Please see Linear Logic in wikipedia: the symbol ⅋ ("par") is used to denote multipicative disjunction, whereas ⊗ is used to denote multiplicative conjunction. They are duals of each other. There is also some discussion of this connective.
Variously, one can denote the "par" operation using the symbol $ \sqcup $, which is described in the paper (pdf) "Linear Logic Displayed" as "join-like."
The logical rules for ⊗ and ⅋ are as follows:
If Γ: A: B Δ ⊢Θ, then Γ: A⊗B: Δ ⊢ Θ; conversely, if Γ ⊢ Δ: A and ⊢ B: Θ, then Γ ⊢ Δ: A⊗B: Θ.
Dually, if Γ ⊢ Δ: A: B: Θ, then Γ ⊢ Δ: A⅋B: Θ; conversely, if A: Γ ⊢Δ and Θ: B ⊢, then Θ: A⅋B: Γ ⊢ Δ.
Multiplication distributes over addition if one is a conjunction and one is a disjunction:
A⊗(B⊕C) ≡ (A⊗B)⊕(A⊗C) (and on the other side);
(multiplicative conjunction distributes over additive disjunction)
A⅋(B&C) ≡ (A⅋B)&(A⅋C) (and on the other side);
(multiplicative disjunction over additive conjunction)
- A⊗$0$ ≡ $0$ (and on the other side);
- A⅋⊤≡⊤ (and on the other side).
Linear implication $A\multimap B$ corresponds to the internal hom, which can be defined as (A⊗$B^⊥)^⊥$. There is a de Morgan dual of the tensor called ‘par’, with A⅋B=($A^⊥⊗B^⊥)^⊥$. Tensor and par are the ‘multiplicative’ connectives, which roughly speaking represent the parallel availability of resources.
The ‘additive’ connectives & and ⊕, which correspond in another way to traditional conjunction and disjunction, are modeled as usual by products and coproducts.
For a nice exploration in linear logic: see this:
Game semantics for linear logic was first proposed by Andreas Blass (Blass (1992).) The semantics here may or may not be the same as proposed by Blass.
We can interpret any proposition in linear logic as a game between two players: we and they. The overall rules are perfectly symmetric between us and them, although no individual game is. At any given moment in a game, exactly one of these four situations obtains: it is our turn, it is their turn, we have won, or they have won; the last two states continue forever afterwards (and the game is over). If it is our turn, then they are winning; if it is their turn, then we are winning. So there are two ways to win: because the game is over (and a winner has been decided), or because it is forever the other players turn (either because they have no move or because every move results in its still being their turn).
This is a little complicated, but it's important in order to be able to distinguish the four constants:
In ⊤, it is their turn, but they have no moves; the game never ends, but we win.
Dually, in 0, it is our turn, but we have no moves; the game never ends, but they win.
In contrast, in 1, the game ends immediately, and we have won.
Dually, in ⊥, the game ends immediately, and they have won.
The binary operators show how to combine two games into a larger game:
In A&B, is is their turn, and they must choose to play either A or B.
Once they make their choice, play continues in the chosen game, with
ending and winning conditions as in that game.
Dually, in A⊕B, is is our turn, and we must choose to play either A or
B. Once we make our choice, play continues in the chosen game, with
ending and winning conditions as in that game.
In A⊗B, play continues with both games in parallel. If it is our turn
in either game, then it is our turn overall; if it is their turn in
both games, then it is their turn overall. If either game ends, then
play continues in the other game; if both games end, then the overall
game ends. If we have won both games, then we have won overall; if
they have won either game, then they have won overall.
Dually, in A⅋B, play continues with both games in parallel. If it is their turn in either game, then it is their turn overall; if it is our
turn in both games, then it is our turn overall. If either game ends,
then play continues in the other game; if both games end, then the
overall game ends. If they have won both games, then they have won
overall; if we have won either game, then we have won overall.
So we can classify things as follows:
In a conjunction, they choose what game to play; in a disjunction, we
have control. Whoever has control must win at least one game to win
In an addition, one game must be played; in a multiplication, all
games must be played.
To further clarify the difference between ⊤ and 1 (the additive and
multiplicative versions of truth, both of which we win); consider A⅋⊤
and A⅋1. In A⅋⊤, it is always their move (since it is their move in ⊤,
hence their move in at least one game), so we win just as we win ⊤.
(In fact, A⅋⊤≡⊤.) However, in A⅋1, the game 1 ends immediately, so
play continues as in A. We have won 1, so we only have to end the game
to win overall, but there is no guarantee that this will happen.
Indeed, in 0⅋1, the game never ends and it is always our turn, so they
win. (In ⊥⅋1, both games end immediately, and we win. In A⊗1, we must
win both games to win overall, so this reduces to A; indeed, A⊗1≡A.)
Negation is easy: To play A ⊥, simply swap roles and play A.
A game is valid if we have a strategy to win (whether by putting the
game in a state where we have won or by guaranteeing that it is
forever their turn). The soundness and completeness of this
interpretation is the theorem that A is a valid game if and only if ⊢A
is a valid sequent. (Recall that all questions of validity of sequents
can be reduced to the validity of single propositions.)