Logic textbooks traditionally teach about four kinds of categorical statements (i.e., propositions) when discussing the Aristotelian square of opposition: they are A, E, I, and O propositions. The first two are universal statements whereas the last two are particular utterances. I will now explain briefly how I propositions work. These are particular affirmatives:
Example of an I proposition:
A) Some trees are oaks.
But what happens if we contrapose an I proposition? This would entail switching subject and predicate, then adding the complement (a term not belonging to the class) of the switched subject and predicate.
But what happens if we contrapose an I proposition? This would entail switching subject and predicate, then adding the complement (a term not belonging to the class) of the switched subject and predicate.
Here is the contraposition of an I proposition:
B) Some non-oaks are non-trees.
What?
Notice that B) is similar to the utterance:
C) Some non-Greeks are non-humans.
C) is the contraposition of Some humans are Greek.
An interesting result of contraposing an I proposition is that the true value of the statement differs after one does the contraposition; it is not preserved with I propositions.
Whether one is doing logic, theology or some other discipline, reasoning correctly is important.
An interesting result of contraposing an I proposition is that the true value of the statement differs after one does the contraposition; it is not preserved with I propositions.
Whether one is doing logic, theology or some other discipline, reasoning correctly is important.
31 comments:
Converse, in logic, the proposition resulting from an interchange of subject and predicate with each other. Thus, the converse of “No man is a pencil” is “No pencil is a man.” In traditional syllogistics, generally only E (universal negative) and I (particular affirmative) propositions yield a valid converse.
https://www.britannica.com/topic/categorical-proposition
More false statements by Dan. Plato would want a word with him and so would the writer of Genesis.
https://www.tandfonline.com/doi/full/10.1080/23311983.2020.1868687
Sure, go ahead and call a boy, a girl, and vice versa.
See also https://fosterheologicalreflections.blogspot.com/2024/01/obversion-of-categorical-statements-in.html
Thanks for the additional info, y'all.
Duncan, I love and appreciate the info from Pitt.edu, but one thing I will add is that contraposing conditional statements is a little different from contraposing categorical statements/propositions. The two are related, but different animals. Nevertheless, if one contraposes a conditional statement, the inference is valid and the statewents are logically equivalent. Contraposition as a rule of inference is also known as transposition.
I'm going to be brief due to the memorial, but I know language existed before Genesis was written, even if one thinkks the book was written early. After all, I believe it existed in oraal form before being written. However, Genesis depicts language as being largely non-arbitrary. For instance, there's a reason Eve is called Eve or that Adam has his name. The same applies to animals like the ape or serpent.
Duue to the passage of time, etymology becomes obscure and we might even not know what a word originally meant. However, God made the original pair, male and female and these words now have fixed meanings. Or to use an example from units of measurement, hour, minute, meter, yard all have fixed meanings. To call an inch a yard would show one is confused and his/her language, confusing. My point is that we have reasons for using words, they are not merely arbitrary. Those who insist on this wrongheaded idea normally have agendas. Even Plato recognied that there is a reason we apply "cat" to a feline and "dog" to a canine. To call "Yahweh" Baal would be equally unacceptable.
"Some" is a particular term: it might seem vague on its own, but when used categorically, the vagueness seems to be attenuated as in "Some cats are felines." Logic books point out that "Some" in that case means at least 1.
Duncan, for the picture you submitted, I'm not sure how that relates to contraposition, nor doe the Armstrong as far as I can tell.
I did a word search on the BOBCAT file: I looked at 44 occurrences of the word "some" but never came across a categorical statement containing the term. Maybe I needed to look at the 45th occurrence or you could tell me which page makes the point you mentioned.
I conceded that "some" is vague on its own, but IMO, "Some dogs are canines" is less vague than "Some" alone or "Some X is Y," etc. You tell me how one can differentiate a particular affirmative statement from a universal affirmative or universal negative without using "some."
The British and American systems of measurement are two different things as you know. So, it's not surprising if the British Imperial gallon and American customary gallon differ: they're not the same unit of measurements. However, it would be confusing if someone called an American pint a gallon and the gallon, a pint.
See https://opentextbc.ca/basickitchenandfoodservicemanagement/chapter/imperial-and-u-s-systems-of-measurement/#:~:text=The%20only%20difference%20between%20the,the%20table%20in%20Table%207.
That's like the speed of light: miles in America, but kms in British and scientific lingo.
See page 231 here: https://books.google.com/books?id=McPADwAAQBAJ&pg=PA231&dq=particular+affirmative+%22vague%22&hl=en&newbks=1&newbks_redir=0&sa=X&ved=2ahUKEwjstY6dyo6FAxXSv4kEHSldD644ChDoAXoECA0QAg#v=onepage&q=particular%20affirmative%20%22vague%22&f=false
https://www.jewishvirtuallibrary.org/nehushtan
https://www.thetorah.com/article/nehushtan-the-copper-serpent-its-origins-and-fate
Mutual exclusivity:In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails, but not both.
In the coin-tossing example, both outcomes are, in theory, collectively exhaustive, which means that at least one of the outcomes must happen, so these two possibilities together exhaust all the possibilities.[1] However, not all mutually exclusive events are collectively exhaustive. For example, the outcomes 1 and 4 of a single roll of a six-sided die are mutually exclusive (both cannot happen at the same time) but not collectively exhaustive (there are other possible outcomes; 2,3,5,6).:
https://en.m.wikipedia.org/wiki/Mutual_exclusivity
@servant, thanks. I agree with everything stated. As another example, All trees are biological organisms and Some trees are not biological organisms cannot both be true at the same time or with respect to the same referent, etc.
I agree that "some" by itself is vague. However, my point is that the vagueness might not disappear but it's attenuated/lessened when one utters a particular affirmative or particular negative proposition. I guess we could say, "At least one dog is a canine," but we traditionally say "Some dogs are canines" instead. If you read the material I posted from De Morgan at Google Books, he made a similar point.
Here's another thing to consider: logic aims for precision. It is a work in progress.
Well, no flattery intended, but you have strengths that I don't. The questions often make me reconsider thongs or reevaluate ideas.
Things, not thongs ☺
Why does autocorrect do that?
Come on Google, it's 2024 let's get that edit button already.
I know this is a flawed example
There is only one tree
[subject] is a tree
So according to the logic of some, the mammal and a mammal must be some sort of confusing theological rubbish - This tree I speak of then must be a “substance” to which all the other “members” belong?
What if “tree” is used in several meanings and means different things based on the context
Unknown, you haven't really given a flawed example because formal logic allows for statements like, Some S is P or "Fred is a man."
But one popular definition of substance coming from ancient times is a bearer of properties. Aristotle distinguishes between primary and secondary substance. The idea is that trees are substances insofar as they bearcertain properties, but this does not exclude using tree in a different sense.
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