The Propositional Calculus

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Consistency follows from soundness. Proving soundness is not deep. We ordinarily just do induction on the length of derivations.
— TonesInDeepFreeze
Some simplified detail might be fun.

I hope I say all this right (I'm pretty rusty):

Df. A model M is a model of a set of sentences G iff every member of G is true in M.

Df. A set of sentences G entails a sentence S iff every model of G is a model of {S}.

Df. A system T has the soundness property iff for every set of sentences G and sentence S, if S is derivable from G then G entails S.

Thm. If all the axioms of T are logically true, and T has the soundness property then set of theorems of T is consistent. Proof outline: If the set of theorems of T were inconsistent, then there is sentence P & ~P derivable from the axioms. But the axioms are true in all models, and soundness provides that the axioms entail P & ~P, but that is impossible since P & ~ P is false in every model.

Outline of proving a system has the soundness property: Base step: Prove that all the axioms are logically true. Inductive step: Suppose we are at step n in a derivation and soundness has obtained. For step n+1, show that the rules of inference are truth preserving.

This is found in any textbook in introductory mathematical logic.
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You can stipulate that your account applies to one way of using a word or a phrase, though there may be others

This is interesting. Sometimes I encounter people who say symbolic logic is dogmatic because it demands that 'if then' must be taken as the material conditional even though English speakers use 'if then' in other senses. And it is true that ordinarily English speakers don't have the material conditional in mind. If you asked 10000 people whether "If London is in Asia then Groucho Marx was an aviator" is true, you'd be lucky if even one person said it is true.

So, it is helpful when an intro textbook makes a disclaimer that use of the material conditional in symbolic logic is not to be construed as a claim that the material conditional captures the many everyday senses of 'if then'. And, as far as I recall, Kalish, Montague and Mar (KMM) does not make that disclaimer.

Another thing I wish were different in the book: Truth tables are not mentioned until page 87, after the propostional calculus has all been specified. I think it's much better to explain the truth tables before specifying the proof calculus. That way, the student can see how the proof calculus is truth preserving. Otherwise, the student is first all wrapped up in a bunch of rules while the student doesn't know the motivation for those rules.

I wish there were an intro textbook just as precise as KMM and Mar, but with a more streamlined proof system. The boxes method is intuitive, and helped me a lot as a beginner. But I would like to see a textbook with a more streamlined natural deduction system that uses line accumulations instead. (I think I posted such a system earlier in this thread?)

That said, KMM is still my favorite intro symbolic logic textbook. It certainly set me up with a solid foundation.
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Quine's Methods of Logic

Quine is always quite the pleasure to read. Church too (for me, the intro chapter in 'Introduction To Mathematical Logic' is the definitive primer). Those are two of my heroes. Smullyan also is a great writer. And I particularly like Boolos. And for textbooks, Enderton is great.
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A forum on philosophy ought have threads on the basics of logic.

I think this is the first time I've ventured into the category of 'Logic & Philosophy of Mathematics'.
Its description: What are logic and mathematics? How are they related? How do they relate to human reason and to the world?

The last question grabs my attention. To answer that, for sure, a thread on the basics of logic would come in useful. I'm not so certain about mathematics...what is the 'philosophy of...'?
https://plato.stanford.edu/entries/philosophy-mathematics/

Moving on.

I liked the word "informal" in your previous post, it's just that propositional calculus is a formal system. It's a branch of mathematics.

If you want to raise the logical literacy of the forum, perhaps it would be better to aim at that dialect called "philosophical English," a dialect spoken by people familiar with formal systems. The traditional early chapters of a logic textbook try to show how the logical constants capture some of what we mean by familiar idioms.

I'm interested in being logical and literate but not to the point of formal truth tables.
I've studied this at an introductory level, read the books, then gave them away. For me, the applicability and relationship to the 'real world' seemed too narrow. The kind of 'truth', and its relationships within this artificial domain and symbolic language, seem to imply a kind of universal certainty. 'Truth preserving' perhaps...but only in a set, forced way.

What do you mean by a 'dialect called 'philosophical English'?

Seems ↪Srap Tasmaner is correct that we cannot have a less formal discussion of propositional calculus. It's either too rich for some or too poor for others. I think that a shame.

I've no intention of writing another logic text that will satisfy TonesInDeepFreeze. End of tread, I suppose.

About writing another logic text. You might like to look at this, the first out of 5 'best books' :
[the article has an excellent introduction]

I chose Logic Primer by Colin Allen and Michael Hand for the reason that I taught from it for over a decade at the University of York. One of the interesting things about teaching logic at a university is that no logic teacher at a university is happy with anyone else’s textbook. This is why there are so many logic textbooks: everyone gets hyper-frustrated with the text they’re teaching and ends up writing their own. Now, I’m quite lazy, and I didn’t. I stuck to this book, though actually I changed it in lots of ways. When I teach with it, I reorder it, I delete sections, I add in new sections and new definitions of terms, so in practice the students are learning from my annotated version of the text.

But this is why so many logic textbooks are written. The solution to that problem has arisen in our Web 2.0. I’ll mention it for reference, namely that there is now a logic textbook which is open-source and freely editable, called forallx. It’s online, and more and more logic teachers are saying ‘I’ll take that, and I can edit it in any way I like and use it.’ Anyone can freely access not only the original version of the text, but also any of its modifications. So there’s a Cambridge version of this textbook, a York version, a Calgary version, a SUNY version, a UBC version and probably many more I don’t know about. But the underlying formal language and system is the same in all of those.

Oh, and just for @Agent Smith ( perhaps you've read it already?)

[ ... ] The next book is Mark Sainsbury’s Paradoxes. I love this book. Whole university courses are taught around this book. It’s an absolute classic.
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What are logic and mathematics? How are they related? How do they relate to human reason and to the world?

I've since realized that's an inadequate description for the category. It's also, perhaps primarily, for problems in logic itself.
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I've since realized that's an inadequate description for the category. It's also, perhaps primarily, for problems in logic itself.

That sounds interesting. So, how then to re-write?
Re:' problems in logic'... do you mean mistakes in how we apply logic? Solving logical problems?
Do logical problems even make sense?
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And it is true that ordinarily English speakers don't have the material conditional in mind.

Oh I'd have to go look, but, if memory serves, Grice defended material implication as a faithful representation of conditional reasoning in natural languages and did not join any campaign (not even Strawson's) either to reform logic or to abandon natural language for more precise pastures. And in my view, if it was Grice's view, it deserves deep consideration.
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Material implication in classical and intuitionistic logic is a static relationship that holds between sets , as in "Smoking events might cause Cancer events", where the condition always exists ,even after the consequent is arrived at, due to the fact it is talking about timeless sets rather than time contingent states of processes.

(Smoking "might" cause cancer is due to the fact ~A OR B => A --> B , which doesn't have a conjunction of events in the premise)

For resource-sensitive logical implication that is truly material in denoting conditional changes of state over time, see linear logic for expressing "If I am in the state of smoking then I might arrive at a state of cancer". It has the same form as the above rule, but the premise can only be used one when arriving at a conclusion.

The 'might' here can also be avoided by defining only one axiom of implication in which smoking is the premise. Otherwise the resulting logic expresses multiple and mutually exclusive possible outcomes of smoking, i.e possible worlds are built into the syntax.

For a programming language with native linear types, see Idris.
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You have a better understanding of logic than I do, but I seem to run into more problems related to clarity. Like, introducing terms without providing a definition or conveying their meaning. I believe when terms are introduced without clear meaning they form a roadblock preventing the conversation to progress. I can’t grant an argument if a premise contains a term that I don’t understand. I can’t even grant that the statement is propositional.

For example, most at least implicitly understand that the term “Taller” has relationality built in. If I say the tree is taller, then I am saying it is taller in relation to something else (e.g., than it was last year, than the surrounding trees, etc). If I introduce the term “taller” but in a non-relational sense (e.g., “The tree is taller [full stop]”, you would, presumably, require a conceptual analysis to understand what that could mean. What is more, most of the time no effort is made to define terms or to convey our sense of them. Vagueness and ambiguity often go unchecked, relying instead on the assumption that our interlocutor shares our interpretations.

I enjoy the argumentative stage of discourse, but i seem to dwell mostly in the clarification stage. Do you or any other logician take a similar view? It seems necessary to be in agreement on all terms before arguing one way or another on an issue. Otherwise, how would you know whether or not you agree without a doxastic view of it? I want to give you some examples demonstrating my approach in these situations.

If asked whether or not I believe there is a God, I require you either provide a definition, or convey what you mean by the term “God”. If you take a pantheistic meaning of “God” (e.g., God just refers to the universe or cosmos), then I do believe the universe exists. If you take an abrahamic meaning of “God” such as the God referenced by Christian, Judaic, and Islamic faiths, with properties defined in terms such as omnipotence, omniscience, omnipresence, omnibenevolence, then I believe any propositions stating that such a God exists are false. I think such properties are mutually incompatible, and derive contradictions as demonstrated by the problem of evil. On a separate issue, I don’t understand, and therefore cannot grant any statements made by moral realists if they introduce normative terms on a stance-independent construal.

One last thing, couldn’t the terms “proposition” and “statement” be differentiated with regard to truth value? A proposition being a statement capable of being true or false. A statement being an utterance which expresses a complete idea (not necessarily declarative, possibly interrogative, imperative, etc).
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A statement being an utterance which expresses a complete idea (not necessarily declarative, possibly interrogative, imperative, etc).

In propostional logic, we consider only declarative statements.
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Material implication in classical and intuitionistic logic is a static relationship that holds between sets , as in "Smoking events might cause Cancer events", where the condition always exists ,even after the consequent is arrived at, due to the fact it is talking about timeless sets rather than time contingent states of processes.sime

Where do you find that explanation of the material conditional?

The material conditional is that the conditional is false when the antecedent is true and the consequent is false, and true otherwise,
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Grice defended material implication as a faithful representation of conditional reasoning in natural languages

That's interesting. I'd like to understand more about that.

This is my guess how (most?) English speakers think (contrary to the material conditional) about 'if then' in everyday life:

(1a) "If London is in England, then vodka is a beverage."

False, because there is no relation between the true antecedent and the true consequent.

(1b) "If London is in England, then Westminster Abbey is in England."

True, because the true antecedent implies the true consequent.

(2a) "If London is in England, then marble is soft."

False, because there is no relation between the true antecedent and the false consequent, and the consequent is false anyway.

(2b) If The Beatles recorded "Help", then Eric Clapton played on it.

False. There is a relation between the antecedent and the consequent, but the sentence is false because The Beatles having recorded "Help" doesn't imply that Eric Clapton played on it, and it's false anyway that Eric Clapton played on "Help".

(2c) "If Paris is on the moon, then I'm a monkey's uncle."

Some people will take that as true, as an idiomatic instance of ex falso quodlibet.

(3) "If New York is in Asia, then vodka is a beverage."

False, because there is no relation between the false antecedent and the true consequent.

(4) "If New York is in Asia, then marble is soft."

False, because there is no relation between the false antecedent and the false consequent, and the consequent is false anyway.
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Im am aware. Only interested in statements which are truth apt (i.e., propositional). I was only offering a possible distinction between statements and propositions. Im pretty sure its more technical than the one I offered, but I do hear both terms used interchangeably.
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Just out of curiosity, was your intention simply just to inform me that only declaratives are considered in prop logic? Or was your intention to launch a criticism towards some apparent error you think I’ve made? Im fine with either, although if you were attempting the latter, it would make for a fine example regarding the concerns I mentioned earlier.
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I said nothing to suggest that I was criticizing you. You mentioned that there are expressions other than declaratives, which is of course true. (And, one can devise systems of logic for interrogatives too.) But since the context has been propositional logic, and to answer any potential question whether propositional logic considers expressions other than declaratives, I noted that it does not.
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I was only offering a possible distinction between statements and propositions.

Of course. It's important in the study of propositional logic that we understand that 'statement' is considered only in the sense of declarative statements.

There is variation though among logicians as to the meanings of 'proposition' and 'statement'. (But not so much extending to including interrogatives and such.) What is meant by 'proposition' and 'statement' may depend on the particular logician's or philosopher's framework.

Usually, we take 'sentence' to mean the syntactical object - the string of symbols.

Then it's a question whether we take 'proposition' as a synonym for 'sentence' or whether we take 'proposition' for what is expressed by a sentence.

Same for 'statement' - whether it just means a sentence or whether it means what is expressed by a sentence.

So, 'sentence', 'statement', 'proposition'. We just have to be careful what we mean in context.

/

For example, Church takes 'sentence' in the usual syntactical sense, but for him a proposition is a different abstract object. ('Introduction To Mathematical Logic, pg 26)
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I presumed as much. I only mentioned a few of the alternative types of sentences which express non-propositional statements to provide an example showing that not all statements are propositions. Im sure there are more distinctions between the terms than that one, though. Im not saying you did anything to suggest a criticism, but without further context I wasn’t sure. I just try to be charitable and give people the benefit of the doubt when what they say can be interpreted both positively and negatively. I wouldn’t view a criticism to be necessarily negative either, I appreciate a proper critique.
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What are your thoughts regarding my concerns with the lack of clarity here in the forum? Was I able to articulate my concerns in a clear enough manner for you to understand? Just seeking some feedback.
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introducing terms without providing a definition or conveying their meaning

Of course, definitions are crucial.

But how demanding we should be must depend on context.

Since, for example, this thread is about a subject of mathematical logic, different contexts range from just philosophy about mathematics, to a blend of philosophy about mathematics and mathematics itself, to just the mathematics itself. Then there are degrees of formality, from very liberal informality to rigorous formality.

Forum-wide, usually mathematics is not the subject, but still there may be degrees of formality, from liberally speculative philosophy to more rigorous technical aspects of philosophy.

So what context do you have in mind regarding definitions?

Most informally, we know that of course we can't be bogged down by defining every word of English we use, and even if we could, we'd encounter circularity (English is not a formal language in which there are undefined primitives and then a sequence of definitions.)

For philosophy, I would agree that there should be an expectation that a poster should provide definitions for special philosophical terminology where there is a reasonable need to know the specific definitions. But there's still a limit - since we are not posting entire treatises, we don't have the time for everything.

For mathematics, in principle, every mathematical statement should be formalizable (this is called 'Hilbert's thesis'). But that's only in principle; in actual discourse, we have to be allowed informality, as long as we know, in the background of our reasoning, that could formalize it all if we had all the time and patience to do it (I nickname this 'Bourbaki's thesis'). So, yes, mathematicians, at least in principle, must be able to define all terminology down to the primitives. But, again, in a forum we don't have time to define everything down to, say, the sole primitive ('e' for epsilon, i.e. "member of") of set theory.

On the other hand, there are cranks. Cranks often talk as if they are making mathematical statements (not just philosophical statements about mathematics) as they are using mathematical terminology. But their usage is incorrect, usually ludicrously so. And they have no concept even of what a mathematical definition is, or what the specific definitions are of the terminology they use. For me, as far as definitions, that is the worst of a forum such as this; and it's not just this forum, but all over the Internet.

It seems necessary to be in agreement on all terms before arguing one way or another on an issue. Otherwise, how would you know whether or not you agree without a doxastic view of it?

I know what 'doxastic' means, but I don't know what you mean by "a doxastic view of it" in that context.

therefore cannot grant any statements made by moral realists if they introduce normative terms on a stance-independent construal.

I know what 'moral realism' and 'normative' mean, and maybe I have a bit of a sense of what 'stance-independence' means, but I don't know what is meant by 'introduce normative terms on a stance-independent construal'.

/

"It is not our business to set up prohibitions, but rather to arrive at conventions." - Rudolph Carnap
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So what context do you have in mind regarding definitions?

I actually don’t think definitions need be a requisite, though they are useful insofar as they capture the standard meaning of a term. I try to avoid committing to a definition since it requires a semantic thesis or a theory of public meaning. Im happy to hear your sense of a term and work from that understanding so long you are able to convey your meaning (through any means: ostensively, semantic primitivity, family resemblance, interpretive dance, etc.) so that I form a concept. Of course not every term requires the authors conveyance. It is necessary when idiosyncrasies, proprietary definitions or plain gibberish is detected.

I know what 'doxastic' means, but I don't know what you mean by "a doxastic view of it" in that context.

Im very impressed. I smuggled in a couple terms that Im uncertain that I understand (certainly not well enough to use them), and this is one of them (the other being “Abrahamic” in context with religious tradition—maybe I got that one right). You not only caught it immediately, but were intellectually honest about it. Sometimes I test if people acknowledge not understanding the term, or pretend to. I was going for doxastic attitude (“an epistemic attitude held towards a proposition”) such as belief, disbelief and suspended belief. I meant to say that without a concept of a term (without a concrete image or relative abstractions), I cant say whether or not I believe, disbelieve, suspend believing any statement containing it. I don’t know if its proposition, or coherent, or contradictory, or vacuous.

know what 'moral realism' and 'normative' mean, and maybe I have a bit of a sense of what 'stance-independence' means, but I don't know what is meant by 'introduce normative terms on a stance-independent construal'.

Stance-independence, in the context of normative terms, is a metaethical view regarding the meaning of such terms as good, bad, proper, improper, etc. A construal is a way in which something must be in order to be understood. I don’t understand normative terms in a stance-independent sense, other than what realists claim to refer to (spooky metaphysics). I understand normative terms on a stance-dependent construal (antirealist). Saying something is ‘good’ on a stance-dependent understanding of the term, is to say that goodness is understood in accordance with the desires of an agent, or with a given standard. To say “friends are good” is to say “I desire friends,” and to say “the heart is functioning properly” is to say “the heart is functioning in accordance with medical standards”.

It’s important to remember that on a stance-dependent construal, normative terms must be indexed to an agent or a standard. So if you ask me “Is committing racist war crimes good since the Nazi desired it?” Im committed to say yes—but only in the sense that Im uttering the tautology “Committing racist war crimes is desired by the Nazi because its what the Nazi desired”—not ‘good’ in accordance to my desires.
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“the heart is functioning properly” is to say “the heart is functioning in accordance with medical standards”.

Isn't there a problem with the 'naturalistic fallacy'? The medical standards may be too low or otherwise in error. In that case we could say without contradiction that someone's heart is functioning in accordance with medical standards but is not functioning properly. So they do not mean the same thing - if they did, it would be self-contradictory to say one and deny the other.
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Material Implication (abbrev. Imp)

p $\to$ q :: ~p v q

Commutation (abbrev. Comm)

p & q :: q & p

p v q :: q v p

Distribution (abbrev. Dist)

p & (q v r) :: (p & q) v (p & r)

p v (q & r) :: (p v q) & (p v r)

Association (abbrev. Assoc)

p v (q v r) :: (p v q) v r

p & (q & r) :: p & (q & r)

Double Negation

p :: ~~p
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As regards how contradictions trivialize the meaning of true (re ex contradictione sequitur quodlibet aka the principle of explosion).

True stands in contradistinction to false, the concepts being specifically designed to categorize propositions, perhaps with the intention of separating wheat from chaff. So if every proposition is true, we can no longer do that; in short truth/falsity lose their utility as proposition-classifying properties,
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I actually don’t think definitions need be a requisite, though they are useful insofar as they capture the standard meaning of a term. I try to avoid committing to a definition since it requires a semantic thesis or a theory of public meaning.

Like, introducing terms without providing a definition or conveying their meaning. I believe when terms are introduced without clear meaning they form a roadblock preventing the conversation to progress. I can’t grant an argument if a premise contains a term that I don’t understand. I can’t even grant that the statement is propositional.

What is more, most of the time no effort is made to define terms or to convey our sense of them. Vagueness and ambiguity often go unchecked, relying instead on the assumption that our interlocutor shares our interpretations.

It seems necessary to be in agreement on all terms before arguing one way or another on an issue.

If asked whether or not I believe there is a God,I require you provide a definition.

So I took the time to write a post about that. Then you say the opposite, that definitions are not required. .
So I don't understand you.
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Regarding definitions, Im saying one ought not be required to provide a definition for a term so long as they convey what the term means. Definitions require a semantic thesis or a theory of public meaning (to claim that the best understanding of a term is the meaning communicated by the public at the time), necessary and sufficient conditions (for example, a definition for “women,” in the context of biological sex, would have sufficient conditions such as: physiological or phenotypical proximity to the archetypal female human; as well as necessary conditions such as: a natural genetic predisposition to produce large gametes), the definiendum must be defined by the definiens (in the defining statement “a woman is a being with physiological or phenotypical proximity to the archetypal female human, with a natural genetic predisposition to produce large gametes”, “being, physiological or phenotypical proximity to the archetypal female human, natural genetic predisposition to produce large gametes” is the definiens defining the definiendum “woman”), and other such criterion. Im trying to make things simpler by just conveying my sense of the term and understanding yours.

I was being lazy and equivocated two senses of the term “definition”. Thank you for pointing out the equivocation. I’ll edit it real quick.
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Definitions require a semantic thesis

Except in formal mathematics, definitions are purely syntactical.

necessary and sufficient conditions

Indeed.
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I used “women” because it is a notoriously difficult term to define, thus a good example to portray how complicated definitions can become. In case your wondering why, it wasn’t just to be complicated; there was utility.
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Isn't there a problem with the 'naturalistic fallacy'? The medical standards may be too low or otherwise in error. In that case we could say without contradiction that someone's heart is functioning in accordance with medical standards but is not functioning properly. So they do not mean the same thing - if they did, it would be self-contradictory to say one and deny the other.

I don’t believe so, since the aim in medicine is to maximize health and well-being, without necessarily committing to views such as “One ought to be healthy” or “Health is good” despite the ubiquitous use of normative language within systems of healthcare and medical vocabulary (e.g., disease, disorder, etc.). This is because terms such as “health” and “well-being” are cashed out in relation to the desires of an agent. So, when we say “Disease is bad” in medicine, what that actually translates into is “Disease is [generally] undesired”. Furthermore, when asked what is meant by ‘undesired’ there, it refers to the preferences commonly regarded, but not limited to, by the public. If probed further, one would have to regard individuals from a case by case perspective, such as: “Patient ‘x’ prefers not to have disease ‘y’, because disease ‘y’ increases stimuli that patient ‘x’ associates with pain, and patient ‘x’ desires to avoid the experience of pain”.

So, when statements such as “The heart is functioning properly” are made in medicine, what is implicitly being said is “The heart is functioning in accordance with agreed upon standards for maximal health and well-being” which is essentially making the following argument: “If you desire health and well-being (e.g., longevity and less pain), then your heart should function in accordance with medical standards”. It is up to you, the individual agent, whether or not to make the assertion “I desire health and well-being”, (an objective claim regarding your own psychology) or to agree with the inference “Then your heart ought to function in accordance with medical standards” (an empirical claim backed up by facts extracted from clinical research data).

A variant of the naturalistic fallacy occurs when explanations for terms such as “Good” are reduced to naturalistic properties (Moore gives examples such as “Desires” or [Mills] “Pleasure”) which identify ‘Good’ with its object. This is not occurring in the examples above. What is said is not that “Health and well-being ARE ‘Good’” ontologically speaking, but rather that “Health and well-being ARE WHAT IS MEANT by the term ‘Good’” semantically speaking. Another variant of naturalistic fallacy occurs when crossing the Is/Ought divide (deriving the way things ought to be from the way things, in fact, are). This too is avoided by use of the conditional statement (if p, then q). Since the term ‘Good’ refers to ‘what is desired’, then the conditional statement reads NOT as “If you desire ‘x’, then ‘x’ is good”, but rather it reads “If you desire ‘x’, then you desire ‘x’”, which is a tautology we would all believe to be trivially true. The above doesn’t appeal to nature (medical intervention often preventing natural occurrences such as a virus). The fallacy can and sometimes is committed upon meta-ethical investigations regarding terms such as “Proper function, Malfunction, Disfunction, Disease, etc.” when the individuals being asked attempt to, indeed, fallaciously explain such normative terms reductively (to say ‘Good’ is some object or objective property, rather than a percept or subjective property). This problem seems to be an issue with moral realism.
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A succinct summation: Classical Logic

An archived SEP article.
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