X is stupid means your feeling about X, — Corvus

First, I didn't say anyone is stupid (other than Donald Trump in the previous post). Anyway, claiming that someone is stupid may not be just expressing a feeling. Donald Trump is a terribly stupid person, no matter what my feeling about that is.

one could be stupid on something, but genius in other subjects — Corvus

Of course. And not just in subjects but also in life. For example, Trump is a genius as a demagogue and as a big time grifter.

Why should you suppose that other people will agree with a psychological reflection of someone without critical objective ontological infallible evidence? — Corvus

First, I didn't say the poster about the book is stupid. I said what she wrote is stupid and that she's an ignoramus (obviously regarding logic; she could be wonderfully informed about certain other subjects).

Second, I did explain very clearly how what she wrote is stupid and ignorant.

Third, it is a ridiculous standard that one should not be believed unless one were INFALLIBLE with evidence.

Fourth, talking about throwing around jargon. 'ontological'. Oh come on, descriptions about people aren't ontology.

I'll stand corrected, but I think I said she is stupid. I said that what she wrote it stupid. — TonesInDeepFreeze

I meant to include the word 'don't'. I meant to say that I don't think I said she is stupid. And that is correct. I did not say she is stupid. I said that what she wrote is stupid.

should you not allow that other people could have different opinions about anything of their own? — Corvus

Of course. And we should allow that one may have good grounds for thinking that certain of those opinions are stupid and to say so. Moreover, I gave detailed explanation why I consider certain of her remarks to be stupid. More importantly, they're ignorant.

Surely you don't think all opinions are equally intelligent and don't deserve to be called out for being stupid opinions. Or do you?

Just because she had different opinion about logic, that doesn't mean what she wrote is stupid — Corvus

I already answered that. You skipped my answer so that you could merely reassert. What she wrote is stupid not merely because her opinions differ from mine, but on account of the reasons I specifically explained.

When someone describes other people as stupid without justified ground, it reveals more about the describer and his psychological state and motives, than the other people who were described as stupid. — Corvus

First, I didn't say she is stupid. I said that what she wrote is stupid. It's possible that she is intelligent but has blocks in understanding a certain subject. I sometimes write and say and do stupid things. And I have blocks in understanding certain things. But I am not stupid. And she might not be stupid.

Second, that someone says that another person wrote stupid things or even says the person is stupid, doesn't in and of itself entail very much about psychological states or motives. When I say that Donald Trump is a stupid, ignorant, grotesquely dishonest man, that's a true statement about Trump, and it hardly says much about me other than that I am willing to state an obvious fact.

I feel that the Informal Arguments books have far more practical ideas than the simple traditional or symbolic logic. — Corvus

First, I think informal logic is vitally important. I think it crucial to basic critical thinking in everyday life and in fields of study. I wish it were required starting in elementary school.

Second, my point went right over your head like a 747. You had conveyed some terribly mixed up ideas about logic and you said you got those ideas from a logic book. My point is that I bet the book didn't say those things but instead you misconstrued or misremembered the book. But if the book really did say those things, then, yes, that book is quite bad.

those [symbolic logic] books are full of boring dry useless contents, — Corvus

(1) The point of the books is to explain the subject. The books don't have to be entertaining.

(2) The books may be boring to you but not to many other people.

(3) I already commented on the utter usefulness of symbolic, including that you are now using a computer built from symbolic logic.

(4) Symbolic logic improves reasoning skills not just abstractly but in everyday life.

(5) Logic even makes me laugh sometimes. There is sometimes even true wit in the way the formulas are constructed and the way the proofs and arguments unfold. Even the prose of the authors. Halmos, Quine, and Boolos and others. Smullyan! Even the arch dryness of Alonzo Church. The ingeniousness and surprises delight my mind and my soul. The achievements of logicians and mathematicians inspire me. Their intelligence, creativity, standards of rigor and honesty are to be cherished. The enlightenment that logic embodies inspires me. And there is also the lore - the historical twists and turns, the biographies, the rivalries, the jokes, the epigrams, and all that stuff. The history of logic from ancient to modern, from all over the world and from many cultures.

(6) And there is philosophy of mathematics, and philosophy about logic, and philosophy of language. I can't even fathom the richness. And understanding those does require understanding something about the math and logic the philosophers are talking about. Also the application of formal logic to philosophy itself as a tool for clarifying arguments, for making the questions, problems and arguments rigorous, for providing certain objective contexts for philosophers to propose ideas, to inquire about them and to debate them.

(7) And the pure mental pleasure of understanding something you didn't understand before. The pure pleasure of an idea or proof finally making sense to you after you have struggled to grasp it for so long. And the pure mental pleasure of being introduced to new ideas; constructivism, intuitionism, free logic, multi valued logic, dialetheism ... ideas you never could have imagined. And they're all there just waiting for you to open the books, to read the journal articles, and if you're lucky, to take a class with a good teacher, and if you're really lucky, to take a class with one of the great minds of our modern world.

(8) Gaining the vocabulary and understanding of concepts so that you can talk about them with other people. So that you can present and explain ideas to other people, and so that you can learn from people that know more than you do. So that you can have a buddy to work on problems with.

you will realise that philosophy is far more than dog fighting with symbolic logic jargons. — Corvus

(1) Who do you think you're talking to? You are quite presumptuous to think you need to tell me that philosophy is more than symbolic logic. Moreover, I haven't posted anything that could be remotely suggestive that I think philosophy is just symbolic logic.

(2) 'dog fighting'. To what are you referring? Our discussion? You have been using terminology in a way that doesn't even make sense. I have given you corrections you could use.

(3) 'jargons'. The terminologies of symbolic logic are not mere jargon. The terminologies are from rigorous definitions. And logicians don't just throw around a bunch of words. Rather, they use terminology meaningfully and communicatively.

You must try to look at the problems with your own reasoning first, and if needed, create your own definitions, — Corvus

First I apprise myself of existing definitions. Then if I have concept that I haven't seen defined, then I do construct a definition. I do that in logic and math pretty often. And I understand the rules for definitions, especially as I understand their purpose - the criteria of eliminability and non-creativity.

and apply them to the real philosophical issue in the world — Corvus

I'm not a philosopher (I'm not a logician or mathematician either). Though I do think about certain questions and sometimes try to formulate them explicitly.

Your arguments sound to me the whole world should exist for symbolic logic and its traditional concepts. — Corvus

That essentially is a HUGE strawman. I have never written anything that remotely suggests that "the world should exist for symbolic logic" You are ridiculous to say that my arguments even "sound like" that.

I am not suggesting that my comments supplant yours.
— TonesInDeepFreeze

LOL — fishfry

I don't know what your point is there.

fishfry's posts in this thread about ordinals have been generous and instructive. His posts deserve not to mangled, misconstrued, or strawmaned.

I do not presume to speak for fishfry, but I would like to state some points, and add some points, in my own words too.

fishfry:

On matters of logic and mathematics, any divergence I take from you or qualification I mention is not intended as a criticism of you personally. (I am not suggesting that you have claimed or not claimed that I have intended my remarks on logic and mathematics to be personal criticisms of you.) My personal criticisms have only been about the dialogue itself.

I am not suggesting that you would be remiss if you didn't adopt my formulations and definitions.

I am not suggesting that my sometimes more detailed formulations supplant your sometimes less formal explanations (though in some instances I think your formulations are not correct).

When I simply add detail or additional commentary, I am not suggesting that your original remarks are thereby incorrect.

I am not suggesting that you would be remiss by not reading or not replying to anything I write (except rebuttals that are defenses against your incorrect criticisms of my claims).

When I state something that you have already stated, I am not suggesting that you had stated to the contrary nor that you hadn't already stated it.

I am not suggesting that you would be remiss not to the include the details I include.

I am not suggesting that threads such this one require the detail that I include.

I am not claiming that my formulations are pedagogically superior, as my intentions are not purely, or even primarily, pedagogical.

I am not suggesting that my comments supplant yours.

I am not suggesting that posting should be expected to keep a level of precision as we may expect in professional publication.

If I misconstrue a poster, then it is unintentional. I do not intend to cause a strawman. If I have misconstrued a person, then they can let me know. But they would be incorrect if they claimed my error was intentional or that I had intentionally set up a strawman. (And I am not suggesting that you have or have not claimed that I have set up a strawman in previous threads.)

I try to write mostly at face value. But in any communication it is often not clear whether a person meant to imply more than they literally said or not. If I have seemed to imply something that I did not literally write, then anyone can ask me whether I meant to imply it or not.

/

I am posting because I like talking about mathematical logic.

I like expressing the concepts and explaining them.

Sometimes my explanations are not understandable for people who are not familiar with mathematical logic, and in that case, I still enjoy having explanations and formulations available possibly for posters to revisit or even I enjoy just the fact that my remarks are on the record.

I enjoy making formulations that are as rigorous as feasible in the confines of a thread.

Posting sharpens my knowledge of the subject and improves my skill in composing formulations.

When I post corrections to other posters, I find some small satisfaction in seeing that the correction is available to be read.

I hope that some readers might benefit from my posts.

I believe many of my formulations do provide insight and rigor and at least examples showing that rigorous formulations of certain notions exist.

Sometimes I enjoy the interaction with posters.

I enjoy reading some posters.

I benefit from any true corrections or suggestions presented to my own posts.

And with cranks, I find entertainment and satisfaction in providing counter to them.

/

ordinals are logically prior to cardinals, in the modern formulation. — fishfry

That is correct and it is important. It points to the fact that the set theoretic treatment of ordinals and cardinals is rigorous as it proceeds only step-by-step through the theorems and definitions.

the equivalence class of all sets having that cardinality. The problem was that this was a proper class and not a set — fishfry

That is correct and it is important. For every S, there is the equivalence class of all sets that have a bijection with S. But that equivalence class is a proper set. So for a rigorous set theory, without proper classes, another definition of the cardinality operator needed to be devised. The numeration theorem ("for every S there is an ordinal T such that there is a bijection between S and T") is a theorem of ZFC, and it allows us to define card(S) = the unique ordinal k such that k has a bijection with S and such that no ordinal that is a member of k has a bijection with S.

two sets having the same cardinality -- meaning that there is a bijection between them -- and assigning them a cardinal -- a specific mathematical object that can represent their cardinality. — fishfry

That is correct and it is important. A set theoretic operator (such as 'the cardinality of') can be defined only by first showing that for every S, there exists a unique T such that T has a [fill in a certain property here]. For example, with the operator 'card' (meaning 'the cardinality of') we first prove:

For every S, there is a unique T such that T is an ordinal and T has a bijection with S and no member of T has a bijection with S.

card(S) = the unique T such that T is an ordinal and T has a bijection with S and no member of T has a bijection with S.

After von Neumann, we identified a cardinal with the least ordinal of all the ordinals having that cardinality — fishfry

I see the main point there, but the formulation is not clear to me. I suggest this sequence:

df: k is ord-less-than j <-> k e j

When context is clear, we just say "k is less than j" or "k < j" or "k e j".

df: k is ord-least in S <-> (k e S & ~Ej(j e S & j ek))

When context is clear, we just say "k is least in S".

df: k is the least ordinal such that P <-> (k is an ordinal & Pk & ~Ej(Pj & j ek))

df: the cardinality of S = the unique T such that T is the least ordinal having a bijection with S

So the cardinality of S is card(S).

df: S is a cardinal <-> (S is an ordinal and there is no ordinal T less than S such that there is a bijection between S and T)

Any nonempty collection of ordinals always has a least member — fishfry

That is correct and important. No clear understanding of ordinals and cardinals can be had without it.

conflating order with quantity [is] an elementary conceptual error. In an order relation x < y, it means that x precedes y in the order. x is not "smaller than" y in a quantitative sense. — fishfry

That is correct and important especially in context of being presented in this thread with misconceptions about this. The less than relation on ordinals is simply the membership relation. That is, membership ALONE is the basis for the less than relation on ordinals. But a general quantitative relation for sets is formulated with cardinals that are based both on bijection and instantiated to specific sets with regard to the ordinal less than relation.

cardinality, which is an equivalence relation based on bijection — fishfry

I see what you intend, but to be precise, a relation is a set of tuples. But cardinality is not a set of tuples. The equivalence relation is among the cardinalities but is not the cardinalities themselves.

Two sets may have the same cardinality, without there being any notion of ordinal at all — fishfry

I agree with the intent of that and I think perhaps some authors say things like that in a sense that does not require ordinals, but I find it not quite right.

There are two different notions:

(1) 'the cardinality of S' to mean the least ordinal k such that there is a bijection between S and k.

and

(2) 'S and T have the same cardinality' to mean there is a bijection between S and T

In (1) 'cardinality of' is a 1-place operation.

In (2) 'same cardinality' is a 2-place predicate.

That's okay, except:

(2) S and T have the same cardinality iff there is a bijection between S and T.

In that sense, we don't need to rely on ordinals.

(3) S and T have the same cardinality iff card(S) = card(T).

In that sense, we do rely on ordinals.

Which of (2) or (3) is the definition of 'same cardinality'? We wouldn't know unless the author told us, and whether the definition relies on ordinals would depend on what s/he told us the definition is.

And (3) is better than (2) in the sense that (3) uses 'cardinality' compositionally from (1) while (2) takes 'cardinality' noncompositionally. And (3) better fits the usage such as: "What is the cardinality of S? It's the least ordinal k such that there is a bijection between S and k. Okay, so S and T have the same cardinality iff the cardinality of S is the cardinality of T."

So I would just define:

S and T have the same cardinality iff card(S) = card(T).

Then take (2) as a theorem not a definition.

And if we want to leave out ordinals, then just say "There is a bijection between S and T".

/

To clean up two misconceptions that have been expressed in responses to fishfry:

Incorrect: The notion of ordinals presupposes the notion of cardinality.

A definition of 'is an ordinal' does not refer to 'cardinal' nor 'cardinality', and it doesn't even refer to 'bijection'. That is a plain fact that can be verified by looking at any textbook on set theory.

A definition of 'the order type of' does not refer to 'cardinality nor 'cardinality'. That is a plain fact that can be verified by looking at any textbook on set theory.. (The term 'ordinality' has been used. I am not familiar with it. Perhaps 'the ordinality of' means 'the order type of'?)

Incorrect: We should not use 'least' if we don't mean quantity.

It is typical of cranks unfamiliar with mathematical practice to think that the special mathematical senses of words most conform to their own sense of the words or even to everyday non-mathematical senses. The formal theories don't even have natural language words in them. Rather, they are purely symbolic. Natural language words are used conversationally and in writing so that we can more easily communicate and see concepts in our mind's eye. The words themselves are often suggestive of our intuitions and our conceptual motivations, but proofs in the formal theory cannot appeal to what the words suggest or connote. And for any word such as 'least' if a crank simply could not stomach using that word in the mathematical sense, then, if we were fabulously indulgent of the crank, we could say, "Fine, we'll say 'schmleast' instead. 'schmardinality' instead'. 'ploompty ket' instead of 'empty set' ... It would not affect the mathematics, as the structural relations among the words would remain, and the formal symbolism too.

It sounds too harsh to describe someone stupid just by reading her few lines of the book reviews. — Corvus

I'll stand corrected, but I think I said she is stupid. I said that what she wrote it stupid. And I said she is an ignoramus and a nutjob* (also see her list of conspiracy theory sources).

* That she is a nutjob doesn't in and of itself entail that her comments about logic are incorrect. Her comments about logic are incorrect anyway. Pointing out that she is a nutjob is just to anecdotally celebrate the great comedy of life.

There could be just differences in opinions. — Corvus

Not true. Her posted opinions broadcast that she has fundamental misunderstandings.

my ideas about logic — Corvus

I can't reasonably say that you shouldn't have ideas about the subject. But, as far as I can tell, you know virtually nothing about formal logic, so I don't how you would form ideas, especially ideas that resonate with wildly overopinionated views of a poster.

I would have thought everything is clear on the sufficient and necessary conditions for the premises. — Corvus

You could present your notion again so that it is salient for me, keeping in mind that I'm asking specifically for a definition.

true definition, you asked. I was meaning the right and proper definition that fits for the better premise. — Corvus

What is "the right and proper definition that fits for the better premise"?

Are you familiar with the basics of the subject of definitions in logic, even if only in non-formal sense? (For formal senses, I know that you know nothing about that, but maybe you'll get to it eventually. I recommend Suppes's 'Introduction To Logic' as the best explanation of formal definitions I've found.)

not possible to have a 100% true definition in many cases — Corvus

Inserting in your definition of 'true definition:

"not possible to have a 100% right and proper definition that fits for the better premise in many cases"

I don't know what that is supposed to mean.

adding another definition i.e. dogs bark, and cars meow — Corvus

Those aren't definitions. They're predications. And I don't see why you think they address the invalidity of the original argument.

wide definition (dogs are animals.) — Corvus

That's not a definition. It's a predication.

I would have thought anyone would know what sufficient and necessary definitions as better premises are like. — Corvus

I have never seen that notion mentioned in my readings in logic, though I can't rule out that such mentions exist.

True definitions are what philosophers are seeking to find and come up with in their thinking and debates process. — Corvus

Yes, some important philosophy is concerned with that. But (1) we can ask such philosophers what they mean by 'true definition', (2) I would like to know your own definition. You merely deferred it to 'right and proper that fits the premises', which is hardly any more defining than 'true'. And it turns out that you conflate definition with predication, (3) In formal logic, we do need a formal, not merely open-ended philosophical, definition of 'definition'.

comment on Valid arguments doesn't have to have false conclusions — Corvus

No, I said they don't have to have true conclusions.

But it would be judged as an inconsistent argument even if valid? — Corvus

I'm sorry, but you don't understand even the basic concepts of validity and inconsistency.

Arguments aren't inconsistent. Sets of statements or statements themselves may be inconsistent.

Statements can be:

True (or at least true in a given model) [semantic]

False (or at least false in a given model) [semantic]

Logically true. True in every model [semantic]

Logically false. False in every model [semantic]

Contingent. True in some models and false in other models. [semantic]

Consistent. Doesn't imply a contradiction [syntactic]

Inconsistent. Imply a contradiction [syntactic]

Arguments can be:

Valid. Conclusion is entailed by premises.

Invalid. Conclusion is not entailed by premises.

Sound. Valid and all the premises are true.

And I don't recall seeing another word for this, so I use 'irrefragable':

Irrefragable. Valid and all the premises are logically true.

mostly the points from Critical Thinking and Informal Arguments books — Corvus

There are books that are as mixed up about the concepts as you are?

In another post, she writes:

"Professor Paul Kreeft offers the following in his book "Socratic Logic", "I have never found anyone except a professional philosopher who actually used symbolic logic in an actual conversation or debate.""

I love that. She's writing on a computer that is enabled by extensive research and application of symbolic logic. Symbolic logic is at the very heart of the invention of the modern computer. It is hard to even imagine the concepts of computer science without symbolic logic or equivalent formalisms.

Reminds me of Whoopi Goldberg one year at the academy awards when she said (paraphrasing), "What good every came from the space program? It has given us nothing", as she was being broadcast live via SATELLITE!

I read it all the time, "Symbolic logic is stupid. It's a waste of time. It's just a bunch of philosophers and mathematicians too absorbed in their formulas to recognize the real world. Nobody needs it". As that is typed into a computer of which its invention, advancements, design, and implementation are filled to the brim with symbolic logic and the programming languages that have come from symbolic logic.

it is possible for the arguments to come to the true conclusion, had the premises came up with the complete set of sufficient and necessary propositions — Corvus

What is your definition of 'the complete set of sufficient and necessary propositions'?

I was wondering whether the diagram method is only OK for simple arguments with just 1 or 2 premises. — Corvus

Venn diagrams are profoundly useful. But they are limited. There are logical arguments they don't test.

Maybe there's a theorem somewhere that says exactly what is the class of arguments decided by Venn diagrams.

But we do know that truth tables decide all propositional arguments.

you just need to add more definitions into the premise making it sufficient and necessary condition. — Corvus

What definitions would you add? What sufficient and necessary condition?

the OP argument problem stems from the premise that there is limited scope for the definition of dogs and cats. — Corvus

It has nothing to do with domains or definitions.

The problem stems from the fact that the argument posted is invalid, as has been explained.

it demonstrates how insufficient premises render wrong conclusions in the argument, even if they look valid. — Corvus

The argument doesn't look valid. It is clearly invalid.

true definitions — Corvus

What is your definition of 'true definition'?

She wrote about the Hodges book:

"Here's what he does explain:
(1) Logical arguments begin from true premises.
(2) Logical arguments begin from premises and come to conclusions which are free of obvious contradiction."

I will stand corrected, but I don't see how that could be a correct paraphrase of anything Hodges wrote.

(1) Logical arguments don't require true premises. A SOUND argument requires true premises.

An argument is valid iff any model in which the premises are true is a model in which the conclusion is true. Validity does not require that the premises be true in any particular model or even any model at all.

(2) Logical arguments don't ensure contradiction-free conclusions. Rather, a logical argument ensures a contradiction-free conclusion if the premises are contradiction free. It is not required that the premises be contradiction-free. And the word 'obvious' mindlessly thrown in there clouds principle she's stating (though the principle is incorrect anyway).

She's an ignoramus.

I thought she was not saying Logic is stupid. — Corvus

I didn't say she said logic is stupid. I said that what she said about logic is stupid.

Rather, she was saying that the books don't mention some important points in Logic. — Corvus

SHE claims those are important points. Her remarks reveal that doesn't understand what logic is about.

She quotes a few philosophical texts — Corvus

Yes, typically crank: Argument by compilation of selective quotation.

Meanwhile, perhaps you would pick a quote that you think supports her criticisms of the books.

Before she even gets to the quotes she fires her loose cannon with a claim that is tantamount to saying that not just are the logic books wrong and amiss but that they are pernicious to mankind itself. She's a nutjob.

What is your the other symbolic logic book before the K/M — Corvus

I'd rather not say, because I think it's not a good book. It wasn't the book in particular that gave me chops, but rather any book with a good number of exercises also would have given me the chops.

I didn't say that anyone denied it. I said it shouldn't be denied.
— TonesInDeepFreeze

I just laughed, man. — fishfry

It is literally true that I did not say anyone denied it. And I haven't said that I blame you if you think I meant to imply that you denied it. And I don't blame you if you think that I meant to imply that you denied it. And above (though cross-posted) I say again that I did not mean to imply that you denied it.

I think we're two of a kind. — fishfry

Yes, you are the rational, logical, reasonable, accommodating, conciliatory, factually correct, patient and pedagogically sagacious one. You are the Gallant of The Philosophy Forum. And I am the irrational, illogical, unreasonable, unaccommodating, non-conciliatory, factually incorrect, impatient, and pedagogically unwise one. I am the Goofus. And you're the better looking one too. The roles of hero and villain have been clearly scripted. Now for casting.

Peace — fishfry

and Love, man.

Your pickiness with everything I write annoys me. — fishfry

It is not picky for me to say that you start your post with exaggeration: (1) There is a vast amount of what you write that I don't respond to, let alone with disagreement, correction, or qualification. (2) My points are not mere pickiness. That is only your own characterization.

And whether something annoys you, you blow it way out of proportion, and often seemingly taking it to be improperly motivated against you.

Especially because half the time you're actually wrong on the facts. — fishfry

If half the time I'm wrong, then the other half I'm right or at least neutral. Moreover, I have not been wrong half the time or anywhere remotely close.

I started my post by saying that I basically agree with your post. Then I said I'm adding only a technical qualification. And I even said that most writers don't use the method I am mentioning but that others do and that it can be done. That's pretty damn mild.

Then after one of your posts, I said that my point should not be denied. I didn't say that you denied it. My point was that no one should deny it. Then when you asked whether it had been denied and that you didn't mention it so as not to complicate things. Then I stated explicitly that I did not mean to imply that you denied it. And I will even add here that I don't blame you for thinking that I did mean to imply that you denied it. But after I've said it now about three times, you may correctly infer that I mean what I say when I say that I did not mean to imply that you denied it.

It's in your obfuscatory and unnecessarily argumentative mind that anyone denied it. — fishfry

I didn't say that anyone denied it. I said it shouldn't be denied. And if that was not clear, I followed up in reply to say I did not intend to imply that it was denied.

And there was no obfuscation.

Rudin, the number one, main, classic real analysis text, that defines the extended reals exactly as I did — fishfry

Yes, I said in my original post:

Yes, in many (probably most or even just about all) writings, the points of infinity are just arbitrary points, and they are not specified to be any particular mathematical objects. — TonesInDeepFreeze

I introduced my point to grant that. Then I said that we can also handle it another way. It's not unreasonable for me to say that.

You are wrong on the pedagogy AND wrong on the facts. — fishfry

(1) I didn't make a claim about pedagogy. (2) t's not pedagogically inappropriate to add, essentially a footnote in this case, a certain technical qualification. (3) Posts don't need to restrict themselves to what is pedagogically best anyway.

EDIT: And I'm not wrong on the substance.

Throughout the thread, it seems to me that regularly take technical and heuristic disagreements, corrections, and even mere technical qualifications as attempts to undermine you personally.

I did not intend to imply that you personally denied it. But rather that it should not be denied.

You said what the extended reals are. I noted a qualification.

I did not say that you were personally amiss for not including that qualification, nor that you were not reasonable to deem it as too much detail for your purposes.

Merely, I added stated the qualification, and said that it should not be denied, while not meaning to imply that you personally denied it.

You must have driven your teachers crazy. — fishfry

I did. In fifth grade, the teacher showed a wall map of the acquisitions of U.S. territory. The map omitted the Gadsen Purchase and included it in the Mexican Cession. I said aloud in class that the map is wrong. She said it's not. I explained that it omits the Gadsen Purchase and that land was not obtained by the Mexican Cession. She said to be quiet. I told her that I would be quiet when she told the class that they should understand that the map is wrong. My parents were called. They told me not to argue with the teacher.

I did not think this was an appropriate context in which to mention the one-point compactification of the real line. — fishfry

Doesn't have to be that. Could be just to choose any two mathematical objects that are not real numbers for +inf and -inf. For example, +inf = w ('w' for omega, standing for the set of natural numbers). Then the system is a certain specific mathematical object. Not a major point in context of this thread; but it is a technicality that should not be deined.

So what would your replacement be? — Banno

I'd cobble together some of my remarks here with some other stuff. Whatever I did, I would make clear that 'infinity' and 'infinite' are not be be conflated, and explain that as I have here.

Infinity. — Banno

It is not required that the extension points have infinite cardinality.

wo meaningless symbols — fishfry

I agree with the basics in your post.

One technical point though:

Yes, in many (probably most or even just about all) writings, the points of infinity are just arbitrary points, and they are not specified to be any particular mathematical objects. But in some treatments, the points are specified to be certain objects, so that the set of reals with extensions is a definite set.

I think the expectation is that folk will look at the related articles for more detail. — Banno

That's a terrible excuse. One shouldn't initiate further study by first publishing a dictionary entry that conflates important concepts.

it's a quantity, not a quality — Banno

What does 'it' refer to there?

Yes, there are points of infinity on the extended real line. So if by 'infinities' we mean such points and others in different number systems and such, fine.

But you said they are cardinalities. It is not required that such points have infinite cardinality, though in some treatments they might. Cardinality is a different subject.

That Wolfram article is poorly conceived.

It defines 'infinity' in the sense of points such as on the real line.

But then it mentions infinity with regard to infinite sets. Notice that 'infinite' is the word there, not 'infinity'.

'infinity' is a noun. So it is a name for a certain object, such as the point of positive infinity on the extended real line. Note that that point does not itself have to be an infinite set, even if in some treatments it may be.

'infinite' is an adjective not a noun. It is not name of a certain object and so it is not the name of a certain cardinality. Rather, it is property of certain sets and a property of certain cardinalities.

The Wolfram article sets up confusion by glibly conflating 'infinity' with 'infinite'.

As I've said previously in this thread and elsewhere, and now again:

'infinity' - a noun - does not refer to a cardinality. It couldn't even do that, since there are many infinite cardinalities. But 'infinity' may refer to things like a point on the extended real line.

What refers to cardinality is 'infinite' - an adjective. The predicate 'is infinite' applies to sets, as a set either is or is not infinite.

.I'm looking at this again with a fresh start.

First, we should put aside quibbles about (a) Anderson running as Independent and (b) the mistaken claim that Reagan was far ahead in the polls. We should just take the problem at face value and take as stipulated the hypotheses that 'Republican' includes both Reagan and Anderson and that Reagan was far ahead in the polls.

/

The validity of modus ponens is:

(a) When the premises are true then the conclusion is true.

The validity of modus ponens is not:

(b) When there is good reason to believe the premises are true then there is good reason to believe the conclusion is true.

So I don't think the example belies the validity of modus ponens.

But we might claim that if (b) fails then modus ponens is not reliable for informing our belief, but we do expect that modus pones is reliable for informing our belief, as indeed we have not just good reason, but irrefragable reason, to believe modus ponens is reliable for informing our belief. So it is a puzzle.

/

McGee actually wrote not simply about good reason for belief, but about was in fact believed. His argument can be fairly paraphrased:

(1) People believed and had good reason to believe: If either Reagan wins or Anderson wins, then if Reagan does not win then Anderson wins.

(2) People believed and had good reason to believe: Either Reagan wins or Anderson wins.

(3) People did not have reason to believe: If Reagan doesn't win then Anderson wins.

We can add (a) if people did not have reason to believe, then, a fortiori, they did not have good reason to believe, and (b) other than unjustifiably optimistic Anderson supporters, people did not believe that if Reagan doesn't win then Anderson wins.

But I don't know whether the particular wording changes the puzzle.


/

'has reason to believe' and 'has good reason to believe' are intensional:

Suppose there is a spy who stole documents from Interpol and that Smith is that spy. And Jones knows about the caper but little of its details. Then:

Jones has good reason to believe "the spy is the spy". But Jones does not have good reason to believe "Smith is the spy".

If we take out 'has good reason to believe' and leave only 'believed' then we have:

(4) People believed: If either Reagan wins or Anderson wins, then if Reagan does not win then Anderson wins.

(5) People believed: Either Reagan wins or Anderson wins.

(6) People did not believe: Reagan doesn't win then Anderson wins.

That mentions belief, but intensionality is not present. It is just three statements about what people believed.

(4) and (5) are quite unlikely true if by 'people' we mean typical people, even typical people well informed about the campaign, even just journalists and political scientists. Such people never had such thoughts as "If either Reagan wins or Anderson wins, then if Reagan does not win then Anderson wins" and "Either Reagan wins or Anderson wins"*.

* There it does matter that we say "A Republican wins" rather than "Reagan wins or Anderson wins", since the former was believed.

But we should be generous to McGee by revising to (a) "People who were informed about the campaigns and understood formal logic and were presented with such a proposition would have believed". Then there is no harm in taking "People believed" to stand for (a). And then (4) and (5) are true.

All that is shown in this version is that people believed certain premises but not a conclusion that follows from those premises. That's just a factual matter. It doesn't belie modus ponens.

/

Does temporality bear on the puzzle?

McGee's version uses future tense. 'wins' stands for 'will win'. Keeping consistent tense:

(7) People believed and had good reason to believe: If either Reagan will win or Anderson will win, then if Reagan will not win then Anderson will win.

(8) People believed and had good reason to believe: Either Reagan will or Anderson will win.

(9) People did not believe* and did not have reason to believe: If Reagan will not win then Anderson will win.

* I added 'did not believe' because it is true and fits the the pattern.

Recast in past tense, and hypothesize that the people mentioned lost access to information about the election starting with news about the voting results:

(10) People believed and had good reason to believe: If either Reagan won or Anderson won, then if Reagan didn't win then Anderson won.

(11) People believed and had good reason to believe: Either Reagan won or Anderson won.

(12) People did not believe and did not have reason to believe: If Reagan didn't win then Anderson won.

Still obtains as a puzzle. So I don't see temporality as bearing on the puzzle.

/

I mentioned "R -> (R v A))". Most people don't believe it, since they don't even know about, but they would believe it if they knew about formal logic, so they do have good reason to believe it.

Another poster mentioned material implication with its clause "'False antecedent then false consequent' is true". I bet most people don't believe that, since they've never heard of it, and they wouldn't believe it even if they knew about it. And a lot of people who have heard of it don't buy it. But there are people who do believe it, especially if they accept the first chapter in a logic book, so we can limit to those people.

Anderson ran as Independent, but he was a Republican. It doesn't matter anyway, since we don't need to mention 'Republican', as we could just say 'Reagan or Anderson'. Moreover, we could say 'Reagan or x' for any x whatsoever. We could say:

If either Reagan wins or Donald Duck wins, then if Reagan doesn't win then Donald Duck wins.
Either Reagan wins or Donald Duck wins.
Therefore, If Reagan doesn't win then Donald Duck wins.

or

If either Reagan wins or Carter wins, then if Reagan doesn't win then Carter wins.
Either Reagan wins or Carter wins.
Therefore, If Reagan doesn't win, then Carter wins.

But with that argument, there's no puzzle.

/

The actual factual error in the problem is the claim that Reagan was way in the polls. Actually the polls were close between Reagan and Carter.

There might be something lurking in the notion of 'good reason' that has to do with degrees of good reason, which also relates to degrees of confidence in beliefs. And Pfhorrest broaches the matter of lack of certainty. I'm not inclined to it, but maybe a solution does lie in that direction.

I'm not inclined to quibble with the givens of the problem or appeal to lack of certainty. That seems not to face the structure of the problem head on.

I guess we could say that there is good reason to believe the conclusion and that there is good reason not to believe the conclusion. Which in its form is not a contradiction.

The reason for believing that the conclusion is false is a good reason. So maybe its a better reason than the reason for believing the conclusion is true. So maybe its such a better reason that it makes the reason for believing the conclusion is true really not a good reason. But the reason for believing the conclusion is true is that it follows from a sound argument (true premises and modus ponens), and you can't get a better reason than that! Thus, still a puzzle.

If there is good reason to believe those premises, then there is reason (even good reason) to believe the conclusion — Pfhorrest

That seems right, of course. But from a different view, there is not a good reason to believe the conclusion, since there is an overwhelming better reason to believe that if Reagan does not win, then Carter wins, so that Anderson does not win. That there is both good reason to believe the conclusion and not good reason to believe the conclusion is the paradox.

I don't take logical inference to be "ALL about" [all-caps added] good reasons for belief. Logical inference can take place in a machine that doesn't even have beliefs or reasons for belief. Modus ponens and other deductive forms have settings other than grounds for belief. — TonesInDeepFreeze

To emphasize that point. The validity of modus ponens bears upon grounds for belief, but the validity of modus ponens can be (and often is) understood irrespective of grounds for belief. The validity of modus ponens is that if the premises are true then the conclusion is true, And that "the premises are true then the conclusion is true" is true of modus ponens no matter whether we even wish to raise the subject of grounds for belief.

Inferences may be drawn irrespective of belief. One could draw inferences from modus ponens all day long without even giving a thought as to what one thought are grounds for belief of anything. Indeed, in a formal sense, an argument is an ordered pair <P c> where P is a set of sentences (or formulas but that complicates this with a technicality) and c is a sentence (or formula). A valid argument is an argument such that in all models in which all the members of P are true are models in which c is true. Then sound systems of logic are ones such that proofs only result in valid inferences. And a valid inference is one such that, again, in all models in which all the members of P are true are models in which c is true. There is no requirement that we mention "reason to believe" or anything like that. So inference isn't "ALL about" reasons for belief.

the quote in the OP is claiming that having good reason to believe the premises doesn't constitute having good reason to believe the conclusion. — Pfhorrest

He said there is good reason to believe the premises, but not a reason to believe the conclusion. And that is true*. The part about "constituting" or anything like it, is not in the quote. We might think it is fair to think he intended that (I don't know; it's a fine point); but he didn't actually say it.

* It's true given the background premise that the polls showed Reagan far ahead. But that premise is false, since Reagan was not far ahead in the polls. No matter for the analysis though, as we may take it hypothetically that Reagan was far ahead or that we had good reason to believe he would win on any other grounds.

about whether they have good reason to believe them. — Pfhorrest

Yes, and I took account of that in followup posts. Actually, he mentions both 'good reason to believe' and 'reason to believe'. I would guess he didn't mean that difference as playing a role, but it might be good to see what happens with the distinction and without the distinction.

That's also what logical inferences (like modus ponens) are all about: — Pfhorrest

"all about" is sweeping. I don't take logical inference to be "ALL about" [all-caps added] good reasons for belief. Logical inference can take place in a machine that doesn't even have beliefs or reasons for belief. Modus ponens and other deductive forms have settings other than grounds for belief.

Most of the rest of your post is explanation of what I understood when I read the first post in this thread. That's okay though, as other readers may benefit from it.

Should we then refer to these terms as different types of infinite? — Bradaction

I wouldn't. I would say they are different predicates of the form: x is infinite & Rx.

I think there is something to what you say. But I don't know whether we need the notion of domains for it.

She has good reason to believe she will receive the apple.
She believes that (A -> (A v O)) is valid.
So she has good reason to believe she will receive the apple or she will receive the orange.
She believes that (A v O) -> (~A -> O) is valid
So she has good reason to believe that if she doesn't receive the apple then she will receive the orange.

But she doesn't have good reason to believe that if she doesn't receive the apple then she will receive the orange.

And that's a puzzle.

But why doesn't she have good reason to believe that if she doesn't receive the apple then she will receive the orange? Because she has good reason to believe that if she doesn't receive the apple then she will receive the banana. (That's where your line of thinking comes in.)

So the banana comes up regarding her beliefs, but it doesn't come up in the argument itself.

So how can that be used to solve the puzzle?

This makes me want to abandon my suggestion that maybe its more about disjunction and intensionality than about modus ponens and intensionality. Maybe it's something about deduction and intensionality in genera (or maybe even more generally about inference and intensionality in general?)

When I first read the claim given by the author that Reagan was decisively ahead of Carter in polling, I felt something was wrong, but I let it slide. Then when I took a moment to really think about it, I realized that it is wrong. Indeed it is famous that the polls were close yet Reagan won so decisively.

are vertical infinity, horizontal infinity and infinity, all potentially different terms that could be given to different types of infinity? — Bradaction

'is infinte' can be qualified any way you can come up with a definition of your qualifier.

is countably infinite

is uncountaby infinite

is infinte in correspondence with the y axis

is infinite in correspondence with the x axis

Etc.

Suppose instead of "R v A" our second premise is "R v C". Then there's no puzzle.

But why did we adopt "R v A"? Because Reagan looked bound to win. So we got it from the theorem

R -> (R v A)

So maybe it's not modus ponens that should be in question, but "R -> (R v A)".

I'm not saying we should doubt the validity of "R -> (R v A)". But maybe it's the one not mixing well with intensionality and not so much podus ponens. I think that might be right. Because we can can do it this way, without modus ponens:

1. R
2. R v A
therefore 3. ~R -> A

In that space of assumptions, (apple, orange) — fdrake

The domain is {apple, orange, banana}.

{apple, orange} is a subset of the domain. {apple orange"} is not a "space of assumptions". It is not a set of assumptions. It's a set whose members are two different pieces of fruit.

if it's not an orange it must be an apple. not(apple) implies orange holds in that domain — fdrake

If x is in {apple orange} and x is not the orange, then x is the apple. And if x is in {apple orange} and x is not the apple then x is the orange.

if you don't receive an orange, you would need to eliminate the possibility of receiving a banana. You can't do that. — fdrake

No, we can reason from an assumption that you won't receive the banana. That the banana is in the domain doesn't entail that we can't assume that you won't receive the banana. Let 'W' stand for 'we receive x'.

The domain is {apple orange banana} but that doesn't stop us from reasoning from the premise:

W(apple) or W(orange)

Let the domain = {0 1 2}. Let 'W' stand for 'x is the number of pens' in my pocket.

Suppose I know I have a pen in my pocket, so I have the premise:

W(1) or W(2)

Or the author's example:

Let the domain be {Reagan, Anderson, Carter}. (By the way, Anderson ran as an Independent, not as a Republican, though he was a Republican.)

Let 'W' stand for 'x wins'.

Then we have the premise:

W(Reagan) or W(Anderson)

What you can do is eliminate the possibility of receiving a banana if you have already assumed, or it is true that you will have received, a roundish fruit. — fdrake

Yes, just as we assume a Republican will win.

But they can't exclude the banana, so they have no reason to believe (in the OP's terms) that they wouldn't receive a banana (analogously, a democrat, Carter, would win). — fdrake

No, they have very good reason: the polls. But it doesn't matter about the factual givens anyway. For sake of argument we accept that we have good reason to believe that a Republican will win and moreover that we assume a Republican will win.

it's evaluated over the candidates — fdrake

I already answered that.

Again, we can take it merely propositionally.

R for 'Reagan wins'
A for 'Anderson wins'
C for 'Carter wins'

(R v A) -> (~R -> A)
R v A
therefore ~R -> A

I don't have a solution, but below is one way to lay out the problem by "brute force".

In case it matters, we note that the text mentions both 'good reason to believe' and 'reason to believe'.

1. we have good reason to believe (R v A) -> (~R -> A)
2. we have good reason to believe R v A
therefore 3. we have reason to believe ~R -> A

That does not prove the invalidity of modus ponens. But it is a puzzle.

Mentioning both 'reason to believe' and 'good reason to believe' suggests degrees of reasons to believe. Or perhaps the author didn't mean to imply degrees. In that case we have one of these two:

1. we have good reason to believe (R v A) -> (~R -> A)
2. we have good reason to believe R v A
therefore 3. we have good reason to believe ~R -> A

That seems to preserve the puzzle.

1. we have reason to believe (R v A) -> (~R -> A)
2. we have reason to believe R v A
therefore 3. we have reason to believe ~R -> A

That doesn't seem as strong a puzzle, but still a puzzle.

Or take the modal operator outside the scope of the argument itself and we have three versions. Of the three, the first is closest to the author's text:

1. (R v A) -> (~R -> A)
2. R v A
therefore 3. ~R -> A

we have good reason to believe 1.
we have good reason to believe 2.
we believe that modus ponens is valid, so we have reason to believe 3.

or

1. (R v A) -> (~R -> A)
2. R v A
therefore 3. ~R -> A

we have good reason to believe 1.
we have good reason to believe 2.
we believe that modus ponens is valid, so we have good reason to believe 3.

1. (R v A) -> (~R -> A)
2. R v A
therefore 3. ~R -> A

we have reason to believe 1.
we have reason to believe 2.
we believe that modus ponens is valid, so we have reason to believe 3.

But if it's just 'believes' then there is a chink in the armor:

1. (R v A) -> (~R -> A)
2. R v A
therefore 3. ~R -> A

we believe 1.
we believe 2.
we believe that modus ponens is valid, so we believe 3.

If someone claimed that they believe 1 and 2, but not 3, then I might say, "I don't think you really do believe 2."

Change to 'knows', and the puzzle is even weaker:

1. (R v A) -> (~R -> A)
2. R v A
therefore 3. ~R -> A

we know 1.
we know 2.
so we know 3.

If someone said they know 1 and 2 but not 3, then I might say, "Then wake up and smell the coffee: you're just not following through to accept knowledge implied by what you do know."

So I think the specific nature of the intensionality does have something to do with this puzzle.