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If A then maybe B, but not necessarily.
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Some Cs are Ds.
There exists at least one x, such that C(x) and D(x).
To see this correctly, install this add-on.
[; \exists x, C(x) \wedge D(x) ;]
(Or look here, while it lasts.)
(This takes C and D as logical predicates, such that ‘C(x)’ means ‘x is a C’ and ‘D(x)’ means ‘x is a D’. It’s trivial to recast this such that C and D are sets, and ‘Some Cs are Ds’ is translated to ‘There’s at least one thing that is a member of both C and D.’)
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There’s not really a way to express this in formal logic, because there isn’t really any information there. Facetiously, one could express it as “A→(B∨¬B)” (A implies B or not B) but that’s tautologically true.
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Depending on what you mean by this, you could express it as “C∩D≠∅” (the intersection of the sets C and D is not empty)
ETA: My version of no. 2 is logically equivalent to Derleth’s, which can be written without the use of LaTeX as “∃x, C(x) ∧ D(x)”.
I interpreted the statement as something for which some fuzzy logic would be suited, such that the probability of A implying B was known and could be expressed compactly in that logic’s notation.
But that both assumes Unicode support and isn’t nearly as pretty as mine. 
Just to add on if you want to google for more info, this is called “existential instantiation” and there are logical rules that use it.
Yours is only prettier if you have the extension installed. But seriously, thanks for introducing me to TeX The World (sorry, [; extrm{\TeX\ The World} ;] :p). It’s a neat idea.
Wow, these sorts of things really aren’t standardly supported by the latest version of html yet?
You can render TeX in HTML quite nicely. But it takes a lot of work, and access to a page’s CSS, which a post on a vBulletin board does not provide. And that’s why it’s done in an automated fashion, with Jacascript packages like MathJax or browser addons.
Or click here:
(I think there’s a way to place those generated images from “math.eretrandre.org” into an html document without an extra click, but probably not for a user message in a message board like SDMB.)
Oops !! The link works on preview, but not after post!
Double oops!! The link works if one removes the message-board generated “” after each preview!!
I can’t enter MathML here in a way that will make it render correctly in modern (or, indeed, any) browsers. This is a limitation of the message board software, nothing else.
(Besides, composing MathML by hand is horrible; it’s designed to be produced and consumed by software.)
For a certain subset of TeX, and provided you don’t mind ugly typography (or else are willing to render the TeX as an image), sure. Web browsers aren’t suitable for displaying complex mathematical typography. To paraphrase Samuel Johnson, it is like a dog’s walking on his hind legs: it is not done well; but then you are surprised to find it done at all.
As others have pointed out, there’s no meaningful way of representing this in normal logic. But some logical systems, such as defeasible logic and fuzzy logic, allow you to represent such notions. For example, I’ve seen “PROBABLY (p → q)” for “if p, then probably q”. In one of my papers on defeasible reasoning, we used the notation “p ⇒ q” to mean “if p, then typically q”. For example, “bird ⇒ flies” means “Birds typically fly”, we can then define other “defeating” rules such as “heavy ↝ ¬flies” (“Heavy things might not be able to fly”). Taken together, the two rules could be interepreted as “Birds typically fly, unless they are heavy.”
As others have pointed out, on the obvious reading of (1), it has no real content. I think it could be written in modal logic notation as “A–>◊B” (where ◊ translates a “possibly”), but that still is not really saying much. (I do not agree that it says " A–>probably B". I see no sign of an implication there that B is more likely to be true if A is true than if A is false.)
[I don’t know if the “possibly” symbol will show up correctly in everyone’s browser, but it is meant to be a little hollow diamond shape.]
I think, however, (1) might be interpreted as saying (or trying to say) that B may be (but is not necessarily) true only if A is true. That can most succinctly be expressed as “B–>A”.
[;
eg\square(A \rightarrow \diamond B) ;]
But that sentence is ambiguous, and can be translated into modal logic in at least two ways.
Agreed with njtt. I beleive the OP’s item (1) should be interpreted as “A is a necessary, but not sufficient, condition for B.”
This may be my very first ever verifiable factual contribution to the Straight Dope.
It’s not correct to say 1) can’t be expressed using formal logic. One way to express it is as follows:
A -> (<>B & -B)
Read “->” as an arrow, “<>” as a diamond, and “” as a box."
Arrow means if-then, diamond means “possibly,” and box means “necessarily.” (Dash means “not,” ampersand means “and”.)
What is it you’re working on that makes you wonder about these two translations?
Is A -> (<>B & -B) actually any different from A–>◊B (which I already suggested)? :dubious:
And do either of them really differ from just ◊B? If not, the mention of A (and, indeed, of “if” and “then”) is superfluous.
I still think a more plausible (and charitable) reading of the intended meaning is B–>A.
It is different. “<>B” might be true while “<>B & -B” is false–for example, if B means “2 + 2 = 4.”
They’re both completely different from simply <>B.
Let A mean “I am jaywalking across a busy street” and B mean “I am about to get run over a car” Suppose that I am standing in the middle of a perfectly open and flat field with no cars in sight. In such a scenario, A → <>B is true, while <>B is false. (We have to be careful here about defining the scope of our possibility operator here, but the details don’t need to be worked out in the context of a discussion board forum post…)*
I don’t think so. With no further context provided, what we have to go on are the uses of the words “maybe” and “necessarily,” and there’s hardly a more straightforward translation of these two terms than into the diamond operator and the box operator.
*A really, really clear-cut case, albeit one that relies on the idea that a conditional with a false antecedent is always true, is this: let A mean “2+2=5” and B mean “2+2=7”. In that case, A -> <>B is true, while <>B is false.