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Many logical operators seem to have multiple variants:

  • Material implication: ⇒, →, ⊃
  • Negation: ¬, ~, !
  • Conjunction: ∧, ·, &
  • Disjunction: ∨, +, ∥
  • etc.

Are there differences in their usage (e.g., certain ones more appropriate for certain fields or traditions over others)? Do they come in groups (e.g., if you use ~ for negation you should use · for conjugation; using ~ and ∧ together would be strange)?

This might not exactly be a logic question, but more of a sociological one pertaining to logic, that is what (if any) are the implicit rules that govern the use of these symbol variants, the violation of which would result in prefessional philosophers thinking, "Ah, this person is an amateur"?

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    Welcome! The short answer is variants do come in groups, and they are used by different communitites. In computer science, for instance, logic operators even vary among different, implemented computer languages. In, uh, visual basic for instance (&&, ||, ==, <>) may be used, while in SQL one might see (AND, OR, IS, IS NOT) to represent similar imperatives. And logicians have similar parallelisms. Some draw a bar over a variable and other's prefer ' ¬' in front. – J D Aug 15 at 14:42
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⊃, ~, &, ⋀ (big wedge symbol for "for all"), ⋁ (big vee symbol for "there exists")

tend to more frequently occur in old texts. These may also omit the universal quantifier symbol and write just (x) or the formula with free variables instead.

⇒, →, ¬, ∧, ∨, ∀, ∃

are more modern variants.

-, !, ·, &, +, |

are mostly fond in more technical applications (computer engineering, programming).

~, - &, ∥

may also appear when typesetting or rendering special symbols is technically difficult.

These are rough groupings; there do exist all kinds of combinations though.

For a neutral modern text, I suggest to go with

  • Negation: ¬
  • Conjunction: ∧
  • Disjunction: ∨
  • Material implication: →
  • Biimplication: ↔
  • Universal quantifier: ∀
  • Existential quantifier: ∃
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