Words of Encouragement

“Man is by nature a social animal; an individual who is unsocial naturally and not accidentally is either beneath our notice or more than human. Society is something that precedes the individual. Anyone who either cannot lead the common life or is so self-sufficient as not to need to, and therefore does not partake of society, is either a beast or a god. ” – Aristotle

Recently, I have been in a rut. I worked under the assumption that I could perform actions in our world, with an absence of self-expression – just put my nose to the grindstone with as little interaction as possible. This put me in a state of depression because I was under the constant pressure to avoid being wrong, in order to avoid confrontation. I could never come to 100% certainty that my actions would result in the intended goal, and this would incapacitate me. Since I did not want to do something wrong, I would do nothing.

However, I have come to a simple, trivial revelation: action and self-expression are inexorably linked. One logically implies the other. Whenever you do anything in this world, you are making a statement, viz. “I assume (it’s possible) this course of action will result in the end sought after.”

I argue that, on some level, most people are aware of this, and sometimes respond. Chances are, there will always be a person who will tell you that you are wrong. Most of the time, however, this person never gives any viable alternative. Its a case of diagnosis without prognosis. I say: never take these people seriously.

If you don’t assume its possible for the action to produce the intended effect, then why perform the action? All action implies ends and means. Sometimes, the end is only a mere possibility, this is called a chance, and sometimes, its difficult to know your odds. However, if you don’t work under the assumption of this possibility, then this results in no course of action.

Therefore, when a person gives you a critique, observe closely whether they give any viable alternative. If they do not, ignore their ramblings all together, for non-constructive criticism may be entertaining sometimes, but its not useful.

When attempting anything, don’t think too much about others expectations. Most of the time, you are in a situation where the person cannot give you a precise answer. Sometimes, a person is ambiguous because they don’t know the answer themselves. In an attempt to give you their best answer, you end up with an unsatisfying answer that doesn’t address your problem. Most of the time, it’s not worth trying to sort this out with the person in question, because most of the time, the person is not entirely aware of this inadequacy, and don’t want to readily admit their own inadequacies.

Instead, think: “How do I think this should be done?” Look at it as a matter of self-expression. If a person gives you non-constructive criticism, then they are providing nothing but their own statements of self-expression. However, in this case, their statements are of a negative nature, e.g. “It should not be done the way you’re doing it, but I don’t know how you should be doing it.”

No matter how strongly a person responds, unless you are physically harming them, ignore them! Don’t let them incapacitate you! Chances are, you will always bother someone, even if no words come out of your mouth. As I said, all action is self-expression. You are making a literal statement, and just like statements using words, there is always a chance that they will vociferously respond to your actions in a negative manner.

Scholactics on Numbers

Here is a Scholastic view on the concept of number. Of course, since quantity is a fundamental category, quantity itself is undefinable, since the whole point of categories is that they are the most basic, irreducible concepts that humans have. This is in Latin, and can be understood if placed into Google translate, but its very difficult to decipher. Unfortunately, the logic museum only has a partial translation of this whole piece, which doesn’t get this far. This comes from Summa Totius Logicae, tract. 3, c. 1. It was previously thought to be authored by Aquinas, but this is now thought to be incorrect, and the author is unknown. This is especially interesting to me since I am in the process of reading Frege.

Quantitas dividitur in continuam et discretam. Dicitur autem discreta quantitas, cujus partes inter se ita se habent, quod sunt separatae, et ad unum communem terminum non copulantur: partes enim hujus numeri qui est decem, ad nullum communem terminum copulantur: non enim in numero qui est decem invenitur aliqua particula ad quam copulentur aliae particulae, cum omnes particulae ejus sint separatae una ab alia. Continua vero quantitas dicitur, cujus partes ad unum communem terminum copulantur, quia omnes sunt conjunctae, et non sunt actu separatae, sed sunt separabiles, ut infra dicetur. Dividitur autem quantitas discreta in numerum et orationem. Est autem numerus multitudo ex unitatibus aggregata. Aliter autem definitur numerus sic: numerus est multitudo mensurata per unum. Ad videndum autem praedictas definitiones, sciendum est quod unum convertitur cum ente et unum est principium numeri. Unum autem primo modo sumptum, nihil aliud est quam ens indivisum. Addit autem unum supra ens, negationem seu privationem divisionis. Et quia omne ens est unum isto modo sumptum, ideo unum sic sumptum non solum est in genere quantitatis, sed in omnibus generibus sicut et ens; et ideo unum est de transcendentibus, et multitudo causata per unum isto modo sumptum, non est numerus qui est species quantitatis; sed est de transcendentibus: dicimus enim esse quatuor Angelos vel tres personas in divinis, et tamen nec in Angelis nec in Deo est quantitas. Unum autem quod est principium numeri, addit super unum quod convertitur cum ente non rem aliquam, sed concernit illud addendo sibi duas rationes: scilicet quia dicit non omnem indivisionem, idest non dicit omne ens in quantum est indivisum, sed dicit ens indivisum quantitatis continuae, et dicit rationem mensurae discretae. Quia enim numerus, qui est species quantitatis, causatur ex divisione continui; supponatur quod divideremus unam lineam in multas partes: cum quaelibet pars lineae quae sic dividitur sit indivisa, et linea sic considerata est unum: unde unum nihil aliud est quam continuum indivisum. Unum ergo quod convertitur cum ente, dicit ens indivisum, quodcumque sit illud. Unum autem quod est principium numeri, dicit ens continuum indivisum; et numerus ex talibus unitatibus aggregatur, ubi sunt multa continua divisa ab invicem, et in se indivisa. Secunda ratio quam addit unum quod est principium numeri, supra unum quod convertitur cum ente, est ratio mensurae discretae. Ubi nota, quod mensurari discreto potest sumi dupliciter. Uno modo, ut idem sit quod certificari apud intellectum, quot sunt aliquae res, in quantum aliquis per unum aliquoties replicatum certificatur de numero illorum quae numerat; et isto modo sumpta mensura est proprietas accidentalis ipsius numeri, et convenit etiam uni quod convertitur cum ente. Alio modo sumitur mensurare pro eo quod est facere tot res formaliter sicut albedo formaliter facit album; et talis mensura est de ratione unius vel numeri. Habemus ergo quid est unum ex cujus aggregatione fit numerus, et qualiter sit mensura essentialiter vel accidentaliter. Ad videndum autem quid sit multitudo, sciendum quod, ut supra dictum est, primum quod intellectus noster intelligit est ens. Secundo vero intelligit negationem entis, prout intelligitur aliud non esse hoc ens. Ex his duobus statim intelligit divisionem; unde divisio est distinctio per ens et non ens. Tertio intelligit unum, quod privat divisionem: est enim unum ens in quo non est praedicta divisio; et sic intellectus unius posterior est intellectu divisionis, sicut intellectus privationis posterior est intellectu habitus quem privat. Quarto intelligit multitudinem: quae dicit duas negationes: quarum una est, quod hoc non sit illud: altera est, quod quodlibet eorum non sit divisum, et per consequens sit unum. Et propterea multitudo definitur per unum; quia nunquam dicuntur multa, nisi quodlibet eorum sit unum, seu ens indivisum. Et sicut modo sumpsimus ens et divisionem, et unum et multitudinem in transcendentibus, sic sumatur in quantitate; ita quod sicut sumebatur ens, ita hic sumatur continuum, licet aliqua differentia sit inter ea, ut supra dictum est. Sciendum quod haec multitudo quae est in quantitate, est multa continua, quorum unum non est aliud, et quodlibet eorum est indivisum in se, sive unum, quod idem est. Et sic patet prima definitio numeri, scilicet: numerus est multitudo ex unitatibus aggregata. Patet etiam secunda definitio, scilicet, numerus est multitudo mensurata per unum: nam quia unum multoties replicatum causat multitudinem, per hoc certificamur de multitudine quanta sit discrete, et sic unum est mensura multitudinis, et patet de numero et cetera.

LK Doesn’t Accept Subjective Value

Link

First of all, why in the world does LK find Austrian Economics so fascinating? If his goal is to deconstruct their theories, he is palpably bad at it. Diagnosis without prognosis is all well and good, and I engage in it often, but at least libertarians make clear what their alternative is. He has no discernible alternative. I can only conjecture that he thinks the current status quo is fine and the state needs more power to accomplish its goals.

To brush off the potential of some inane philosophical arguments right off the bat, value can be subjective or objective. There are no other options. Value, as meant in this context, is defined by Merriam-Webster as “relative worth, utility, or importance,” i.e. the value of any conceivable object is the utility this object posses. We can use George Reisman’s definition of marginal utility from Capitalism, “The law of diminishing marginal utility states that the utility or, equivalently, the importance or personal value that an individual attaches to a unit of any good diminishes as the quantity of the good in his possession increases.” We will use the Stanford Encyclopedia of Philosophy‘s definition for incommensurable and incomparable, “this entry uses the term “incomparable” ordinarily to describe two or more concrete bearers of value of which no positive comparative evaluative judgment is true. In contrast, this entry uses the term “incommensurable” to describe the way in which two or more abstract values stand in relation to one another.”

First things first, since LK fails to define nor understand modern Austrian subjective value theory, I will sum up the main tenets as I see them. Values are comparable, only within a single anthropomorphic agent. They are incomparable between agents. They are not quantitatively commensurable, but are only subject to the relational categories of “more” and “less” (ordinal, not cardinal). Furthermore, its only possible to apply these relations to actualities, not potentialities, as in those actions which have actual existence (or hypothetical actual existence), not mere possible actions which have not yet been actualized. Actualization refers to the action’s completion, i.e. the attainment of the end sought. Potentialities may refer to other means for same end which were not chosen, or an entirely different course of action with different ends which was not chosen. Therefore, “more” and “less” only refer to a single instance of choice between one actuality and one potentiality that is presented to a single anthropomorphic agent [1]. Applications of this principle will be shown.

On to LK’s criticism of Murphy,

“The Austrian argument against progressive taxation, from Robert P. Murphy’s Lessons for the Young Economist (2010), is as follows:

“If preferences are subjective to each individual, and cannot even be measured or quantified for each individual, then obviously it would make no sense at all to try to combine or aggregate individual preferences into ‘social’ preferences. Unfortunately, even professional economists often engage in just this type of reasoning. Many people (try to) justify progressive income taxation, for example, by claiming that ‘a dollar means more to a poor man than to a rich man.’ The idea is that taking $1 million from Bill Gates won’t lower his utility very much, whereas handing out $1,000 to a thousand different homeless people will greatly boost each of their utilities. Therefore, the typical argument goes, total or “social” utility has been increased by the redistribution of some of Bill Gates’s wealth.

… For now, we point out that the typical justification for it is absurd. You can’t add up different amounts of utility from various people. In fact, if you use the alternate term preferences it will be more apparent why combining them from different people is an impossible task.” (Murphy 2010: 42).

The trouble with this is that, just because it is impossible to aggregate the subjective utilities of different people or find some objective unit of measurement with which to make objective interpersonal utility comparisons, it does not follow that an economic argument for progressive income taxes, on basis of diminishing marginal subjective utility, has failed.”

Here is the thesis of the blog post, in which I assume I will come out of this understanding why this particular argument fails. With the typical obscurity in his explanations, when he says it’s impossible, I don’t know whether he is concurring, or just taking as given for the sake of argument. Obviously much rests on this. I will take this as him concurring, though I will probably be chastised.

“If any Austrian economist accepts the “law” of diminishing marginal utility (or accepts it merely as a general principle), it follows that a very rich person should, generally speaking, derive less utility from an extra dollar than a person who is very poor, even if one cannot measure the utility in some objective quantity like “utils.””

There is no refutation here. LK cannot possibly know whether any person “derive[s] less utility from an extra dollar than a person who is very poor,” without the actualization of the particular action. If a rich person buys a sandwich from a poor person, we can say that the rich person valued (derived more utility) the sandwich more than the money in the cost of the sandwich. Also, the poor person valued the money in the cost of the sandwich more than the sandwich. The terms “more” and “less” cannot be assigned unless referencing an actualized choice and to say any person “values” an extra dollar is meaningless unless referring to the actualization of an action. Saying a person “values” a dollar has the exact same connotation and intention as saying a person is actually engaged in the action of the possession of a dollar.

To say a poor person values an extra dollar more than a rich person would involve the inter-comparison between two individual actions of two different anthropomorphic agents. Physical science simply has no objective way of expressing this relation. It can describe if Joe ran more than Bob in terms of distance. It can describe if Joe is older than Bob in terms age. Joe values something more than Bob, in terms of… what? The brain’s production of dopamine? All in all, the examples of age and distance have physical existence, while value only has conceptual existence. This conceptual existence is a construct for the purpose of describing actualized actions. When LK says that a rich person values a dollar less than a poor person, this can never be a scientific statement, but one of LK’s own normative ethical theories or arbitrary personal opinion.

“If indeed there is good reason to think that the value of an additional unit of income to a person who is already very rich is considerably less than the value of an additional unit to someone who is poor, then redistribution of income to promote happiness and reduce hardship has sound economic justification.

And indeed empirical evidence seems to show that as wealth rises, the happiness that one derives from additional income falls or levels off after about $70,000 (US) (e.g., a fascinating discussion of this topic here).”

He forgot to temper his argument with the word “probably,” because his argument rests on the approximation of his own personal conception of value. Of course, if it were approximately true that “the value of an additional unit of income to a person who is already very rich is considerably less than the value of an additional unit to someone who is poor” than this would yield exceptions, which involves problems with his ethical theories. Apparently, any individual values of individual people must yield to the broader good, which is determined by LK’s subjective opinion.

Moreover, this is approximately true according to the statistics of what measurements? No physical evidence is presented in the linked post. The post refutes some journal article, “Note that past studies found that after annual income reaches around $70,000 (in adjusted dollars), gains in happiness become too small for continued acquisition of wealth to be a significant factor in making you happy.” No link to these past studies are provided. These past studies normally rely on polls, which ask people questions, where people answer according to their own subjective conceptions of happiness. The numbers assigned to happiness are applied by the individual people according to their own subjective notions, then aggregated as if they were equivalent units. Of course, no justification for this is supplied. These units don’t have any physical existence and are therefore not physical evidence of anything except what historically existed inside the brains of those who were asked during the moment they were asked. There are so many more problems in addition to this making these polls unscientific for LK’s purposes.

Curiously, it was none other than the Austrian economist Friedrich von Wieser who prided himself on having provided a solid economic justification for progressive taxation on the basis of diminishing marginal utility. Modern Austrians apparently choose to forget this embarrassing fact.”

Indeed, Wieser argued that this particular type of taxation has no “offence to the spirit of the private constitution of the economy itself.” His argument, unlike LK’s, does not rely on the inter-comparison of the differing actions between differing anthropomorphic agents. In addition, Weiser, nor anyone, has a justification of precisely how much taxation should be applied as a function of an increase in income, because, as already explained, this isn’t possible. Therefore, any rate or function assigned is according to LK’s opinion. Also, why would Austrians want to refute a relatively obscure author from decades ago? Is this so embarrassing? Austrians are not a homogeneous blob. Its not a big deal. Modern Austrianism has become affiliated with Mises, with the older Austrians providing its historical foundation. It has changed throughout the years.

“The argument against progressive taxation that we cannot objectively “measure” the utility lost by the rich man as compared with that gained by the poor man does not necessarily refute the argument from diminishing marginal utility: for it requires a highly unrealistic assumption, as we shall now see.

 

If we were to take two poor people both with the same income, and imagine one becoming extremely rich while the other remains poor, the argument against progressive income tax could only work if the utility the rich man derived from a unit of money while poor was vastly – and indeed unrealistically and extremely – greater than that of his fellow, so that as each additional unit of money the man received – even to very high levels like millions or billions of dollars – the diminished utility still remained so high that it exceeded that of his fellow who still remained poor.”

When he says “unrealistically and extremely,” there is no demonstration, and no possible demonstration. Again, he has strong opinions that he does not back up with any physical evidence. Also, what is LK’s “argument from diminishing marginal utility”? He supplies Weiser’s argument, but does he ever supply his own argument? Who knows? Probably not LK.

“Now of course individuals display variation and can and do have different degrees of subjective utility in terms of the satisfaction that they derive from any good x (and even from a unit of money), but to believe that all or most rich people derive greater utility from one unit of their money than a poor person from one extra unit again requires the ridiculous assumption that, if (hypothetically) or when (in reality) they were poor, all or most of these rich people derived a degree if utility vastly – and indeed unrealistically and extremely – greater than that of other poor people. “

“satisfaction that they derive from any good x” is not a physical quantity. “a unit of money” is not a measurement of the type of value thus far considered. He is confusing objective exchange value with subjective use value, which are entirely separate concepts. Money has no intrinsic relation to psychological value and LK fails to show this relation.

He also entirely misses the point, probably on purpose. “to believe that all or most rich people derive greater utility from one unit of their money than a poor person from one extra unit again requires the ridiculous assumption.” The point of subjective value theory is the absence of this assumption. The praxeologist is much more reserved and does not make any assumption about the relation between differing actions between differing anthropomorphic agents. Spinning this to make it seem as if they make this assumption is a strawman, which is very characteristic of LK’s style.

“This, quite frankly, violates everything we know about human psychology, neuroscience and evolution. Human beings are all products of Darwinian evolution; they have the same fundamental biochemistry and neural processes in the brain; the mind and all its emotions, like happiness, satisfaction and pleasure, are causally dependent on brain processes. People do display individual variation in many traits – such as height, eye colour, and no doubt in what economists call utility – but not to the extent that average people have such a vast difference between them as would be required in the case we have imagined above.”

Indeed, he may be correct when he says, “happiness, satisfaction and pleasure, are causally dependent on brain processes,” but does he provide any measurements of any brain processes? Without these physical measurements, again, the argument is not supported.

“But we need only think of height here. Most human beings have a height between 5 feet and 6 feet, and even exceptions (apart from highly usually things like dwarfism and gigantism) do not deviate too far from this range. Height is a product of genetics and environmental influences. There is every reason to think that the propensity to feel emotions like happiness, satisfaction and pleasure – the emotions that the word “utility” in an economic sense describes – are a product of genetics and environmental influences too, with individual variation, but not so vast that the utility felt by two average people while poor is so vastly different that one million dollars or $100 million – under the principle of diminishing marginal utility – given to one man would still not reduce his utility from one extra dollar to a level below that experienced by the other poor man from one extra dollar.”

Even though there is “every reason to think,” LK does not provide any of these reasons. Perhaps if they were so self-evident to his genius mind, he could elaborate on them. Height being relatively invariable does not syllogistically conclude that any of the subjective psychological states mentioned are relatively invariable. Also, relative to what? These heights are relatively giant in the microscopic scale. Height is physically measured in space, numbers assigned to psychological states are arbitrary. Again, “given to one man would still not reduce his utility from one extra dollar to a level below that experienced by the other poor man from one extra dollar.” Reduce to a level below in terms of what exactly? No physical evidence is given for the assertion, as usual. Need I point out more evidence of his lack of empirical justification?

“In short, Austrians, like neoclassicals, if they really accept the “law” of diminishing marginal utility without the ridiculous assumption we have identified above, then the economic argument for progressive taxation from diminishing marginal utility is hard to refute.

Of course, they might make a moral argument from Rothbardian natural rights or Hoppe’s argumentation ethics, but this is clearly a different type of argument from the one based on subjective utility.”

In short, LK failed to point where Austrians even make such an assumption. The theory of subjective value makes the opposite point in that the economist is unable to make any of these assumptions. Ironically, LK’s error is the one he attributes to those he is criticizing. He assumes that it is possible to make the inter-comparison of differing actions between differing anthropomorphic agents. However, no demonstration is given, nor any physical evidence supplied.

There is no reason to bring in the ethical theories of Rothbard and Hoppe, since this has nothing to do with the economic argument presented by Murphy. Many of those who are Austrian economists unmentioned by LK have different ethical theories. Again, LK chooses a select few as if they were a single homogenous blob.

[1] Value scales are mere explanatory tools under unrealistic conditions.

Every Philosopher has a Normative Criteria

I was thinking about Russel’s criticism of Aquinas, as shown on Wikipedia,

“He does not, like the Platonic Socrates, set out to follow wherever the argument may lead. He is not engaged in an inquiry, the result of which it is impossible to know in advance. Before he begins to philosophize, he already knows the truth; it is declared in the Catholic faith. If he can find apparently rational arguments for some parts of the faith, so much the better; if he cannot, he need only fall back on revelation. The finding of arguments for a conclusion given in advance is not philosophy, but special pleading. I cannot, therefore, feel that he deserves to be put on a level with the best philosophers either of Greece or of modern times.”

I partially agree and disagree with Russell.

I full-heartedly disagree that he doesn’t “deserve” to be put on a level with the best philosophers. He was one of the best, in my humble opinion. Although I’m unsure whether Russell would agree, he was obviously the foremost expositor of Aristotle’s ideas during the time in medieval Europe known as the “dark ages” because of how regressive the period was. Imagine if he, nor other scholastics, historically existed. We would have a much different academic world if this was the case. Aristotle may not be as respected as he is now. Perhaps he would have even been completely lost?

His theories of sense and memory formation are (I think) original, or at least partially original. According to Wikipedia, or some other source I can’t remember, his are the most close and accurate description of how the mind works according to modern cognitive science. I will try to reproduce this source if pressed, though I can’t seem to find it off hand.

In addition, his contemporaries described him to be a genius. Again, I think I remember this from a lecture.

I partially agree that “Before he begins to philosophize, he already knows the truth.” However, I don’t think that this is as uncommon as one might first superficially understand. To a degree, most have a goal when philosophizing. I don’t know if there is a school of thought who purely “set[s] out to follow wherever the argument may lead,” because choosing a particular form of argument is itself not established by the physical world. Logic is a normative discipline, not one grounded in observation of physical reality. Like ethics, it is prescriptive, not descriptive.

One is not physically bound to the “law of contradiction”, for example. If I break this, so what? The consequences are social, not physical. People would most likely ignore me if I don’t make any sense. Therefore, I find that following the rules of logic are indispensable for my life because communication is necessary for my survival. Of course, this explanation grounded on human survival is one that may or may not be correct, but I digress.

Back to matter at hand, the reason that I partially agree with Russell is because I think theology is boring. The subject has nothing to do with logic or science. If desired, perhaps it can fall into a broader notion of science, as in any subject that follows from a set of principles. However, isn’t it true that every theologian basically has their own principles? I admit that I am not well versed in theology, coming from a not very religious family. It seems to me that the method behind theology is arbitrary. If I explained God as a flying spaghetti monster, on what ground can you refute me? I think I share my contention with HL Mencken when he said, “Theology is the effort to explain the unknowable in terms of the not worth knowing.”

Now I am sure that Russell probably had a more nuanced view than I am presenting. However, this sentence strikes me as plainly incorrect: “The finding of arguments for a conclusion given in advance is not philosophy, but special pleading.”

Take Frege, Russell’s precursor, as my case in point. His goal was to ground numbers in the realm of logic. He most definitely had this conclusion in advance. Frege thought that is should be true that mathematics had a basis in logic, so his entire philosophy set out to accomplish this task. Yet, I am pretty sure Russell would think Frege deserves to be considered on the level of the best philosophers.

A Simpler Demonstration About the Nonexistant Liar Paradox

I remain convinced that syllogistic logic is immune to the liar. I don’t know why its immune to the liar, but there is simply no way to form a valid syllogism in which the liar makes an appearance. Here is where the confusion arises in my view.

In symbolic logic based on the works of Frege, a letter represents either judgable or unjudgable content. For example, the noun “house” is not judgable content. “The house exists” is judgable content. If A is judgable content then,

⊢A

is valid. In traditional form, it would simply be “A is true.” However, if A is unjudgable content, then

⊢A

is invalid. This would amount to “house is true,” which is nonsense.

Frege’s goal was to put logic into terms where it could more easily be analyzed. Modern systems are based on his. In Russell’s Prinipia Mathematica, he writes the law of excluded middle as

⊢. p V ¬p.

In traditional logic, Aristotle said, “Of two contradictory judgments, the one must be true and the other false.” (Metaphysics, Book III, chapter 8) In symbolic logic, this would amount to

⊢A⊕⊢¬A

which is different than Russell’s formulation. In the truth table

A	¬A
T	T
T	F
F	T
F	F

the first and last rows are excluded. In Russels formulation, only the last row is excluded. However, since the law of contradiction applies, the first row is also excluded in his system, so its no problem for Russell’s system. The point is, all this his only makes sense if A is judgable content. If it were not, then the truth table could not be applied to A.

For some reason, the liar paradox is often given in regular language. We can try to construct the paradox by saying

All said by the liar is false.

A is said by the liar.

A is false.

 

Through an incorrect version of the law of excluded middle we have

If A is false, A is true.

A is false.

A is true.

 

Now we have a contradiction. However, this is not the correct law of excluded middle. Applying the correct law of excluded middle we still would still have

All said by the liar is false.

A is said by the liar.

A is false.

 

With the correct law of excluded middle we have

If A is false, ¬A is true.

A is false.

¬A is true.

 

Since ¬A was never asserted by the liar, this is where the train ends. There is no paradox.

 

Mathematical symbols are also used in instances explaining the liar paradox. We may construct

A is false.

A = “A is false”.

Through substitution we have

“A is false” is false.

A is true.

Hence, a contradiction. However, in traditional logic, there is no principle allowing the transition from the second to third step. In traditional logic, “is equal to “A is false”.” is predicated of A. This is not the same as a mathematical “=” sign. The copula “is” has a much different signification in traditional logic. Its meaning is entirely different. In traditional logical form, it would be expressed as a classic figure 3 syllogism.

A is false.

A is equal to “A is false”.

Some (that which is) false is equal to “A is false”.

or, if the premisses are switched around,

Some (that which is) equal to “A is false” is false.

There is no paradox here. The reason that the subject is not plainly “false” or plainly “equal to “A is false”” is because in traditional logic, subjects are always substances, and they are never accidents. This distinction was invented by Aristotle, for more information go here: http://plato.stanford.edu/entries/aristotle/#Hyl You could have also added parenthesis in the premises if desired. A is (that which is) false. They have identical meanings. Its needed in the conclusions because, if omitted, the sentence sounds awkward, but it would still be technically correct, grammatically, if the parenthesis was omitted.

Substitution is valid in Fregian function and argument type constructions. e.g. f(x) and f(y). However, the function and argument never strictly replaced the subject and predicate. For all intents and purposes for analysis, it did replace the subject and predicate form, but the subject and predicate did not cease to exist. Instead, Frege ingeniously subsumed them within his system for the sake of brevity. For easier and clear analysis, he lumped all content into a subject. If the content was judgable, then he merely asserted it, or not. As explained in Begriffsschrift, “Such a language would have only a single predicate for all judgments, namely, ‘is a fact’.”

 

Gupta and Belnap in The Revisionist Theory of Truth, express the liar paradox on page 5 as follows

“(The Liar) The Liar is not true.
Now it is easily seen that the T-biconditionals for English imply a
contradiction. For we have
(2) The Liar = ‘The Liar is not true’,
and the T-biconditional for The Liar is
(3) The Liar is not true’ is true if and only if The Liar is not true.
By substitutivity of identicals we deduce from (2) and (3) that
(4) The Liar is true if and only if The Liar is not true,
which immediately yields a contradiction.”

However, the principle labeled “substitutivity of identicals,” is never explained. Obviously, the error here is that this principle is invalid or that this principle is imaginary.

More of my Untenable Thoughts on Euclidean Geometry

The parallel postulate is an axiom, and the real world is more easily described by non-Euclidean geometry. This is not to say that Euclidean geometry doesn’t apply to the real world.

 

To say that only Euclidean or non-Euclidean geometry applies to the real world are both metaphysical statements. There is no evidence that denies Euclidean geometry from applying to the real world, because nothing says its IMPOSSIBLE for you to describe space and light as Euclidean.

 

In fact, no evidence can prove or disprove whether Euclidean geometry applies to the real world or not. It would be silly to attempt this. Its an arbitrary designation made for modeling.

 

However, to say that the mind perceives things as Euclidean is correct. You cannot visualize any non-Euclidean geometry, without visualizing it as Euclidean geometry. That is mental capacity of our mind, and is an inductive observation.

 

I can prove this by asking you to draw an instance where the parallel postulate is invalid. Since its not possible for you to draw this, then its sufficiently proven.

 

Schopenhauer had similar arguments shown here: http://en.wikipedia.org/wiki/Schopenhauer%27s_criticism_of_the_proofs_of_the_Parallel_Postulate

 

So when people say the universe is Euclidean, it is the same as saying the mental capacity of the human mind only perceives Euclidean space. Objectively, the universe is not necessarily Euclidean, non-Euclidean, or any geometry at all. Any geometry is a mental analytical conception.

 

This is where the confusion arises. Mises never used the term “axiom” because the basis of praxeology is formed of statements about the nature of how humans perform actions, which is inductive.

 

Everything derived from this is apriori, and is based on the analysis of the causal factors of human actions.

 

This isn’t like geometry because these are not constructions for the purposes of modeling reality. They are observations about reality itself. They are inductions from introspection that are validated by literally asking any person.

Traditional Logic on the Problem of Multiple Generality

Some say that some statements are not addressed well by traditional logic.

e.g.

Some cat is feared by every mouse.

and

All mice are afraid of at least one cat.

 

Now I could not find this specific controversy on http://plato.stanford.edu/, so I only have the wiki, http://en.wikipedia.org/wiki/Problem_of_multiple_generality.

 

The converse of these propositions are both treated in traditional logic. The first is a singular proposition, and yes, singular propositions exist in traditional logic. The Wikipedia article is wrong when it says, “The syntax of traditional logic (TL) permits exactly four sentence types.”

 

Traditional logic also permits indesignate types and singular types. The converse of the singular proposition,

Some cat is feared by every mouse.

becomes,

One feared by every mouse is some cat.

That was easy!

 

The second is a general proposition.

All mice are afraid of at least one cat.

becomes,

Some afraid of at least one cat are mice.

 

The subject and predicate do not depend on the structure of the sentence. Not every colloquial sentence is formed as subject copula predicate. In all singular propositions, the subject is always the singular individual. Sometimes in propositions, you may reform it if you want. “All mice” is a generality. “At least one cat” is a generality. “Every mouse” is a generality. “Some cat” is a singular individual. In addition, we are dealing with the predicate as either “afraid of” or “feared by”. We can change a subject into a predicate through the use of these principles. We may change actions into passions. However, by themselves, the sentences are in logical form and need no more additions. If we ever wish to reform propositions, it sometimes depends on the context. The main objective is to always try and convey the real intended meaning. Sometimes, its not even possible. We can change the first sentence from

Some cat is feared by every mouse.

to

Every mouse is afraid of some cat.

We have now changed the logical subject, but there is no change in meaning. We can do this in these instances because actions can be converted to passions and vice versa. The converse of this is,

Some afraid of some cat is every mouse.

 

The way you decide to present the proposition in logical form shows how you think these are logically divided. Sometimes logical divisions are natural, and sometimes logical divisions are artificial. Dividing a category of “afraid of some cat” into a subset of “every mouse” is an artificial division, for example. “Some cat is feared by every mouse” is a singular proposition, so its not based on any division. This is the type of insight that traditional logic gives, a way to organize knowledge in the mind.

 

According to the Wiki:

“Using modern predicate calculus, we quickly discover that the statement is ambiguous.”

However, the the two propositions were not the same propositions. Its not possible to convert one into the other. “Modern predicate calculus” gives us,

“For every mouse m, there exists a cat c, such that c is feared by m”

or

“There exists one cat c, such that for every mouse m, c is feared by m.”

These, for some reason, are both formed from the original proposition. Logically, the proposition was always in the form,

Some cat is (feared by every mouse)

Where the object in parenthesis is the predicate.

(Some cat is feared) by every mouse

Here, what are they designating with the parenthesis? Its not the predicate, so I have no idea.

 

All in all, once I comprehend the appeal of this system, I will blog about my giant revelation.

Robbins and Greaves on the Fundamental Postulates of Economics

Stated by Lionel Robbins in An Essay on the Nature and Significance of Economic Theory (1935):

“The propositions of economic theory, like all scientific theory, are obviously deductions from a series of postulates…The main postulate of the theory of value is the fact that individuals can arrange their preferences in an order, and in fact do so. The main postulate of the theory of production is the fact that there are [sic] more than one factor of production. The main postulate of the theory of dynamics is the fact that we are not certain regarding future scarcities. These are not postulates the existence of whose counterpart in reality admits of extensive dispute once their nature is fully realised. We do not need controlled experiments to establish their validity: they are so much the stuff of our everyday experience that they have only to be stated to be recognised as obvious (pp. 78-9).”

Enumerated below he have:

1. Individuals can arrange their preferences in an order, and in fact do so.

2. The fact that there is more than one factor of production.

3. The fact that we are not certain regarding future scarcities.

Note: In (2) I changed “are” to “is” because I think it was a grammatical error.

 

Greaves in Understanding the Dollar Crises (1973) gives these postulates in different forms explained in pages 27 through 29:

1. All men seek to improve their situation from their viewpoint.

2. The factors available for improving men’s situations are scarce.

3. Men make mistakes.

 

I think Robbins’ formulation of the first two postulates are worse than Greaves. Regarding (1), individuals don’t actually have an arrangement of preferences in their mind at all times. When they act deliberately, they always only choose between two options in each instance. During an instance of action, they don’t consult and refer to their psychological preference scale. They decide on the merits of each individual instance (I am speaking from introspection).

Therefore, a person is only presented with an individual option in each case. After the presentation, the choice is made between the old option and the new option. After the choice, one option remains as the person’s necessary reality. There is a sequence of events across time:

1. Presentation of new option

2. Choice between two options

Now across time, nothing is 100% ceteris paribus. However, we may pretend that it is ceteris paribus as a thought experiment. Lets say someone is doing A and they are presented with option B. Option B may be present in the subject’s memory, or it may be presented by their sense, either through observation or by another person informing them. At this time point, they choose option A or B, so either A is above B or vice versa.

Let’s say A is above B. Now they are presented with C and they choose A again. Now given these conditions, we now know that A is above both B and C. However, we don’t know whether B is above C or if C is above B. We can never know this given these conditions. Furthermore, if we rid ourselves of the ceteris paribus condition, then at this point in time, B could now be above A. Things could have changed during the time elapsed when the person is presented with option B and when they are presented with C.

All in all, constructing scales is contingent on the ceteris paribus (CP) condition. If valid, then we can know that A is above both B and C. Now, let’s say when presented with C, the person chooses C. We can now construct their full scale of A, B, and C. It would be, C is above A, and A is above B. The ordinal preference scale would be:

1. C

2. A

3. B

The point is that preference scales can only be constructed given the CP condition and the conditions regarding the actual choices taken by the subject. At least three specific conditions went into constructing this preference scale: ceteris paribus, A was chosen above B (choice 1), and C was chosen above A (choice 2). So far we have one CP condition and two choice conditions.

These conditions are not even sufficient for universalizing the construction of scales. Remember we could not construct this scale when A was chosen above C, since we then cannot know the relation between B and C. After the initial option gives the relation between two preferences, if presented with a third option, the person must choose this new option for the construction of the scale to continue. This is the final condition required for the construction of value scales. Stated differently, under every choice after the first choice, the person must choose the new option. We can now universalize these conditions for any construction of preferences scales.

To construct a preference scale of n preferences, it requires the CP condition, n-1 option conditions, and the condition that after the first choice, the new option is always chosen.

Greaves’ first postulate is far more modest. We said earlier that a person is presented with the new option, then they make a choice. His postulate gives us the ability to say:

When a person chooses, this person prefers the option chosen over the option not chosen.

Robbins’ postulate doesn’t justify us doing this. Greaves’ postulate gives this justification and is therefore a more fundamental postulate. One thing that should be noted is that Greaves’ postulate does not take time into consideration. In reality, a person can make a choice and regret their choice upon reflection, because in reality, things are not ceteris paribus. Therefore, this postulate is only strictly universally valid for the brief instance of time when choice is made. Sometimes, a person could potentially regret their decision even after a millisecond.

Let’s say a person is drunk in Vegas. They are gambling and decide its a good idea to spend a lot of money for very low odds. They spend their money and immediately regret their decision. However, it still takes time, however brief, for them to regret their decision. During the instance of the choice, they preferred the low odds over their money.

 

Robbins’ second postulate is much different than Greaves’ second postulate. They are opposites in a certain sense. Robbins says there is always more than one factor, while Greaves says there is never an infinite number of factors. Robbin’s postulate is false in many instances, only Greaves’ postulate is universal for all instances.

Let’s illustrate Robbins’ example with haircuts. We have a small isolated town with one barber. Let’s condition this scenario making it impossible that anyone can move out of the town. He then enjoys a monopoly in this small town. Strictly speaking, however, this is not true. Hazlitt explains in Free Market Economics: A Basic Reader (1974),

“There has been a tremendous literature within recent years deploring the absence of perfect competition; there could have been equal emphasis on the absence of perfect monopoly. In real life competition is never perfect, but neither is monopoly.”

You see, even if this barber is the only place trading haircuts for money, they still may choose the option of cutting their own hair. They also may choose the option of just not cutting their hair. Therefore, the barber is still competing with the feasibility of self grooming and the feasibility of not grooming.

The only way in which Robbins’ postulate is invalid is if there is absolutely no other options. For example, Robinson Crusoe lives on an island and has no tools for cutting his hair and its not possible for him to make these tools. In this instance, there is absolutely no other options. Since there is only a single option, there can only be a single reality. However, this either rests on Crusoe being aware and convinced that no other options exist or completely oblivious to other options, i.e. his subjective determinations or subjective experience. Therefore, Robbins’ formulation is contingent on one of these conditions.

On the other hand, Greaves’ second postulate is a statement about the objective, and is not contingent on any subjective determinations or experience. Its statement is equivalent to the physical conservation laws accepted in academia, e.g. energy may not be created or destroyed. If true, our universe has a total supply of energy. The amount of useful energy may seem almost infinite to us now, but is subject to ultimate limits. In most instances, we usually consider scarcities that are prevalent. Most people are unaware of the ultimate scarcity of energy, but many people are aware about the amount of useful electrical energy personally available to them.

 

Robbins’ and Greaves’ third postulate is about subjectivism. I think Robbins formulates this postulate better than Greaves. In my opinion, the uncertainty about future scarcity is based on a more fundamental statement about certainty. Certainty is a modal determination about the future. Either something in the future is necessary, possible, or impossible.

The law of causality tells us that certain causes produce the same effects under the same conditions. These give us necessary causal relationships, e.g. under the correct conditions, gravity always causes things to fall, or water is formed through the chemical combination of hydrogen and oxygen under the correct conditions.

Scientific principles give us impossibilities. For example, the Pauli exclusion principle says its impossible for two identical fermions to occupy the same quantum state simultaneously, or the conservation of energy tells us that its impossible for the total energy of an isolated system to change.

All other predictive statements are mere possibilities. These can be quantified as probabilities. However, probabilities are of two kinds, viz. probabilities based on historical measurement and probabilities based what Nancy Cartwright terms “singular causes.” These are probabilities like the wave functions describing the quantum state of a system of particles. These wave functions are calculated from the fundamental properties of the particles themselves.

The other kind of probability is taken from the measurement of historical occurrences. Since these probabilities are statements about history, they are not predictions and hence not statements about future possibility. These kinds of probabilities only serve to support predictions made by the former kind of probability derived from singular causes. I quote Cartwright from Nature’s Capacities and Their Measurement,

“My position is opposed to the tradition of Hume. I begin with the assumption that the causal language of science makes sense, and that causality is an objective feature of our scientific image of nature. That does not yet separate me from the Humean tradition. Hume too took causal claims to have an analogue in reality. He began with singular causal claims, looking for some special connection between the individual cause and its effect, a connection that would be strong enough to constitute causation. He failed to find anything more than spatio-temporal contiguity, so he moved to the generic level. This marks the first stage in the Hume programme: (1) for Hume, singular causal facts are true in virtue of generic causal facts. But the programme was far bolder: at the generic level causation was to disappear altogether. It was to be replaced by mere regularity. This is the second thesis of the Hume programme: (2) generic causal facts are reducible to regularities. This book challenges both theses. It begins with the claim that, even if the association is law-like, neither regular association nor functional dependence can do the jobs that causality does. Working science needs some independent notion of causal law. What kind of a concept could this be? I maintain that the Hume programme has things upside down. One should not start with the notion of generic causation at all. Singular causal claims are primary. This is true in two senses. First, they are a necessary ingredient in the methods we use to establish generic causal claims. Even the methods that test causal laws by looking for regularities will not work unless some singular causal information is filled in first. Second, the regularities themselves play a secondary role in establishing a causal law. They are just evidence—and only one kind of evidence at that—that certain kinds of singular causal fact have happened.” (page 1)

Therefore, the third postulate says that predictions about scarcity are always uncertain, i.e. they are always to do with future possibility, and never to do with future necessity or impossibility. Since measurements of historical occurrence tell us nothing about any primary, singular, causal facts, they give no statement about the future at all. They are “just evidence.”

Tying in Greaves’ formulation, since predictions about scarcity are always uncertain, mistakes are possible. However, it doesn’t follow that men will always make mistakes. It follows that sometimes, men make mistakes. In certain situations, men rarely make mistakes, e.g. people don’t usually make mistakes about the supply of money in their bank account. In other instances men often make mistakes, e.g. their supply of money as valued by their stocks.

Greaves nor Robbins tell us why the predictions about scarcity are always uncertain. The reason is that the singular causal mechanisms behind choice have yet to be fully elaborated by cognitive science or psychology. Scarcity is ultimately a judgment made by a person using a supply as a factor for their purposes, e.g. a supply of food is low because it will not last for a desired time. Many philosophers are adherents of libertarianism, which posits the existence of free-will. This means choice is not subject to any singular causal mechanisms, rendering prediction about future scarcity impossible, since scarcity depends upon the supply used for a chosen purpose.

Entrepreneurs can use certain general notions, e.g. Henry Ford knew his customer’s would buy cars because people generally like things that make life more convenient. That’s a pretty safe bet. He knew cars were scarce, because mass produced cars did not even exist until that point. Now, coming out with a new car is not guaranteed to sell unless you are satisfying a new niche market previously suffering from scarcity. This is why hybrid cars and Tesla Moters became successful. However, these general notions are far from universal and businesses often fail even if they follow them.

Suppose it were even possible to determine the causal mechanisms behind the choices of individuals. It may still be impossible to keep these measuring devices inside people’s brains across any given time where a prediction is relevant. In order to have any large scale economic benefit, multiple people would need to be probed for extended periods and monitored. This raises ethical dilemmas. Also, the probes may alter the subjects behavior in undetermined ways. Therefore, the third postulate rests one out of three contingencies.

1. The existence of free-will

2. The impossibility of determining the causal mechanism behind choice

3. The impossibility of ever measuring these causal mechanisms for any group of individuals for any extended period of time where prediction is useful.

The Liar Paradox Deconstructed

Edit: The way plato.stanford.edu presents the paradox is also addressed as an addendum to this blog post. It rests on the same errors as the Wiki. http://plato.stanford.edu/entries/liar-paradox/#SimFalLia

 

Personally, I am convinced that the Liar Paradox is a myth. This blog post demonstrates why the Liar Paradox poses no actual problem for the realm of logic. I will show step by step why the argument ultimately rests on a hypothetical syllogism where the conclusion does not follow from its premises. To be thorough, I will deconstruct all the relevant passages in the Wiki.

http://en.wikipedia.org/wiki/Liar_paradox

This so called paradox is shown:

“If “This sentence is false.” is true, then the sentence is false, but then if “This sentence is false.” is false, then the sentence is true, and so on.”

and

“The simplest version of the paradox is the sentence:

This statement is false. (A)

If (A) is true, then “This statement is false” is true. Therefore (A) must be false. The hypothesis that (A) is true leads to the conclusion that (A) is false, a contradiction.

If (A) is false, then “This statement is false” is false. Therefore (A) must be true. The hypothesis that (A) is false leads to the conclusion that (A) is true, another contradiction. Either way, (A) is both true and false, which is a paradox.”

Each sentence is numbered according to the original order at which they appeared in the article:

1. This statement is false. (A)

2. If (A) is true then “This statement is false” is true

3. Therefore (A) must be false

4. The hypothesis that (A) is true leads

5. to the conclusion that (A) is false, a contradiction.

I will now reduce the sentences into their simple logical form as subject and predicate.

1. This statement is false. (A)

Clearly, this signifies the intention of (A).

*1. (A) is “This statement is false”

2. If (A) is true then “This statement is false” is true

Erasing the unneeded “then”

*2. If (A) is true, “This statement is false” is true.

3. Therefore (A) must be false

Rearranging into simple logical form

*3. (A) is false

Switching (1) and (2) so the order of the premisses in the more popular form [Note 1] of the hypothetical syllogism.

*2. If (A) is true, “This statement is false” is true.

*1. (A) is “This statement is false”

*3. (A) is false

Well here we have it folks. The conclusion simply does not follow from the premisses. We may bypass the rest (4) and (5).

4. The hypothesis that (A) is true leads

5. to the conclusion that (A) is false, a contradiction.

The antecedent in the hypothetical proposition of the major premiss (*2) is “If (A) is true.” In this form of proposition, the truth of the consequent is dependent on the truth of the of this antecedent. The minor premiss (*1) did not express the antecedent of the major premiss. Therefore, the conclusion (*3) does not follow. That was easy. Why do people take this so seriously?

Even though I copied and pasted the exact words from the Wikipeida article, some may think that the Liar’s Paradox is ill representative. Therefore, I will form the argument in another way, which I think is more representative of the standard.

1. A is false.

2. ‘A is false’ is true.

3. A is true.

Now we have asserted the contradictory and alas, a coup de grace! But look again, (3) can not possibly follow from either (1), (2), or a syllogistic combination of them. From the law of excluded middle, if a proposition (A) is false, then its contradictory is true vice versa. Also, there is no syllogistic combination of (1) and (2) since they have no term in common. Therefore, the proper argument can only look like what’s below because we can only apply the law of excluded middle on (1) or (2):

1. A is false.

2. ‘A is false’ is true.

*3a. The contradictory of A is true.  (excluded middle from 1)

*3b. ‘A is true’ is false. (excluded middle from 2)

(3) is contradictory to both (3a) and (3b). Therefore to posit (3) along with (3a) or (3b) violates the law of contradiction. Furthermore, (1) and (2) are completely redundant to each other, along with (3a) and (3b). They say the exact same thing. This redundancy provides no additional information. The proposition was asserted because its true. There’s no need to qualify this again by saying the proposition is true. Lets fully illustrate with an example starting with the incorrect argument:

1. “All lions can fly” is false.

2. “‘All lions can fly’ is false” is true.

3. “All lions can fly” is true.

And the corrected argument:

1. “All lions can fly” is false.

2. “‘All lions can fly’ is false” is true.

3a. “Some lions can not fly” is true.

3b. “‘All lions can fly’ is true” is false.

Removing the useless redundancies:

1. “All lions can fly” is false.

3a. “Some lions can not fly” is true.

This corrected “inference” stands on its own. Strictly speaking its not really even an inference. It tells us nothing we did not already know, we merely reformed the original proposition (1) to attain (3a).

Ultimately, however, we know that in fact no lions can fly. The truth of the universal contains the truth of the particular so that we may also say some lions can not fly. However, how did I know this? Because I am cognizant of the concept of the term “lion” from my past experience. So I may merely analyze this intention and discover that no lions can fly. The point is that I don’t even need to analyze the formal argument at all. The truth of all the terms depend on their corresponding real objects. Truth does not depend on the formal structure of syllogism, although its possible to discover new truths through such syllogisms.

Notice how the original argument gave us “This statement is false.” Practically speaking, the only way a person may assert the truth of this determination is if they know what the statement contains. They can only know the statement is false if they know what the terms signify in reality thereby enabling them to determine its falsity. Therefore, the whole proposition “This statement is false” MUST rest on us having knowledge of “The statement.” There is no instance where we would refer to some generality, “the statement” and say “its false” randomly. Therefore, we may in fact bypass everything said thus far in this blog post and say that if a person knows that “the statement” is false, then they obviously must know what the statement is. Therefore, we may merely analyze the statement to see how we may categorically determine the subject and eliminate any need for such useless analyzing of propositions.

Ironically, this greatly simplifies our problems much more than any attempt to simplify a logical system has done. Notice how this sort of method (the direct account of the real objects) is not derived from any system at all. Its derived from the universal notions (e.g. All lions cannot fly) formed throughout life. Clearly, no simplified logical system endows us with these abilities. “The statement” can only be known in reference to reality. This reality can only exists separately by the action of our conceptual faculties.

[Note 1]

Form of the hypothetical syllogism:

1. If A is B, C is D.

2. A is B.

3. C is D.

Very simple stuff. Same as these sources:

https://faculty.unlv.edu/beisecker/Courses/Phi-102/HypotheticalSyllogisms.htm

http://www.wwnorton.com/college/phil/logic3/ch10/hyposyll.htm

http://en.wikipedia.org/wiki/Hypothetical_syllogism

 

Presentation of the Paradox by plato.stanford.edu

Again, here’s the link: http://plato.stanford.edu/entries/liar-paradox/#SimFalLia

For the sake of accuracy, I won’t initially omit any words by the authors. I will show every step. Let me know if I skipped a step.

“Consider a sentence named ‘FLiar’, which says of itself (i.e., says of FLiar) that it is false.

FLair: FLiar is false.

This seems to lead to contradiction as follows. If the sentence ‘FLiar is false’ is true, then FLiar is false. But if FLiar is false, then the sentence ‘FLiar is false’ is true. Since FLiar just is the sentence ‘FLiar is false’, we have it that FLiar is false if and only if FLiar is true. But, now, if every sentence is true or false, FLiar itself is either true or false, in which case—given our reasoning above—it is both true and false. This is a contradiction. Contradictions, according to many logical theories (e.g., classical logic, intuitionistic logic, and much more) imply absurdity—triviality, that is, that every sentence is true.”

I will consider every individual sentence to extract all of the propositions (at least the relevant ones) contained in them.

“Consider a sentence named ‘FLiar’, which says of itself (i.e., says of FLiar) that it is false.”

No proposition is contained here.

“FLair: FLiar is false.”

Here we have our first proposition:

1. FLair is “FLair is false.”

You can see no omissions here are made. However the copula is added because that is indubitably implied in the clause.

“This seems to lead to contradiction as follows.”

No proposition here pertaining to the argument.

“If the sentence ‘FLiar is false’ is true, then FLiar is false.”

Do I see a hypothetical proposition? This constitutes our second proposition:

2. If the sentence ‘FLiar is false’ is true, then FLiar is false.”

Next sentence: “But if FLiar is false, then the sentence ‘FLiar is false’ is true.”

We have now have our next proposition:

3. But if FLiar is false, then the sentence ‘FLiar is false’ is true.

Next clause in the next sentence constitutes a fourth proposition, “Since FLiar just is the sentence ‘FLiar is false’,” as:

4. Since FLiar just is the sentence ‘FLiar is false’.

Next clause: “we have it that FLiar is false if and only if FLiar is true.”

5. we have it that FLiar is false if and only if FLiar is true.

Next clause: “if every sentence is true or false, FLiar itself is either true or false.”

6. if every sentence is true or false, FLiar itself is either true or false.

Next: “in which case—given our reasoning above—it is both true and false”

7. in which case—given our reasoning above—it is both true and false.

Next: “This is a contradiction.”

8. This is a contradiction.

This is sufficient for the argument. I now present the unaltered clauses in the order as presented:

1. FLair is “FLair is false.”

2. If the sentence ‘FLiar is false’ is true, then FLiar is false.

3. But if FLiar is false, then the sentence ‘FLiar is false’ is true.

4. Since FLiar just is the sentence ‘FLiar is false’.

5. we have it that FLiar is false if and only if FLiar is true.

6. if every sentence is true or false, FLiar itself is either true or false.

7. in which case—given our reasoning above—it is both true and false.

8. This is a contradiction.

Now I will simplify the sentences to get rid of superfluous words reducing them to their logical form. I will specifically enumerate all the alterations made.

(1) This was already put in subject predicate form.

(2) I will erase “then” since its unneeded. I will do this for all the propositions alike in this manner. I will also remove the qualifier “the sentence” because it contributes nothing to the intention of “FLiar is false.” Its already known to be a sentence due to it having quotes.

(3) “But” adds no additional intention. It will be removed from all.

(4) “Since” and “just” similarly add nothing and are removed from all.

(5) “we have it that” is extraneous.

(6) “Itself” is unneeded.

(7) “in which case—given our reasoning above” is superfluous. “It” is FLair.

(8) “This” is in reference to the proposition (7). It is replaced by (7).

Simplified propositions:

1. FLair is “FLair is false.”

2. If “FLiar is false” is true, FLiar is false.

3. If FLiar is false, “FLiar is false” is true.

4. FLiar is “FLiar is false.”

5. FLiar is false if and only if FLiar is true.

6. If every sentence is true or false, FLiar is either true or false.

7. FLiar is both true and false.

8. “FLiar is both true and false” is a contradiction.

We now have our the simplified propositions in the order of presentation. All these contain the full intention of the words used in the original paragraph. If there are disputable alterations, you can let me know. All changes were transparently demonstrated without exception. We may make the following observations:

(1) is the exact same as (4), this redundancy serves no purpose and one may be neglected.

(1), (2), and (3) may all be potential premisses. However, (1) has no terms in common with (2) and (3), and therefore cannot be syllogistically combined.

(5) must be derived from at least two of these premises and the only premisses left to combine are (2) and (3).

The only syllogistic form that admits a conclusion from two hypothetical propositions is the pure hypothetical proposition. (See https://faculty.unlv.edu/beisecker/Courses/Phi-102/HypotheticalSyllogisms.htm) The sole and only form for this syllogism is:

1. If C is D, E is F.

2. If A is B, C is D.

3. If A is B, E is F.

Now to employ this form, we use propositions (2) and (3):

2. If “FLiar is false” is true, FLiar is false.

3. If FLiar is false, “FLiar is false” is true.

Giving us:

If “FLiar is false” is true, “FLiar is false” is true.

We may also switch the order of the premisses and get:

If FLiar is false, FLiar is false.

These are clearly not the same as either (5):

5. FLiar is false if and only if FLiar is true.

Therefore, plato.stanford.edu is guilty of processing conclusions which do not follow from their premisses. It may be possible that the rest of their examples may commit these logical errors. This oversight is astonishing to me.

Induction from Particulars vs. Induction from a Collection of Particulars

Aristotle explains induction from particulars in Posterior Analytics, Book 1, Chapter 31,

“It is owing to our inability to apply our senses, that certain questions remain unsolved. For could we but see what takes place, we should not still be seeking a solution. Not indeed that the mere act of sight would give us scientific knowledge ; but sight would be the means through which we should attain the universal. Thus if we saw the perforations in the glass, and the light-particles coming through, the cause of its illuminative power would be manifest. Sense would perceive the individual instance, and the intellect would recognize that this was a universal law.”

Aquinas explains induction from memory in his commentary on Posterior Analytics, Book 2, Lecture 20,

“But from remembrance many times repeated in regard to the same item but in diverse singulars arises induction, because induction seems to be nothing else than to take something from many things retained in the memory. However, induction requires some reasoning about the particulars, in that one is compared to another: and this is peculiar to reason. Thus, when one recalls that such a herb cured several men of fever, there is said to be induction that such a herb cures fevers. But reason does not stop at the induction gathered from particulars, but from many particulars in which it has been experienced, it takes one common item which is consolidated in the mind and considers it without considering any of the singulars. This common item reason takes as a principle of art and science. For example, as long as a doctor considered that this herb cured Socrates of fever, and Plato and many other individual men, it is induction; but when his considerations arise to the fact that such a species of herb heals a fever absolutely, this is taken as a rule of the art of medicine.

…

But someone could believe that sense alone or the mere remembrance of singulars is sufficient to cause intellectual knowledge of principles, as some of the ancients supposed, who did not discriminate between sense and intellect. Therefore, to exclude this the Philosopher adds that along with sense it is necessary to presuppose such a nature of mind as cannot only suffer this i.e., be susceptible of universal knowledge.”

Note: I replaced “experimentum” in the passage with the word induction.