. " Shall is slowly becoming obsolete in informal speech. It is still common in legalese and very formal writing, With "shall", "would send" is used in the conditional clause. "We shall be grateful if you would send us..."
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FandorinIf he should come, I will ask him about it. He might come or not, mightn't he? Should implies less possibility, doesn't it? The First Conditional.

If he should come, I would ask him.
He will not come, will he? Should shows less probability, doesn' it? But..It's unreal. It keeps me confused. The second conditional.

We shall be grateful, if you will send us a telegram. "Will" implies politeness and request, doesn't it? The First Conditional
I would not attempt to put these into the three conditional categories. I think of shall and should in the if clause as special cases.
Sentence 1. If he should come = If he comes = If he happens to come
This turn of phrase sounds like something from my grandfather's generation. I don't think many people in the U.S. use should much in the if clause anymore -- though I could be wrong.

Sentence 2. If he should come = If he came
This has a distinctly odd ring to it. I don't think I would ever use it.
Sentence 3: Also rather odd and old-fashioned sounding, although your description with "politeness" and "request" does seem to apply.
CJ
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