What semantic role does ⺉/刀(sword/cut) tend to have in Kanji?

Question

判 Has the phonetic component 半, but why the semantic component knife?

刊 has the phonetic component 干, but why the semantic component knife?

剛 has the phonetic part 岡, but why the semantic component knife?

I've seen more of these. With 雨 (rain) it often seems to indicate it has to do with weather, with 貝 (shell) it indicates it has to do with money, so does ⺉/刀 also have such an overarching thing that isnt literally cutting, or is it one of those cases where it's different shit per character or the character just went through simplifications?

Answer 1

• 「判」originally meant to split something into two [halves], extended to mean to distinguish (between different things) > evaluate, judge, criticise.「半」(half) is both a semantic and phonetic component.
• 「刊」originally meant to chop (e.g. a tree), then extended to mean carve > publish.
• 「剛」means unyielding, strong, sturdy.「⺉・刀」is a reference to a meaning like powerful > unyielding.

The most common semantic contribution of「⺉・刀」was originally to do with actions with a sharp tool. Semantic extension obscures this contribution slightly.

Some other examples:

• In「到」,「⺉」is only a phonetic component.
• 「絕」(Shinjitai:「絶」), means to cut off or extinguish, and originally referred to the cutting「刀」of silk threads「糸」.「刀」is found on the top-right hand side.
• In「斷」(Shinjitai:「断」),「㡭」is a variant of「絕」, also containing silk threads (abbreviated, 4「幺」).「刀」is the remaining part of「㡭」after 4「幺」is removed; to see this, rotate「刀」90 degrees clockwise and flip it horizontally.
• 「則」（rules, regulations) originally referred to pattern inscription (using a tool「刀」) on bronzeware (「鼎」, now abbreviated into「貝」).
  • Referring back to the notes in the question,「貝」does not always have something to do with money; among other things, it is sometimes an abbreviation of「鼎」. Also,「貝」has nothing to do with「頁」.

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References:

• 季旭昇《說文新證》
• 《王力古{{kr:漢}}語字典》

Answer 2

For this, you'll have to look at the original meaning of the those three characters to see their relation with 刀.

• 判 originally meant to divide up, presumably with a knife.
• 刊 originally meant to use a knife to cut off the outer skin
• The definition of 剛 was originally 彊斷也。从刀，岡聲, meaning strong. According to 林义光, this can be explained as using a knife to cut a net/web. This meaning seems to not have changed too much.

As is typical with Chinese characters, their meanings change over time. Often, they become more generalized or applied to similar situations. 刊 for example from the meaning of cutting off the outer skin became applied to carving, and from carving onto typesetting/printing (think carving printing blocks!).