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From my understanding, the modal quantifier possibly relaxes the behaviour of logical conjunction:

(true)                and (possibly false)   => true
(true)                and (possibly unknown) => true
(possibly true)       and (possibly false)   => true
(possibly true)       and (possibly unknown) => true
(necessarily true)    and (possibly false)   => true
(necessarily true)    and (possibly unknown) => true
(unknown)             and (possibly false)   => unknown
(possibly unknown)    and (possibly false)   => unknown
(necessarily unknown) and (possibly false)   => unknown

My question is: does it have any specific impact on other operations (like exclusive disjunction, implication and equivalence)? What about these examples (which can't figure out myself):

(true) xor (possibly true)    => true | false   ?
(true) xor (possibly unknown) => true | unknown ?
duplode
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AlexDarkVoid
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  • Well, 3-valued matrices are not sufficient for representing even the very basic properties of modal operators, see e.g paragraph 2.2 in [this paper](http://www.jyb-logic.org/papers/02beziau-galley-proofs.pdf). Anyway, you should post your matrices for negation, possibility and necessity first. – Stanislav Kralin Dec 25 '17 at 11:38
  • Matrices for implication are different in different three-valued systems, see e. g. [these notes](https://www.overleaf.com/7062996fbfhyyhmsgwk#/24252822/) (in Russian). As to exclusive disjunction, it may depend on how you define it (is it equivalent to something like `¬(A ≡ B)` or to `(A∨B) ∧ ¬(A ∧ B)`). – Stanislav Kralin Dec 25 '17 at 11:39
  • Also, your modal operators are alethic (like in Łukasiewicz 3-valued system), while informal interpretation of truth values are rather epistemic (like in Kleene 3-valued (strong) system)... – Stanislav Kralin Dec 25 '17 at 11:40
  • Thanks for links. You see, I'm working on a system that extracts and evaluates conditions, expressed in a natural language. In this context modal logic is just an extension on top of Kleene 3VL: *necessary* and *possibly* themselves are modifiers, not truth values. AND, OR, XOR, IMP are used to represent corresponding linguistic constructs with conjunctions. – AlexDarkVoid Dec 25 '17 at 18:30
  • If you say _X and probably Y_ the systems constructs this (in Lisp syntax): `(m3vl:and3 (cons :necessary X) (cons :possibly Y))`. For _either X or possibly Y_ `(m3vl:xor3 (cons :necessary X) (cons :possibly Y))`. _if X then possibly Y_ becomes `(m3vl:imp3 (cons :necessary X) (cons :possibly Y))`. There X/Y can be anything (true, false, unknown), i.e. they're not bound with their modal quantifiers. **X** and *necessary* **X** are considered equivalent. – AlexDarkVoid Dec 25 '17 at 18:30
  • From the first example (comment): even if Y is `false` or `unknown`, it is still enought to satisfy the whole condition (AND). In case of XOR (*either*): it is unclear. Generally, only one condition must be `true`, i.e. `F xor F -> F, F xor U -> U, F xor T -> T, U xor U -> U, T xor T -> F` – AlexDarkVoid Dec 25 '17 at 18:42

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