3 ms·

It's called logical addition which is different than arithmetic addition, so it's best to not conflate the two. It's called "addition" because you "add" another

by dvt 3y ago

It's called logical addition which is different than arithmetic addition, so it's best to not conflate the two. It's called "addition" because you "add" another propostion, creating a disjunction. And by the rules of logic, as long as the original proposition was true, the truth value is preserved no matter how many propositions you introduce (or "add"). A ⊢ ((A ∨ B) ∨ C) ... ∨ n

thaumasiotes 3y ago
> It's called "addition" because you "add" another propostion, creating a disjunction. By that argument, conjunction is also called "addition". Perhaps there's a different reason? The choice of terminology, contrasting "logical addition" with "logical multiplication", obviously indicates that logical addition is supposed to be analogous to addition. What is the analogy?
- andai 3y ago
  I don't know if this relates to formal logic, but in C a true value is defined as not zero, so all nonzero values are treated as the same value. So the only way to get a false is to OR (or add) two zeroes, making the two operations equivalent as far as boolean logic goes. Actually, you can add 1 and -1 to get 0, which breaks the model... Hmm... Guess it only works on natural numbers / unsigned.
- EGreg 3y ago
  The analogy is deep. But actually the relationship is as follows: P(A u B) = P(A) + P(B) - P(A^B) P(A ^ B) = P(A u B) - P(A) - P(B) And to the guy who said xor is addition — no it’s not: P(A xor B) = P(A) + P(B) - 2 * P(A^B) In Probability, when you have a space of outcomes, doing a Union on two disjoint events (sets of outcomes) the probabilities add. Whereas doing an Intersection on two non-disjoint events those probabilities multiply. If two events have no outcomes in common, for instance, the probability of them both occurring is zero. When you “condition” probability on an event X happening, you restrict the space of all outcomes to that set X, and intersect every event with it. When you condiition B[n] = A[n] given that (A[n-1] AND … A[1]) having already happened, you have a sales funnel and each step is independent of the previous, so the probabilities multiply. It is also called a Markov Chain if you have a discrete set of possibilities at each step so you can form a matrix.
  - cubefox 3y ago
    Interesting, I never saw the arithmetic probability formula for xor. Though the original topic was truth values, not probabilities. To make it short, conjunctions in probability are multiplicative when they are independent, and disjunctions in probability are additive when they are mutually exclusive.
    - EGreg 3y ago
      Well I actually came up with it on the spot. Just think of the venn diagram, and add the areas of A and B, which counts A^B twice, and then just remove it twice. :)
- dvt 3y ago
  > By that argument, conjunction is also called "addition". Perhaps there's a different reason? A chain of (arbitrary) conjunctions is not necessarily implied by the first proposition, so we have the much less interesting: A ⊬ ((A ∧ B) ∧ C) ... ∧ n
- cubefox 3y ago
  > By that argument, conjunction is also called "addition". Yeah, I just noticed: a and b = a + b - (a or b) Or with probability P(a and b) = P(a) + P(b) - P(a or b)

by dvt 3y ago

thaumasiotes 3y ago

> It's called "addition" because you "add" another propostion, creating a disjunction. By that argument, conjunction is also called "addition". Perhaps there's a different reason? The choice of terminology, contrasting "logical addition" with "logical multiplication", obviously indicates that logical addition is supposed to be analogous to addition. What is the analogy?

andai 3y ago
I don't know if this relates to formal logic, but in C a true value is defined as not zero, so all nonzero values are treated as the same value. So the only way to get a false is to OR (or add) two zeroes, making the two operations equivalent as far as boolean logic goes. Actually, you can add 1 and -1 to get 0, which breaks the model... Hmm... Guess it only works on natural numbers / unsigned.
EGreg 3y ago
The analogy is deep. But actually the relationship is as follows: P(A u B) = P(A) + P(B) - P(A^B) P(A ^ B) = P(A u B) - P(A) - P(B) And to the guy who said xor is addition — no it’s not: P(A xor B) = P(A) + P(B) - 2 * P(A^B) In Probability, when you have a space of outcomes, doing a Union on two disjoint events (sets of outcomes) the probabilities add. Whereas doing an Intersection on two non-disjoint events those probabilities multiply. If two events have no outcomes in common, for instance, the probability of them both occurring is zero. When you “condition” probability on an event X happening, you restrict the space of all outcomes to that set X, and intersect every event with it. When you condiition B[n] = A[n] given that (A[n-1] AND … A[1]) having already happened, you have a sales funnel and each step is independent of the previous, so the probabilities multiply. It is also called a Markov Chain if you have a discrete set of possibilities at each step so you can form a matrix.
- cubefox 3y ago
  Interesting, I never saw the arithmetic probability formula for xor. Though the original topic was truth values, not probabilities. To make it short, conjunctions in probability are multiplicative when they are independent, and disjunctions in probability are additive when they are mutually exclusive.
  - EGreg 3y ago
    Well I actually came up with it on the spot. Just think of the venn diagram, and add the areas of A and B, which counts A^B twice, and then just remove it twice. :)
dvt 3y ago
> By that argument, conjunction is also called "addition". Perhaps there's a different reason? A chain of (arbitrary) conjunctions is not necessarily implied by the first proposition, so we have the much less interesting: A ⊬ ((A ∧ B) ∧ C) ... ∧ n
cubefox 3y ago
> By that argument, conjunction is also called "addition". Yeah, I just noticed: a and b = a + b - (a or b) Or with probability P(a and b) = P(a) + P(b) - P(a or b)