| ⚠ Given a proof system L, associated functional semantics ML are generally sound (Γ ⊢L p implies Γ ⊨ML p) but unsound (Γ ⊨ML p may not imply Γ ⊢L p): in other words, though invalid rules in such many-valued logic ML are also underivable in L, valid rules in ML may not always be derivable in L. The denominations below generally refer to proof systems, but they must not be confused for the associated models provided.
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| Frege-Russell logic, a.k.a. "classical" logic, "standard" logic... | (Frege, 1879; Russell & Whitehead, 1910)
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| Trivial
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| Nihilistic
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| 3-valued Heyting algebra
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| 4-valued Boolean algebra
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| Strong Kleene | (Kleene, 1938)
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| CC1 | (Angell, 1962; McCall, 1966)
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| De Morgan monoid C6 (model for R)
| (Anderson & Belnap, 1975; Slaney, 1989)
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| FDE, a.k.a. Belnap–Dunn logic | (Dunn, 1976; Belnap, 1977)
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| LP | (Priest, 1979)
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| Linear logic
| (Girard, 1987)
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| M3V | (Mortensen, 1984; Named as such from McCall, 2012)
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| MC | (Wansing, 2005; As dBD in Omori, 2016)
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| C | (Wansing, 2005)
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| CN | (Cantwell, 2008)
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| MRSP | (Estrada-González, 2008)
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| A connexive extension of E | (Anderson & Belnap, 1975; This model can be found in Øgaard, 2020)
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| Non-monotonic restriction of logic C
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| Ω–∅ conditional probabilities
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