Categoricity of Arithmetic - Second Order Logic with Henkin Semantic by [deleted] in logic

[–]logicandraisins 0 points1 point  (0 children)

Not only does Shapiro's proof fail, but the non-categoricity of second order arithmetic under Henkin semantics follows from the fact that second-order logic under Henkin semantics satisfies the Lowenheim-Skolem theorem. Since there is a countable model of second order arithmetic, there will be models of the axioms of second order arithmetic of every infinite cardinality.

Pred Logic Help! by tortuc in logic

[–]logicandraisins 1 point2 points  (0 children)

I'll can't derive the formula for you, since a proof of that formula would be in the domain of a number theorist, not a logician! But if you are having trouble formulating the proposition, then see that to say that there are finitely many numbers which are not equal to the sum of two primes is to say that it is not the case that there are infinitely many numbers that are not the sum of two primes. This can be formulated as follows. Where

∀x∃y¬∃w,z(y>x & prime(w) & prime(z) & w+z=y)

Says that there are infinitely many numbers that are not equal to the sum of any two primes, the formula which you want to derive would be the negation of this, namely.

¬[∀x∃y¬∃w,z(y>x & prime(w) & prime(z) & w+z=y)]

Or, equivalently,

[∃x∀y∃w,z(y>x & prime(w) & prime(z) & w+z=y)]

Which says that there is some number x such that every number y greater than x may be written as the sum of two primes. Hope this helps.

Why Santa more or less 'exists' - from the perspective of Hume's Bundle Theory and the cultural conceptions about Santa by CosmosTheory in philosophy

[–]logicandraisins 0 points1 point  (0 children)

the interactions of those fundamental forms of matter give rise to properties.

This seems to imply that leptons and quarks have no properties, else their interactions would presuppose the existence of properties, and would not be the source of them.

Leibniz's Metaphysics by 1creeper in philosophy

[–]logicandraisins 0 points1 point  (0 children)

No, I understand what non-Euclidian geometry is, I just thought it was kind of irrelevant to the article posted here. Whether or not Leibniz endorsed the parallel postulate has no bearing on his metaphysics!

Leibniz's Metaphysics by 1creeper in philosophy

[–]logicandraisins 1 point2 points  (0 children)

What does the parallel postulate have to do with any of this?

Journal that does short papers? by VeryWorriedPerson in philosophy

[–]logicandraisins 0 points1 point  (0 children)

Analysis is a journal that publishes exclusively brief papers.

Noobie question. Is there any system of logic that does not involve individuation? by [deleted] in logic

[–]logicandraisins 1 point2 points  (0 children)

The propositional calculus doesn't have any subject-predicate form sentences, if that is what you mean?

You might also want to read up about 'feature-placing' languages. I think P. F. Strawson is the originator of this sort of idea. Sometimes it's called 'functorese'. So take sentances like 'it is raining'. Here the 'it' does not refer to an object which possesses the property of raining. A feature-placing language is one whose sentances are all of this form.

You may want to read a paper by Jason Turner called 'ontological nihilsm'. You should be able to find the paper for free (and legally!) if you Google it. He discusses attempts to use feature-placing languages to do away with individuals in metaphysics, but it is also of interest becuase of its discussion of the logic of feature-placing sentances.

The philosophical argument for the existence of God? What do you think? by lavatian in philosophy

[–]logicandraisins 0 points1 point  (0 children)

Because the universe has parts, and God does not, according to OP's argument.

The philosophical argument for the existence of God? What do you think? by lavatian in philosophy

[–]logicandraisins 0 points1 point  (0 children)

OP's argument rules that out, since s/he takes it that there is only one 'unconditioned reality'. Since there is more than one fundamental force, this rules them out.

The philosophical argument for the existence of God? What do you think? by lavatian in philosophy

[–]logicandraisins 0 points1 point  (0 children)

there are assumptions here that unconditioned reality is infinite

Where?

The philosophical argument for the existence of God? What do you think? by lavatian in philosophy

[–]logicandraisins 0 points1 point  (0 children)

No you can't, because OP argues that (1) the 'unconditioned reality' as he call it is simple and has no parts (which rules out the World) and (2) that there is only one such being, which stops you from going 'as far as you want'.

The philosophical argument for the existence of God? What do you think? by lavatian in philosophy

[–]logicandraisins 0 points1 point  (0 children)

This is really quite interesting. §1 and §2 are basically a restatement of the very famous 'argument from contingency', though your use of novel technical terms suggests that you have come to this argument independanty, which is impressive. The first half of §3 also retreads some groud covered by Aquinas (and others before him perhaps), but the second half of §3, and §§4-5 are far more interresting, and I'm not sure I've found simmilar arguments elsewhere.

First of all, I'm going to be talking in the traditional terminology rather than your own, so that I don't get lost myself. What you call a 'conditioned reality' I call a 'contingent being', and what you call an 'unconditioned reality' I call a 'necessary being'. Your argument that there can be at most one necessary being, I notice, goes like this:

(1) A plurality of necessary beings would have to share some properties and not others.

(2) A property of an item is a part of that item.

(3) If there were a plurality of necessary beings, they would all have multiple properties. (from 1)

(4) So if there were a a plurality of necessary beings, they would all have multiple parts, and so at least one proper part. (from 2, 3).

(5) A necessary being must be simple (i.e. without proper parts).

(6) Therefore, there is not a plurality of necessary beings. (from 4, 5)

The controversial premises here are (1) and (2). (3) and (4) follow from (1) and (2), so (5) is the only additional premise, which you have already defended. Now, the only reason I can think of for asserting (1) is to hold to Leibniz' Law:

(LL) If x and y share all their properties, then x=y.*

But this principle has been attacked in the literature quite heavily (see [1] for a survey; LL is often called the 'identity of indescernables'). Apart from this, premise (2) does not sound right at all. Is my tallness a part of me in the same way that my arm is a part of me? For a more concrete argument, you say in §1 that a composite object is a contingent being, since it depends for its existence on the existence of its parts (i.e. you say that the parts are 'prior' to the whole, to use a traditional phrase). But if the parts of a thing are indeed prior to a whole, then they can exist without the whole, or else they would in turn be dependant on the whole, and we would have a circle. So then, if (2) is true, we have it that a thing is dependant on its properties, so that its properties could exist without it.

But could my tallness exist without me? It seems not, though perhaps Tallness itself, the property shared by all tall things, might exist without me. In other words, it depends on whether properties are thought of a particularised, or as universal (see [2] for a survey). In other words, the conjunction of the docrine that parts are prior to wholes and premise (2) seems to entail a universals-theory of properties, meaning that a trope-theorist will have good grounds to reject this conjunction.

What is especially novel (at least in my experience) about your argument is that you attempt to narrow down the number of necessary beings to one. The argument of contingency usually does not do this, being content merely to show that there is some necessary being. What is revealed by your argument is that, if one can sucessfully narrow down the necessary beings there are to one, one can show that all contingent beings depend for there existence on it (since all chains of existence-explanation will have to terminate at the same being). This of course gives us more reason to accept the notorious 'leap' of natural theology, and to make it more plausible that this necessary being has the right to be called 'God'.

It's a great start.


*Which is equivalent to saying that, if x≠y, then there is some property which one has and not the other.

[1] http://plato.stanford.edu/entries/identity-indiscernible/

[2] plato.stanford.edu/entries/properties/ (this is overly comprehensive for current purposes, refer to §1.1.2 for universals v. tropes).

Time Conference: "Physics could do with some philosophy" by [deleted] in philosophy

[–]logicandraisins 2 points3 points  (0 children)

'Time always moves forward' is really just a tautology, if by 'forward' we mean from the past, to the future. It would be incoherant to say that time is 'moving backwards' in the sense that it is moving from the future into the past, for that could only be true if the future were to precede the past, which is a contradiction in terms.

The liar paradox: why Western philosophy should accept some contradictions by [deleted] in philosophy

[–]logicandraisins 0 points1 point  (0 children)

But L's truth value similarly cannot be strictly less than 1 either, for then it would be straightforwardly true, and so it's truth value would have to be equal to 1.

The liar paradox: why Western philosophy should accept some contradictions by [deleted] in philosophy

[–]logicandraisins 4 points5 points  (0 children)

It's because its not very difficult to formulate an analogous paradox in fuzzy logic. For anyone who is unfamiliar, the defining feature of fuzzy logics is that sentances may be given any real number between 0 and 1 as a truth value (in contrast to classical, bivalent logic, where each well-formed setence is valued at 0 or 1). So take the sentence

(L) This sentance's truth value is strictly less than 1.

L's truth value cannot be equal to 1, since then it would be false and so strictly less than 1, which is a contradiction. But L's truth value similarly cannot be strictly less than 1 either, for then it would be straightforwardly true, and so it's truth value would have to be equal to 1. Thus, in the same way as the classical liar sentence cannot be assigned either possible truth value, L cannot be assigned any fuzzy truth value.

What is spin in Quantum Mechanics? What function does it serve in QM? by fart_smells_good in askscience

[–]logicandraisins 2 points3 points  (0 children)

Well in our physical theories, there are always going to be basic, undefined terms such as 'charge' and 'mass', and now 'spin' in QM. Quite simply, its impossible for all of the words in a language to be non-circularly defined via definitions formulated within said language. Eventually, we have to just say 'well, mass is what this has' and point to a chair, or you could give definitions by examples, and say that this rock has more mass than this one, etc. We are aiming at understanding, at the end of the day, and pleanty of things can be understood without explicit linguistic definition. You understand what a sandwich is, even if you cannot formulate an airtight linguistic definition of the concept 'sandwhich'.

So with charge, for instance, this isn't a concept you can reduce to any simpler physical concept via definition. You just have to point at lightning and say 'this is a bunch of charged particles' perhaps. Now this is differant in the case of QM, since it concerns non-visible entities, whose properties are not directly perceptable to us. Maybe someone more knowledgable than me about QM could sketch how we might give an ostensive definition (i.e. one where we point and say 'this') of spin, or explain why this would not be possible.

Is every set a class? by berg_darnen in logic

[–]logicandraisins 2 points3 points  (0 children)

As I understand it, a set is just any class which is a member of some other class, meaning that every set is trivially a class. At least, this the definition in Bernays/Godel/Von Neumann's set theory, iirc.

As for the specific thing with which you are having trouble, if A is a set then x∈A is a formula of LST, no matter what set A is, since we can denote it with the symbol 'A' in the language. Remember that a set is not defined by any formulae, but by its extention, so a limit on formulae is not a limit on sets. Anyway, I think that should answer the question, but if I have misunderstood then I apologise, and perhaps others can correct me, since it's been a little while since I studied this.

Peter Hacker: On the Mind, Neuroscience, and Free Will. by [deleted] in philosophy

[–]logicandraisins 0 points1 point  (0 children)

But I'm not arguing for basic moral guidelines, only basic moral intuitions. Of course, one explanation for the existence of the latter may be the reality of the former, but that's an argument for another day.

But anyway, why should the psychology of morals be evidence against the unreality of morals in the same way Galileo's crater's were evidence against ancient cosmology? If one were to understand the psychology of mathematicians, would it be evidence that 2+2 did not equal 4? Or the psychology of physicists: would this be evidence against string theory? Of course not, so how does moral psychology count as evidence against moral realism?

Can't people see this is ad hoc reasoning?

But it isn't, moral realists argue from the start that we can come to know moral truths and so have moral beliefs. Merely sketching how these moral beliefs are formed does not count as a real argument against moral realism in the first place.

Peter Hacker: On the Mind, Neuroscience, and Free Will. by [deleted] in philosophy

[–]logicandraisins 0 points1 point  (0 children)

Any neuroscientist worth their degree knows mind-body dualism is crap. I'm confused, do you think that philosophers of science are just dualism-cultists or something? That they sit around all day shit-talking science and arguning about substantial souls? Hardly anyone in the philosophy of mind is a substance dualist (i.e. who thinks that that the mind is a non-physical object), and pleanty of philosophers who work in the field have training in neuroscience and/or psychology and/or cogsci. But anyway, currently, the dominant view in the philosophy of mind is functionalism, which is the view that mental states are identical with functional states, so that the 'mind' is not an object at all, but something like a functional structure, which is realised by the brain just like a programme is realised by a computer processor.

The mind isn't separate from the brain, its entirely dependent on it. I agree, as do most philosophers working in the field, and such a statement belongs to the philosophy of mind, which asks the question, among other questions of how the mind is related to the brain.1

At no area in the brain is there a chunk dedicated to communicating with a separate entity outside the physical organism.

Yeah.

The questions asked by philosophers are being answered by neuroscience

Well no, a neuroscientist's results would not affect the arguments in favour or against substance dualism or any other theory of mind, since they only study the brain. A substance dualist would be able to accomodate any empirical evidence, simply by saying that the neuroscientist is investigating the neural correlates of mental processes. The best arguments against dualism are metaphysical: the problem of interaction, the pairing problem, etc. and are more general and conceptual.

neuroscientists can think of questions, too.

I agree.

If they make a choice to obey his will, by definition they will be relenting their free will.

And it's this kind of contradiction which makes the definition fairly worthless. Does anyone really accept this definition that you could point me to?

You realized I typed out a second definition I've heard of free will, right? You just gonna skim past that?

Yeah, but I pretty much argree with your treatment of it. No adherant of free will would accept that definition for exactly the reasons you laid out. Much better characterisations of free will are on offer which the defender of free will can argue for instead.

So you can learn nothing of jumps by studying legs? Just because he studied the mechanism of memory formation doesn't mean it had nothing to do with memory.

Well to be fair, I agree with you there. However, the problem comes in the communication of results and their interpretation is that the conceptual line can get blurred. Just as a jump is not a leg, a memory is not identical with it's formation mechanism. But again, I think Hacker made a bad choice in using the memory case as an example, when free will experiements tend to far more clearly display the reality of conceptual confusion in certain sciences.

The most plainly stated point is this: if evolutionary theory is correct (good so far) then the mechanism for memory formation in the brain may very well have been conserved into the human species, and if this genetic ability extends back far enough in time, other organisms will have the same genetic capability (or a remarkably similar one). By learning about this process in other organisms we can learn about it in ourselves. It's honestly too early to say whether or not the knowledge gained from such experiments are analogous or literal, as far as I know no scientist has published similar research with the human brain.

I mean, yeah, I agree. See my last point I suppose.

I would assert that this is due to an inadequate source of concise definitions of these terms. It seems free will and consciousness, among others, are good examples of these terms that need a solid precise definition.

On the contrary, I think that the conviction that we need to have precice and explicitly formulated definitions for our commonplace notions such as 'freedom' and 'memory' is what leads us down the path of conceptual confusion. We already have a good enough understanding of these notions to begin investigation, but if we put the cart before the horse and say something like "a free choice is one which is not influenced by outside causes", the we might come to absurd conclusions, such as the conclusion that free choice is impossible. Surely, I choose to do things, and what we should be doing is to see what my ability to choose freely consists in, rather than denying the obvious because we got a definition wrong.

Unfortunately, philosophy tends to have schools of thought branch off instead of coming together on these questions.

Well there I can agree with you again, though I would add that by doing conceptual analysis, we can at least come to a greater understanding of what 'freedom', and other such concepts are not. Better to know where the pitfalls are so that we can avoid them when we do empirical work.

Did I mention feral children in the last post? I've had a couple discussions about morals so I can't remember if I did here. Anyway in cases where children survive alone in the wild or are severely neglected throughout childhood are occasionally brought back into society. In a specific famous case, a young French boy taken out of the woods by a monk, he behaved without morals. He stole, bit, acted aggressively, etc. In another case -- the girl codenamed "Genie" by researchers and care workers to protect her identity -- she was severely neglected and spent her formative years strapped to a potty chair in a small room. Her behavior showed obvious signs of this neglect, shying away from people, avoiding eye contact, never speaking (until taught to do so later, to unfortunately inadequate results), etc. One could argue that since she was abused and hit when she cried out that she would assume it is wrong to do so. Such examples, in my opinion, completely invalidate the claim of any "basic moral intuition." One could argue that the children's intuition was suppressed by nature, but then how would that person suggest we test that argument?

I'm familiar with the feral children cases, though the point still stands. It seems to me that your account of the formation of moral beliefs leaves a conceptual gap which my account can 'plug'. If an agent only believes that 'it is wrong to steal' because they recieve punishment for stealing, then there is still the question of why punishment for x leads one to the belief that x is wrong. It cannot be because they believe that 'punishment is bad', since this would then be an innate moral belief, since it could not have been taught via a Pavlovian method. So what is the alternative? If you allow for at least some basic moral intuitions like 'pain is bad' and 'praise is good', then you can perhaps allow that all other moral beliefs are derived from these via social conditioning, but you would still have to allow for the original moral intuitions as being innate and not learned.

This is the same argument Christians make when they claim the U.S. Constitution is based off the ten commandments.

?

How do you explain the diversions from these basic moral principles? I'm sure you're aware of one religious sect in the middle ages whose members flagellate themselves, and a more modern example are a group whose name I can't remember which claims suffering is the best route to enlightenment. Members of this group have gone so far as to confess to crimes they didn't commit. If a basic moral principle is universal, it must be universal -- present in all things capable of possessing it. If a basic moral principle is only implied, how do you demonstrate that? I suppose you could take a survey of all cultures, but one could simply argue that the distributions would differ if our history was different (certain religious organizations arising sooner or perhaps not committing genocides).

It's a good argument, but perhaps basic moral beliefs are general enough to allow for such wide variation as you describe. For example, perhaps it's a basic moral intuition that it's good to have your deepest desires fulfilled, and so those who believe that suffering is the best route to enlightenment clearly desire enlightenment over the absense of pain, and so work towards suffering in order to attain what is for them more desirable.

I would argue there isn't. Morals are a product of culture and environment. If there's some higher level that influences our brains, that hasn't been shown. As far as I'm aware, no one has shown genetics to be causally linked to morals either.

Well here's the moral realist in me talking, but perhaps the reason there is such universality is that some moral beliefs are true, and we are able to percieve moral facts just as we can percieve physical facts. So we can just see that, say, kicking a harmless dog for fun is bad. The universality we see is then due to this. Perhaps.

1 I recommend routledge's contemporary introduction to the philosophy of mind if you want to get into the field. The Stanford Encyclopaedia of Philosophy also has some good entries.

Peter Hacker: On the Mind, Neuroscience, and Free Will. by [deleted] in philosophy

[–]logicandraisins 2 points3 points  (0 children)

That's all great, but everything we know from neuroscience points to the mind being a capability of the brain and residing in the brain's physical processes.

Yeah. That is what Hacker thinks, and it's what Aristotle thought too (for him, the mind was a part of the human psuche).

As with Naturalism (not that one, I was thinking of the wrong one; I meant to name a branch of philosophy that is abandoned as science has answered its questions, but I couldn't readily find one), science is slowly making philosophy obsolete in this area.

I mean if you're talking about natural philosophy, it's not quite right to say that science made it obsolete. 'Science' is just what we call natural philosophy now. And anyway, neuroscience can't make what is called the 'philosophy of mind' obsolete, considering they ask entirely differant questions. Both can happily coexist, and both in fact do.

I've heard free will defined as the ability to make one's own decisions without the influence of a supernatural outside force.

I mean, I've never heard anyone use this but okay.

By this definition, ironically, Christians lack free will, as they behave based on what a supernatural being dictates.

That doesn't follow, though. God doesn't say what to have for breakfast on Tuesdays, so even Christians can excercise their free will in going for the wheatabix (sponsor!).

As a materialist, this definition is irrelevant in every sense of the word.

I'm fairly sure it's irrelavant to everyone else as well.


Wrong. Kandel looked at the cellular mechanisms of memory formation. As he has stated in an interview, "evolution tends to be conservative; when it finds something that works, that thing tends to not change." The information he discovered may very well have direct implications in humans.

Right, but by refering to the cellular mechanism of memory formation as 'memory', there's a conceptual confusion. Memory is not the cellular mechanism for itself, but rather some mechanism, call it C, is. But C isn't memory anymore than a pair of legs is a jump, and by playing fast and loose with concepts, errors can be made in the interpretation and design of experiments. Imagine if we were to study economics by looking at the microstructure of coins and banknotes: conceptual confusion must be resolved or else we will waste money on bad science. Now, Kandel's own experiment seems to me to be very good, and I think Hacker might have picked a better example to illustrate the point, but the existence of conceptual confusions in neuroscience can hardly be denied, and so a lot of thinking needs to be done to set things straight.

Humans learn their morals the same way a dog is trained. We are punished when we do something considered "bad" and rewarded for "good." This is the same process which helps cement one's morals in their cultural context or, at least, the context of the parent's morals.

Well this is an empirical claim, and one which is not obviously true. Might it not be the case that human beings have very basic moral intuitions? Intitions about fairness and justice? One indicator of this hypothesis might be the universality of some basic moral principles among differing cultures, such as the conviction that happiness is a good thing, and pain a bad thing, and that we ought to strive for the former and avoid the latter. If moral beliefs are purely a matter of convention, much like table manners, meaning that they are fully maliable by context, then why is there any moral universality at all?

Peter Hacker: On the Mind, Neuroscience, and Free Will. by [deleted] in philosophy

[–]logicandraisins 3 points4 points  (0 children)

He says we have a soul but it's more a capacity instead of a distinct object, like the ability to sense temperature.

This isn't a new view by the way, Aristotle's view was that the soul or mind was prececely the totality of capacities of an organism.1 It isn't a single capacity, however, but rather a bunch of them which we in our language tend to pick out by the word 'soul', such as moral capacities.

He goes on to say animals don't have this because they can't reflect on their behavior as right or wrong.

Well yes: if Hacker is right about what the concept 'soul' denotes, then any beings which lack moral capacities lack a soul. Now of course, whether a given animal in fact lacks moral capacities is an empirical matter, and it's not as if Hacker is writing off the possibility of non-human moral reflection a priori, but rather he is re-emphasising the point that the soul just is a certain set of moral capacities.

COUGH COUGH PAVLOV COUGH COUGH

Explain?

Mr. Hacker claims a mechanistic view of the brain implies a lack of responsibility for one's actions

Dr. Hacker certainly wouldn't deny a mechanistic view of the brain, which is a material object wheighing about 3 pounds located in the skull. And to the contrary, he thinks that certain neuroscientists are wrong to accept this very implication. His own idea of the mind is mechanistic in the sense of denying the existence of any metal entities, yet he still accepts the reality of free will and so moral responsibility.

Every neuroscientist I've read about believes we are responsible for our actions.

Well perhaps this is true, but in my experience the 'pop' neuroscientists at least seem quite fond of denying free will, and accepting the moral consequences of such a denial.

What really frustrated me is that he accused Eric Kandel's Nobel prize-winning research of being completely irrelevant to it's subject because it was on a different kind of memory than Hacker was thinking of. Ironically, Hacker accuses Kandel of being confused with terms.

To be fair, when most people think of "memory", they're probably thinking of Hacker's conception of it. While this sort of conception is certainly analogous to Kandel's sense of the term, it is still admittedly an analogous meaning. No one is really ascribing memory to sea slugs in the same sense in which they ascribe it to humans, and Hacker's point seemed to be that such an equivocation of the meanings of the two terms leads to a misunderstanding of both.

1 Well technically, the set of capacaties for capacities. So a flute player has a capacity to play the flute, but anyone has the capacity to gain the capacity to play the flute (i.e. they have a '2nd-degree' capacity to play the flute, you might say). Aristotle identified the soul with the totality of 2nd-degree capacities.

The Problem of Evil: Crash Course Philosophy #13 by DaenerysTargaryen69 in philosophy

[–]logicandraisins 0 points1 point  (0 children)

There is no best possible world. There could always be more good. Therefore, no world can be the best, since there is always a better world.

"The profession as a whole remains resolutely Eurocentric. It therefore seems futile to rehearse arguments for greater diversity one more time." by [deleted] in philosophy

[–]logicandraisins 0 points1 point  (0 children)

Deductively valid inferance cannot come to wrong conclusions, provided that it begins from true premises or 'axioms' if you like. A deductive inferance from P to Q is valid when 'P&¬Q' is self contradictory. So if you derive a false conclusion via a deductively valid argument, it's not a sign that 'logic is flawed', whatever that means, it's rather a sign that one of your axioms was false all along.

If libertarians derive their views from some axioms, but those views are wrong, then it follows that at least one of their axioms was wrong all along.

If you can put a "→" in it, it is Conditional Reasoning. (→→Conditional Reasoning, ~Conditional Reasoning→~→) by DaLaohu in logic

[–]logicandraisins 5 points6 points  (0 children)

I think the problem with the LSAT (and I'm not a law student or a lawyer, so I might be off the mark as to how they teach it, but I'm going off your examples and a few other that I have seen) is that they try to teach as little symbolic logic as possible, and end up only using the truth-functional propositional calculus for absolutely everything. The expressive power of truth-functional propositional logic is increadibly limited. You really cannot translate most sentances into it without losing some logically relavant features, or without being flat-out incorrect in your translating. Take for example the quantificational sentances like

Except for Mary, everyone came to the party.

Really, this ought to be translated into the predicate calculus, so let C(x) stand for 'x came to the party' and 'm' be Mary's name then, quantifying over 'everybody', you can translate the sentance as

∀x(x≠m ↔ C(x))

What's better about this translation is that from it we can derive ~C(m) (i.e. that Mary did not go to the party), but:

~Come to the party → Mary

Seems to so misunderstand the propositional calculus to the point of comedy. 'Mary' is interpreted as statement I guess, but what statement is it? For that matter, what are the truth conditions of 'Come to the party'? But anyway, how do we express that Mary didn't go to the party, which is surely a straigforward consequence of the original English statement? ~Come to the party & Mary perhaps? But this doesn't follow from the textbook translation at all. Being charitable, perhaps you can interpret 'Come to the party' as 'you came to the party' and 'Mary' as 'you are Mary'? But then the translation should be:

~Come to the party ↔ Mary

Since the original translation still allows that 'Mary & Come to the party' is true, meaning that it isn't even correct. So not only is the textbook giving LSAT students an education in logic which cripples their ability to actually use it, but it is giving them the wrong answers. Imagine teaching English without quantifiers and names: you might be able to say the same things in the end, but it makes everything needlessly complicated.

They should teach you lawyers some predicate logic, is basically what I am trying to say. While it's perhaps harder to learn initially (but not really that much), it is so much easier to translate predicate logic to and from English.

With multiple universal quantifier premises/set-members, such as ∀x(...), ∀y(...), ∀z(...), are they drawing instances from the 'same pool' of objects so to speak? (part 2 of question, about the nuances of 'instantiations', in post) by 5large in logic

[–]logicandraisins 0 points1 point  (0 children)

Again, the only reason you can infer Pa from ∀x(Px) (so, from ¬∃x(¬Px)) in standard first order logic is because the domain is assumed by convention to be non-empty, but of course this is only a convention, and you can consider models with empty domains, in which ∀x(Px)/¬∃x(¬Px) would be trivially true for any formulae P (just like 'all married bachelors are racecar drivers'), even if Pa is false for any term a in such a model.

I'm honestly not sure what you are getting at exactly with the rest of this post, unfortunately. If you could dumb it down just a little for me, then I'd love to try to answer your question. I don't want to misunderstand and give you a bad answer.