## All A are B vs A are B. Is there a difference?

• 13
For example:
Suppose that all men are mortal. This supposition is necessarily false if there exists a man that is not mortal.
Suppose instead that men are mortal. Is this supposition also necessarily false if there exists a man that is not mortal?
• 2.4k
I don’t think understand the idea of logic?

I can say “All cats are yellow” as a logical premise if I want to.

There are various conjunctions and the differentiation between ALL, SOME and NONE is the convention of logical propositions.

There is a difference between VALIDITY and SOUNDNESS too. I suggest you look them up.

Hope this helps :)
• 13

You can say that all cats are yellow as a logical premise. Did something lead you to believe that I don't believe that?

What I wanna know is if, in logic, "cats are yellow" should be interpreted as "all cats are yellow."

If it's not specified that all cats are yellow, and not specified that some cats are yellow, and not specified that no cats are yellow, then what does "cats are yellow" mean? Does it mean all cats are yellow? Does it mean some cats are yellow? Does it mean no cats are yellow? Does it mean something else? Does it mean nothing? What is the convention in logic?

A valid argument is an argument where the premises forces the conclusion to be true. A sound argument is an argument that is valid and the premises are true.
• 4.2k
Probably contextual, the presence of a universal quantifier like that doesn't do strictly the same thing every time to every utterance.

'Food is good' could be said after a meal, but 'all food is good' might be too strong; the utterer would probably not enjoy having their pet served to them. 'Wo/men suck' can be said to vent frustration about a partner or strange experience, but does not entail having a negative opinion of all men or women. Such utterances can be vehicles for opinion in the same sense 'Fuck off' might mean 'Leave now' or 'I don't believe you' depending on the context.

Conversely, 'people die', 'girls have uteruses' probably do express the same things as 'all people die' and 'all girls have uteruses' in most cases.
• 2.4k
What I wanna know is if, in logic, "cats are yellow" should be interpreted as "all cats are yellow." — Jimmy

Oh, sorry. I just read the OP and was a little puzzled by what was confusing you.

No, it shouldn’t. In colloquial use it easily could be and if that was what was meant it would be clarified with further language.

I am sure ou can see that the use of “are” can be interpreted as “can be”.
• 13

Is there a convention in logic for how "cats are yellow" should be interpreted?
• 2.4k
Not that I can see? I’d expect an “ALL”/“SOME”/“NONE” before. Without a quantifier there is nothing to work with.
• 7.2k
When we don't explicitly state a quantifier (all/some) then it's assumed that the statement is a universal. So,

Cats are yellow = ALL cats are yellow

Dogs are mammals = ALL dogs are mammals

I can't think of interpreting statements like the above any other way. Why did you ask the question? Did you find an exception?
• 13

I was watching a youtube video made by a mathematician. He claimed that it's true that cats are four-legged animals, and that the definition of four-legged is: have four legs.

But suppose that the proposition that cats are four-legged animals is equivalent to the proposition that all cats are four-legged animals.
If the propositions are equivalent, and if it's true that cats are four-legged animals, then it's true that all cats are four-legged animals.
But suppose the aforementioned statements are equivalent and suppose that all cats are four-legged animals, then it's true that if something is a cat, it's necessarily a four-legged animal.
But it's actually false that if something is a cat, then it's necessarily a four-legged animal. For example, there are cats that are not four-legged animals.
And so, If it's true that the aforementioned statements are equivalent, it's therefore false that cats are four-legged animals, and the mathematician would be wrong. This conclusion doesn't really jive with me, so I want to know for sure if it's true that "Cats are yellow = ALL cats are yellow"

But I do think it's true that:
When we don't explicitly state a quantifier (all/some) then it's assumed that the statement is a universal.
• 2.4k
There is no logical symbol for “ARE”. End of story.

Anything else is merely a question of semantic interpretation to set out logical propositions.
• 7.2k
But it's actually false that if something is a cat, then it's necessarily a four-legged animal. For example, there are cats that are not four-legged animals.

I see. You use a specific example here. However take a mathematical statement:
A) All triangles have 3 sides.

Rephrased we could say:
B) Triangles have 3 sides

A is equivalent to B isn't it? There are no exceptions in this case.

Your example is an exception/counter-example to the claim: All cats are four-legged creatures. You're right but it only applies to the claim about cats and not to ALL universal claims as I've shown with my triangles statement.
• 2.4k
What do you call a triangle with four sides? NOT a triangle.
• 13

I don't know what you mean. Interpreting if "a, then b" as "a->b" is also a semantic interpretation since the interpretation is related to meaning in logic. It just happens to be the case that the convention in logic is to interpret "if a, then b" as "a->b". But nonetheless, there is a logical symbol for "if a, then b."
There is also a convention for which logical symbol to use for "all a are b."

Here's an argument using "are," directly from a textbook, and the logical symbol that is used for "are":

All ravens are black.
There exist ravens in Sweden.
Therefore, there exist black animals in Sweden.

K(x): x is a raven
S(x): x is a black animal
H(x): x is an animal is Sweden

For All x: [K(x)->S(x)]
There exists x such that: [(k(x) and H(x)]
Therefore, there exists x such that: [S(x) and H(x)]
• 13

Well, I wouldn't say I have found an exception to the rule that A are B is equivalent to All A are B (if it is a rule). It could just be the case that it's not true that cats are four-legged animals. The conclusion doesn't really jive with me, but it would be true if the rule is true.
• 2.4k
I don’t know what you mean either. I don’t understand what is bothering you.

I can just as easily conclude that:

All ravens are black
Some OR all ravens are in Sweden

I wouldn’t change “bird” to “animal”. What’s the point? Although ...

We end up with a very many possible expressions :

Some OR All animals are ravens (yet there is no mention of “animals” in the example).

All ravens are Black.

Some Or All ravens are un Sweden.

As for the equivalence issue; I can say that Cats are Yellow and this is NOT saying Cats cannot be Black. Cats are ONLY Yellow would be equivalent to ALL Cats are Yellow. That is not to say that Cats are Yellow cannot be interpreted to mean ALL Cats are Yellow (that is why semantic interpretation matters in how you apply the information you’re given). Like above where I don’t simply assume that “birds” are “animals” if that information is not explicitly provided.

In mathematical logic people are not generally bothered much about “meaning” as much as in philosophical uses of logic.
• 13

I edited and changed "there exist birds in Sweden" to "there exist black animals in Sweden", btw, to make the argument consistent with the symbolic argument. I made a whoopsie.

There is no logical symbol for “ARE”. End of story.

Anything else is merely a question of semantic interpretation to set out logical propositions.

Well, that is what's bothering me, because it's false that there is no logical symbol for are.
And that's why I said what I said in the previous post. I provided evidence, from a textbook on mathematical logic, that it's false that there is no logical symbol for are.

Cats are yellow = ALL cats are yellow

What I wanna know is if, in logic, "cats are yellow" should be interpreted as "all cats are yellow." — Jimmy

Oh, sorry. I just read the OP and was a little puzzled by what was confusing you.

No, it shouldn’t.

You two are in disagreement here. Why is Themadfool wrong? Can you deduce it? Can you source it?
• 13

I edited and changed "there exist birds in Sweden" to "there exist black animals in Sweden", btw, to make the argument consistent with the symbolic argument. I made a whoopsie.

There is no logical symbol for “ARE”. End of story.

Anything else is merely a question of semantic interpretation to set out logical propositions.

Well, that is what's bothering me, because it's false that there is no logical symbol for are.
And that's why I said what I said in the previous post. I provided evidence, from a textbook on mathematical logic, that it's false that there is no logical symbol for are.

Cats are yellow = ALL cats are yellow

What I wanna know is if, in logic, "cats are yellow" should be interpreted as "all cats are yellow." — Jimmy

Oh, sorry. I just read the OP and was a little puzzled by what was confusing you.

No, it shouldn’t.

You two are in disagreement here. Why is he wrong? Can you deduce it? Can you source it?
• 13

I edited and changed "there exist birds in Sweden" to "there exist black animals in Sweden", btw, to make the argument consistent with the symbolic argument. I made a whoopsie.

There is no logical symbol for “ARE”. End of story.

Anything else is merely a question of semantic interpretation to set out logical propositions.

Well, that is what's bothering me, because it's false that there is no logical symbol for are.
And that's why I said what I said in the previous post. I provided evidence, from a textbook on mathematical logic, that it's false that there is no logical symbol for are.

Cats are yellow = ALL cats are yellow

What I wanna know is if, in logic, "cats are yellow" should be interpreted as "all cats are yellow." — Jimmy

Oh, sorry. I just read the OP and was a little puzzled by what was confusing you.

You two are in disagreement here. Why is he wrong? Can you deduce it? Can you source it?
• 2.4k
Well, that is what's bothering me, because it's false that there is no logical symbol for are.
And that's why I said what I said in the previous post. I provided evidence, from a textbook on mathematical logic, that it's false that there is no logical symbol for are.
— Jimmy

Show me then? They use “existent,” so I guess that would be closest to “are”. If it’s “false” provide evidence. I’m looking for logical symbols like these:
https://en.m.wikipedia.org/wiki/List_of_logic_symbols
• 13

All ravens are black.
There exist ravens in Sweden.
Therefore, there exist black animals in Sweden.

K(x): x is a raven
S(x): x is a black animal
H(x): x is an animal is Sweden

For All x: [K(x)->S(x)]
There exists x such that: [(k(x) and H(x)]
Therefore, there exists x such that: [S(x) and H(x)]

All ravens are black is translated to symbolic logic as: For all x: [K(x)->S(x)].
Therefore, the implication symbol is used for "are."
Directly translated back to "normal language", For all x:[K(x)->S(x)] means: For all x, if x is a raven, then x is a black animal.
In first-order logic, you are evidently allowed to assume that all ravens are animals.
• 2.4k
IF p THEN q (where p and q are statements) - the rest you’ve tagged on semantically.

You could, by your argument, also insist that IF means (“If it is such that said item ‘p’ is true then it follows in such a way that q is implicated by the being of p.”). That is all you’ve done only in a shorter and more pithy manner - “If p then there is q.” Statment p could be “The yellow hats are always bananas” and statement q could be “Fat cats drink clouds”; neither of which would change the underlying p->q.
• 13

You could, by your argument, also insist that IF means (“If it is such that said item ‘p’ is true then it follows in such a way that q is implicated by the being of p.”).

That is false if we use the conventions of logic. Because if it is that p is true, then it is not necessarily the case that q is true. Therefore it's false that "If it is such that said item ‘p’ is true then it follows in such a way that q is implicated by the being of p"

Also, this is not what I've done. I've quoted an argument, from a textbook, that shows that "are" is written to mean implication in predicate logic. I've given you the evidence of the convention of that: in logic, "are" can be translated to mean implication.

I think you are confusing predicate logic with propositional logic. Are you equated in predicate logic aka first-order logic?
• 13

You could, by your argument, also insist that IF means (“If it is such that said item ‘p’ is true then it follows in such a way that q is implicated by the being of p.”)

That is false if we use the conventions of logic. Because if it is that p is true, then it is not necessarily the case that q is true. Therefore it's false that "If it is such that said item ‘p’ is true then it follows in such a way that q is implicated by the being of p"

Also, this is not what I've done. I've quoted an argument, from a textbook (that deals with this very topic), that shows that "are" is written to mean implication in predicate logic. I've given you the evidence of the convention of that in logic, "are" is allowed to be translated to mean implication (actually, it's allowed in predicate logic at least)

I think you are confusing predicate logic with propositional logic. Are you educated in predicate logic aka first-order logic?
• 2.4k
Then why ask? If your book shows an example and says “are” - in the example given - means implied, then you had your answer before you posted here? I was simply saying “are” is not a logical symbol (in predicative or propositional notation).

All sorted?
• 13

There is no logical symbol for “ARE” — I like sushi

Then why ask? If your book shows an example and says “are” - in the example given - means implied, then you had your answer before you posted here? I was simply saying “are” is not a logical symbol (in predicative or propositional notation).

All sorted? — I like sushi

You were simply stating that there is no logical symbol for "ARE."
This is not equivalent to the claim that “are” is not a logical symbol. I mean yeah, of course, a word is not a logical symbol.

I didn't ask if it's true that there is no logical symbol for "ARE." You said that there is no logical symbol for "ARE" and I, therefore, claimed that you're wrong.

I asked, for example, the following:

Is there a convention in logic for how "cats are yellow" should be interpreted? — Jimmy1

That is, is it true that?

When we don't explicitly state a quantifier (all/some) then it's assumed that the statement is a universal. — TheMadFool

If you are educated in predicate logic or not is actually relevant to that question (given the quote below), but you don't to answer the question: "Are you educated in predicate logic aka first-order logic?"

↪Jimmy1
Not that I can see? I’d expect an “ALL”/“SOME”/“NONE” before. Without a quantifier there is nothing to work with. — I like sushi

Not all is sorted. Since I can't find a source on it, I still don't know for sure if it's true that:

When we don't explicitly state a quantifier (all/some) then it's assumed that the statement is a universal. — TheMadFool
• 2.4k
Apologies. We’ve certainly been yalking cross purposes here! :D

I’d suggest posting in the Logic and Philosophy of Mathematics for answers to your query.

I’m not formally trained in logic of any sort. What I know I’ve learnt off my own back. If you could provide the title of the textbook you’re working from I’d be very greatful. I’ve not been able to find a decent and concise textbook on logic. Meaning a comprehensive overview of all applications of logic - both mathematical and philosophical. I paid around $80 for a third edition of the most prominent title used in universities and received a first edition copy worth$20 ... needless to say I sent it back!

One thing I did discover from logic is that I’d been assuming what mathematics was most of my life and only really got a glimpse of it a few years ago! Quite an extraordinary and “magical” subject!
• 7.2k
You two are in disagreement here. Why is he wrong? Can you deduce it? Can you source it?
6d
• 10
I was watching a youtube video made by a mathematician. He claimed that it's true that cats are four-legged animals, and that the definition of four-legged is: have four legs.

What is a cat ? It is a human concept and human can be wrong. One day, there can be a cat with five legs, we will say cats can have four or five legs or we will say this is not a cat. It depends of the definition.

When you say that "cats are four legs", it can be a definition. You can find a cat with three legs because it losted one, it will still be true. It will still be true if a cat have five legs too. Because when ou say "cat are four legs", it does not mean you have to check every cats, but that the definition is correct, you can do something with it, anticipate that cats will have four legs.

When you say all cats are four legged, if you find one is three legs, it will be false. Because that means you have checked every individual against the definition.
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