But the second premise in the argument II does not excludes the existence of robbers not being american. It says only that exists at least one robber that is american. It doesn't says nothing about robbers that aren't americans.

(∀x) means "to all x", (∃x) means "there exists an x", "⊃" means "implies" and "∧" means "and". So, for instance, the first conclusion (∃x)[Rx∧Ex] would be read as "there exists an x such that x is a robber and x speak english", while the second conclusion (∃x)[Rx⊃Ex] would be read as "there exists an x such that if x is a robber, then x speaks english" (note that this "if ... then" doesn't necessarily means that there is a causation relation between the antecedent and the consequent. It means only that it's not the case that the antecedent is true and the consequent is false)

Oh, of course.
Ax := "x is american"
Ex := "x speak english"
Rx := "x is robber"

Well, you are just saying that truth is everything that isn't non-truth, which is circular, and as you said, even if it's a good definiton, you would need to define "lie" without using the term "truth". How would you do it?

I don't know if I got it. Why isn't your example fallacious? I mean, why from "We like to be happy and not suffer" you can deduce "We ought to make people happy [...]"?

I declared it to be both? I just said that AvB can be an "is" statement or an "ought" statement, then I showed an argument to be considered in each case. I assume here that i don't know what is the nature of "AvB" (if it's an "is" or "ought" statement), but I give an argument to each possibility. Actually, this isn't mine, but it's Russell's argument.
I don't know why this doesn't hit Hume's point, as it clearly shows that it's possible to derive an "ought" from an "is" statement.
Furthermore, if we can't make this derivation because they are from separate domains, different "kingdoms" of statements, then we couldn't derive "is" statements from "ought" statements too. But this argument shows that we actually can:

• 1. John ought to go to school
• 2. Kids and only kids ought to go to school
• 3. Therefore, John is a kid.

Is this wrong?
Thanks for answering!

Nice, thanks for the explanation and the recomendation. I'm downloading the book and I'll read it. I read the summary and it seems to be very good. Philosophy of logic and metalogic are very interesting for me. Hope it helps me. :D

Thank you for answering! I'll read the links you sent. Do you have any recomendation for me to go deeper on propositional logic (and also, what does "first order" logic means? I read a little about it but I think I didn't get the idea)? I read some books of introduction to logic (H. Gensler, I. M. Copi and C. Mortari), but, as the title say, they are just introductions, nothing much deep. I'm interested in those "meta-logic" things, but I don't know exactly what to search for.

Hello. Well, in the books I read it didn't talk about this difference, and it seems, to me, that they are logically the same thing. But, intuitively, I would say that double implication means what the name states, i.e., if P↔Q, then we know that P⊃Q and Q⊃P.
On the other hand, if two sentences P and Q are logically equivalent, then they express the same logical truth (and therefore have the same truth value), and one of them can be substituted by the other without problem. I don't know if it is right, but the difference seems, for me, to be a semantical one. Do you know the difference, or some book or paper that explains more about this? Thanks for the comment!

Thank you! I totally forgot about making assumptions in the argument to build the proof. I took the way you made the first proof and modified a little to fit in the reductio ad absurdum proof. The same with the second. My goal was to proof that the consequent follows the antecedent by reductio ad absurdum, and it's done. Here the way i did:

For 1:

For 2:

Ok, I understood, thanks for the explanation and the link :D
Also, do you have any recomendation of book to read specifically about this? I'm reading Introduction to Logic by irving M. Copi, and he talks about predicative logic, but doesn't says anything about this kind of fallacy.

Correct--we can derive "It is not the case that some A is not B" from "All A is B." However, we still cannot derive "Some A is B" from either of these without the additional premise, "Some A is A." — aletheist

But isn't implicit on any argument that some A is A? I mean, why do we need to put the identity law in a premise? And I really can't understand why "Some A is B" cannot be infered from "All A is B". :(

"No Greek is man" is just to say that it is not the case that some Greek exists and is a man. That's an existential quantifier, not a universal one. — MindForged

It's a negative existential one, but any universal proposition can be transformed into an existential one.
(∀x)[Px] ↔ ¬(∃x)[¬Px]

Based on this one either can't claim to know almost anything or else you have to change our understanding of biology and what creatures exist on Earth in order to credibly say we don't know them to not exist. — MindForged

But this is really a problem. Answer me: how do you know that a winged horse doesn't exist? Unless you define horse as being something wingless, you can't know if there is a horse with wings. You can induce from previous data that is improbable that something like this exist, but it's not a certainty. It's like you saying that black swans doesn't exists because no one has seen any before, but then one day someone sees one.
The problem is that if you define horse as being something wingless, then the proposition "all winged horses are horses" doesn't make any sense, even it being an analytical one, just like the second proposition and the third. They all would be self contradictory.

"All Greeks are man" is a universal statement, while "No Greek are men" is a particular. One is about an abstract domain not involving existence while the other is explicitly about existence. — MindForged

Why woudn't both be universal? Russell says that "No Greek are men" is the same of "All Greek are not-man". For me, it's clear that both propositions "all greeks are man" and "no greek are man (all greeks are not-man)" are universal ones. For it to be a particular one, it would need to use existential quantification and, therefore, assume the subject existence, woudn't? Thanks for answering :D

Oh, i think i got it. The problem is that the existential quantification does affirm that something exists, while universal quantification only states that "for any x, if Px, then Qx", for instance, without assuming the existence of some x. Right?

It's false because we know it's true that winged horses are non-existent — MindForged

Honestly, how do you know that winged horses are non-existent? I'm noy saying that they exists or that I believe that they exist, but you can't affirm that categorically only based on "no winged horse has ever been seen".

It might help to also consider that Russell would also interpet "All winged horses are wingless" to be true. Since nothing is in the set "winged horses". — Ben92

Yeah, I read it and i was kinda doubtful too. For me it doesn't make any sense.

The problem is that it is invalid to cross predicates in this way. A is B is to predicate B of the subject A. A is C is to predicate C of the subject A. B and C are predicates of the subject A. Until we convert either B or C into a subject, and predicate something of that subject, we have nothing to allow us to draw any conclusions about either B or C, because B and C have not been presented as subjects. — Metaphysician Undercover

Hmmm, this makes sense to me. I think that an argument like "All apples are red and all apples are sweet, therefore some red are sweet" would represent this, because both "red" and "sweet" are adjectives. So it's an invalid argument because there are some predicates that can't be set to the subject and simultaneously to each other, right?

No, that is invalid. It becomes more obvious if we reformulate the two propositions as follows.

For all x, if x is B, then x is C.
There exists an x, such that x is B and x is C.

The first proposition clearly does not entail the second. — aletheist

But it's wrong, the argument says "some B are C", not "all B are C". And even if it was correct, why the second doesn't follow from the first? I mean, if for any x, x is B and x is C, then there exists an x that is B and C. Why would it be wrong?

And I don't see how Russell would consider "All winged horses are wingless" to be trivially true. His description theory of names doesn't say sentence with empty terms are by default true, especially contradictory ones. They are deemed false in his theory because they must posit the existence of some thing (winged horses) but we know the thing to not exist. Nothing satisfies the condition "winged horse" so the translation of the previous argument into classical logic would have a suppressed premise, namely:

There exists at least one winged horse.

Which gets the value false, leading to a false conclusion. Have I misunderstood you or perhaps Russell's theory? — MindForged

Well, actually he says it on Logic and Knowledge (p. 229), and I think it's kinda weird.

"If it happened that there were no Greeks, both the proposition that "All Greeks are man" and the proposition that "No Greeks are men" would be true. The proposition "No Greeks are man" is, of course, the proposition "All Greeks are not-man". Both propositions will be true simultaneously if it happens that there are no Greeks. All statements about all the members of a class that has no members are true, because the contradictory of any general statement does assert existence and is therefore false in this case. This notion, of course, of general propositions not involving existence is one which is not in the tradictional doctrine of the syllogism." — Russell

Because empty terms show this argument form to fail — MindForged

What do you mean by "empty terms"? Are you refering to arguments with undefined variables?

All winged horses are horses,
All winged horses have wings,
Therefore some horses have wings.

Clearly the first two premises are true but the conclusion is clearly false, we know there are no horses with wings. — MindForged

But why is the conclusion false? I mean, I know that horses doesn't have wings, but it's inductive, empirical constatation, isn't? It's not logically impossible that a winged horse exists, unless you define horse as something that doesn't have wings. But, if this the case, then both premises are nonsense, because you would be saying something like "all winged things that doesn't have wings have wings". I don't know if i understood...

Well, petitio principii is a fallacy in which an argument's premises assume the truth of the conclusion. It's a kind of circular argument. E.g., "Charles is a good person because he helps the needy, and anyone who helps the needy is a good person". In this case, the conclusion is assumed to be valid in the premise.
On the other hand, a transcendental argument is an argument in which it's negation implies in a contradiction because it's validity is a necessary condition for you to negate it. E.g. "I think, therefore I am". In this case, existing is a necessary condition for me to think, and I'm thinking, so I exist.
For me they seem to be similar, because in both there is a thing which is assumed to be valid and cannot be negated.
No.

Well, points are also adimensional, but displayed in continuum succession they make a line, which is dimensional. A line, on the other hand, contains only one dimension, but displayed parallelly in a continuum succession they make an area, which is bidimensional. The same is valid for volume.

Basically we need to be omniscient

Why? Of course, you would need to know if an atom in a far away galaxy would impact on your actions, but you don't need everything that is happening with this and other atoms, you only need to know whether they'll change your actions.

You are right. When I said about truth tables I was thinking on truth tables of arguments, by which an argument can be proved or invalided by checking if the conjunction of the premises implies in the conclusion. I totally forgot about the simpler cases, like what you pointed.

I'm not being verbose. You said inferences are by definition not deduction. I'm just asking you what definition are you using, because I pointed an exemple of a deductive argument made by an inference.
But then I realized that an inductive argument contains inferences as well, so an inference cannot be a deduction. You could just answer my question instead of being defensive.

What is the definition of inference? Why can't it be treated as a deduction? I mean, from "A→B" and "B→C" you deduce that "A→C", right?

Ok, I think i got it. Thank you for your answer!

Yeah, i'm talking about traditional logic. But by proving it by truth table wouldn't you already be relying on the inference as a valid method, since the truth table is the conjunction of the premises implying in the conclusion? I mean, why wouldn't it be a petitio principii?

Well, this argument isn't valid... An argument is invalid when there is a case where all it's premises are true and the conclusion is false. This is because the entire argument can be written as the conjunction of all it's premises implying in the conclusion. So, if the conjunction of the premises is true and the conclusion is false, then the implication is false, and the argument is invalid, as you can see in this reducted truth table:

About your proof, you can't just assume that a variable if true or false if you don't have this data. Actually, you can do this when both true and false values for the variable in question leads you to the same answer, meaning that the answer does not depend on this variable, like the following exemple:

P ⊃ Q
¬P ⊃ Q
∴ Q

In this exemple, both P and ¬P implies in Q, which mean that anything implies in Q. This is equivalent to (P^¬P) ⊃ Q, which is always true. But this is not the case, because if A if true, it's conclusion isn't the same as it would be if A was false. If you choose values for a variable, you will only know what would be the result of that if this were to be really the truth value, but you don't know what it is.
I don't know if it answers your question. In any case, one thing I like to do is always begin by checking if the argument is invalid. If I can't make the premises true and the conclusion false, then I start using inference rules to prove the argument validity.

*Sorry for any grammar error, i haven't been practicing writing in english.

• 1. V → T
• 2. ¬V → H
• 3. ¬T → ¬H // T
  Proof:
• 4. H → T ..... 3, transposition
• 5. ¬V → T ..... 2,4, hypothetical syllogism
• 6. (V→T)^(¬V→T) ..... 1,5, conjunction

If V implies T and ¬V also implies T (as you concluded above), then T is implied by anything.
For T to be implied by anything, it must be true, because anything implying true is true.
Does this prove The argument's validity?
This is crazy, because in the truth table, the argument seems to be valid, but i found a lot of inconsistencies, like this:

• 1. V → T
• 2. ¬V → H
• 3. ¬T → ¬H // T
• Proof:
• 4. ¬T→¬V ..... 1, transp.
• 5. ¬T→ H ..... 4,2, S.H.

5 and 3 are inconsistent

======================

• 1. V → T
• 2. ¬V → H
• 3. ¬T → ¬H // T
• Proof:
• 4. T v H ..... 1,2, resolution
• 5. ¬T → H ..... 4, disjunctive syllogism

3 and 5 are inconsistent

=======================

How can this be solved?