Some logic questions.

For those short on time, please skip to the summary.

Sorry, I’ll refrain from using logic symbols, in hopes that (1) it’ll be easier for nonlogicians to give feedback as well, (2) I’ll get my message across without the problem of keyboard complications (symbols don’t work with software / are hard to decipher). (3) I’ll make less mistakes because I’m a rookie.

(1) True or false? The equivalence (or biconditional) is the same as: A conditional (If A then B) followed by the matching negated conditional? (If not A then not B). Thus: Saying “B if and only if A” is the same as saying “If A then B, and if not A then not B”

(2) Is there a term used to separate what I consider “circuit logic” from other propositional logic? To explain: I consider circuit logic to be logic based on output results from a given input. (Not necessarily circuit as in computer chip). Put something in: You automatically get something out. For example: In circuit logic, you can’t have a disjunction (“Either A or B”) because you can’t be sure which one it’ll be. Instead, you use something like quantifier logic like “Pxab”: P meaning “prefers” and thus: “x prefers a over b” or “Given the choice between a or b, x will always choose a.”

(3) Can logic use letters to abbreviate a proposition by representing it using a colon? For example: Would it make sense to other logicans if I said. “A: X and Y.” And further on I said “A or B” meaning “(X and Y) or B”

(4) Edward DeBono published a book called “Water logic.” In this book, he said that his main point is to introduce a new symbol to logic which simply means “to” or “leads to” Do you think he’s right? Let me elaborate.

A contributes to B. I can’t say the equivalent in propositional logic such as “If A then B” and I also can’t say the equivalent in predicate logic such as “Most As are Bs.” I can come close in quantifier logic by saying “Cab” with C meaning “contributes to” therefore: “A contributes to B.”

If Edward is correct, I think the most appropriate symbol for such a meaning would be the same as the conditional, but with a curve at the beginning implying “sort of if A then B” or in other words an empty arrow that curves or wobbles like the approximation symbol in math.


SUMMARY

(1) The biconditional is the same as a conditional and its echoe all negated. IE: “if and only if A, then B” is equivalent to the combined phrases “If A then B” and “If not A then not B.” True?

(2) Can there be a separate logic that I’d call “circuit logic?” That is: You get automatic outputs with whatever input?

(3) If “A: X and Y” is followed by “A or B”, then does the last phrase automatically mean: “(X and Y) or B” ?

(4) Is there or could there be a symbol in logic meaning “contributes to?” or else what’s the best way to say the same thing?

Interesting post… I’ll be giving (4) some thought and will contribute what I can later. Thanks for posting.

Gaia -

  1. is incorrect. Firstly, it would only be an example of equivalence, and not the primary statement of it. There are several logical “equivalences” that are established by one equation, which produces these several replacements - the rule of replacement is required.

What you want is this: (a → b) = (~b → ~a).

This is because implication itself (->) is not transitive.

Your 2) might be akin to inductive logic, but i do not understand your example well enough to know. It sounds like a collection of defintions, which is, strictly speaking, “prelogical”. It might be useful.

  1. is a matter of notation alone. There are many systems of notation. The one I use cannot be employed with my current keyboard, so I rarely use it. But as long as you define your symbols, anything goes, i think.

  2. I’m not sure of theb utilty of “sort of” implications. I guess I would have to read the book.

Faust-

As per (1) I’m going to play devil’s advocate if only for my own learning. Feel free to ignore.

I got confused in your statement. It’s certainly important to remember that the transitive rule for implication both reverses and negates the statement. “If a then b” doesn’t equal “If not a then not b” but with those two paired, I still can’t see how these together aren’t logically equivalent to “If and only if a, then b” - and - if they’re logically equivalent, then they state the exact same thing.

My explaination is kind of a jumbled. So this might be more concise.

My goal is to use as few symbols as possible in propositional logic. Supposedly, I can make the logical equivalence of a biconditional using only the conditional.

I’ll make a claim . . .

I say Groups A and B are logically equivalent.

GROUP A: “If and only if p, then q”

GROUP B: “If p, then q. If not p, then not q.”

I challenge anyone to find a derivation that can be made from one group, and not the other group.

Premise: I won’t need a biconditional to say the same thing as a biconditional.

Group A is a biconditional. Group B has no biconditional.

If groups A and B can have only the same derivations, and all the same derivations, then they are logically equivalent?

If groups A and B are logically equivalent, then I can substitute one for the other without a change in meaning.

If I can substitute group A (the biconditional) with group B (has no biconditional) without a change in meaning, then I won’t need the biconditional to say the same thing as the biconditional.

The actual “negated conditional” of A → B is ~(A → B) and not ~A → ~B. You meant to say the same conditional with the terms negated. That’s the part you were incorrect about, as I illustrated.

As to your actual question - equivalence is used mostly for defitnitions - and it is meant to state that the antecedent is true only when the consequent is also true. I’m not sure how your formulation is more elegant that the common “iff” in plain language, or the equivalence sign usually used in notation. I’m not sure what your goal is, here. This relation is usually defined in positive terms. What are you looking for with this more cumbersome formulation?