Some notes on the symbols: ◻ and ◊ are modal operators, meaning "necessarily" and "possibly" respectively. ϕ is just a Greek letter, Phi, which is commonly used in logic to represent any given proposition. ¬ is the symbol for negation. Thus, in the top left corner, ◻ϕ means 'necessarily Phi,' and in the bottom right, ¬◻ϕ is 'not necessarily Phi,' whereas at the top right ◻¬ϕ is 'necessarily NOT Phi.' The triple bar equal sign is a logical biconditional, meaning that the two terms mean exactly the same thing. I imagine you can work out the rest from that.
Traditionally, the Aristotelian relations of contradiction, contrariety, and subalternation are supplemented with an additional relation of subcontrariety, so called because the subcontraries are located under the contraries. As the contradictories of the two contraries, the subcontraries (e.g., Some pleasure is good, Some pleasure is not good) can both be true, but cannot both be false. For Aristotle, this was therefore not a true opposition, since subcontraries are “merely verbally opposed” (Prior Analytics 63b21–30). Within pragmatic theory, the assertion of one subcontrary (Some men are bald) is not only compatible with, but actually conversationally implicates, the other (Some men are not bald), given Grice’s Maxim of Quantity (“Make your contribution as informative as is required”; see the entries on Paul Grice, pragmatics, and implicature).
The article on implicature is also interesting.
So the contraries are "necessarily Phi" and "necessarily NOT Phi," the latter of which is equivalent to "Not Possibly Phi." The subcontraries are "Possibly Phi" and "Not Necessarily Phi," which is equivalent to "Possibly NOT Phi."
I like the way they've graphed this relationship, because you can also see the entailments on the two vertical sides. If Phi is Necessary, it must also be possible; that one is obvious enough. And if Phi is necessarily not the case, then Phi isn't possible: that's a straight equivalence. What might not be immediately obvious to new students of logic is that "NOT Possibly" entails "Possibly not."
Clarity of thought is improved by clear logic. Plus, it's kind of fun.
3 comments:
Logic is something I've been thinking about regularly, though I probably haven't been thinking logically about it. I am attracted by the idea that our schools (or parents) should be teaching informal logic from 4th or 5th grade and that it should inform all of the humanities education students receive. But that's just me, and I thought the Scholastics were kinda cool, so take that as you will.
Informal logic is a lot more useful for day-to-day things, as for example learning to recognize fallacious forms of argument. (Note, however, that a fallacious argument can be perfectly correct and true: it's just that the fallacious logic cannot guarantee the truth of it. A slippery slope argument may very well prove perfectly prescient about how things will transpire, but the logic doesn't guarantee that it will.)
Formal logic is only useful in a limited way in practical matters. It is very useful in clarity of thought. However, logical objects do not exist in reality. A logical proof will be useless in political or ethical arguments, because all you can get to in those arenas is arguments about what is most likely or predictable.
I have yet to see the attraction of formal logic, not that I've looked all that hard for it. But then, I never got far into mathematics. I guess the closest I've gotten, if it is close at all, is playing Go seriously for a few years. There is a beauty to the reasoning of the play, although Go is more intuitive than chess.
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