An Introduction to SYMBOLIC LOGIC

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~ LOGICAL . SCIENCE – A CRITICAL REVIEW/ESSAY OF THE BOOK: “An Introduction to SYMBOLIC LOGIC” by Susanne K. Langer (Dover Books – 1967) © H. J. Spencer (March, 2018) ABSTRACT This book was written by one of my favorite philosophers; the much under-appreciated Professor Langer, who died in 1985 aged 90. She was one of the few academic thinkers to move beyond her training to write on the role of Feeling and the Arts in human culture. Unfortunately, this early book reflects the brain-washing she received as a Ph.D. graduate student at Harvard and is an attempt to boost the reputation of her mentor, Alfred North Whitehead (another great philosopher) who co-wrote the much-acclaimed (and little read) “Principia Mathematica” with Russell. My present review is not an attempt to summarize Langer’s book (written as both a text and as a critical essay) as I have already written on the weak foundations of logic and mathematics [see “ILLOGICALTY”). It is driven by my transformation from initial satisfaction with this book to extreme frustration as it becomes a propaganda text for the empty subject of the logical foundations of mathematics, known as “Logistics” [a word not to be confused with the talents of military men to keep their soldiers well-supplied]. My dissatisfaction is deeply compounded by my prior extremely positive reaction to Langer’s unique awareness of the key role of intuition in most appreciation of all symbolic expressions of the human imagination from linguistics to philosophy itself. My objective here is to discourage people, especially scientists and philosophers, from wasting their time either reading the infamous Whitehead/Russell tome or even trying to learn the arcane symbolism of formal logic. This vigorous discouragement is a response to the exaggerated reputation of logic, carefully cultivated by most academic philosophers, who see themselves as the Kings of Academe, now that they believe that their symbolic logic even underpins the Queen of the Sciences – mathematics; the Evil-Twins of timeless symbology. Much effort is expended to justify Boolean Algebra, readily seen as trivial using the well-known Venn diagrams. I admit that my explicit title of this essay is a weak attempt at satirizing the unnecessary Symbolism of Logic but it does correctly imply (a favorite logical term) that I reject the proposition that logic is a science. Another goal is to alert people to my 20 year-old paper on Universal Semantic Grammar that provides an alternative to the empty syntactical approach of symbolic logic with a more helpful schema for understanding language that can handle tenses, agency, ‘fuzzy’ quantifiers and homonyms. Its definitional capability (lambda calculus) gives it extensible and recursive capabilities as well as semantic preservation for translations across Natural Languages. Its linearity facilitates computer processing and the ready answer to a very broad range of natural language questions, such as: Why? How? When? Who? What?. However, it seems a losing battle to take on the 2000 year reputation of the claimed foundation of ‘rationality’.

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SUMMARY The thesis of this essay is that logic is a trivial topic when limited to finite examples and generates a false aura of intellectualization when its study wanders off into vague generalities and abstractions, mysteriously compounding its importance by deliberately adopting a mathematical style of symbolism. Worse, the mistaken century-old project of logistics exposes the drive for impressive conceptualization and special symbolization of simple arithmetic that is actually grounded in the real activity of counting small sets of certain types of simple objects, not in the empty, specialized definitions of mathematicians such as Peano, Frege and Russell; deliberately introduced to help with logical analysis. A further motivation is to counter the claims of even philosophers of the quality of Langer that science needs to be built on mathematics and therefore on the timeless “Truthfulness” of Logic.

OBJECTIVE

The principal motivation of this essay is to reinforce my belief in natural organism and relationships as the sources of everyone’s intuition about reality and their commonsense version of metaphysics. This is part of my larger project to replace the Judaic/Greek roots of western thought that have brought the world to the edge of oblivion with their badly formed ideas, as I have come more and more to see in the growing cracks in our western edifices of science and philosophy, reflected in the real lack of basic social progress in humanity’s thinking. If we are to evolve beyond ancient tribalisms into a sustainable form of global co-operative civilization then the technological (in actuality, only military) superiority directly resulting from the grasp of western THING-Thinking must be minimized to allow for the deeper and richer experiential and relational modes associated with less instrumental civilizations.

REVIEW

INTRODUCTION Symbolic logic is the mathematical schema that gives linguistic (analytic) philosophers the sense that their approach to philosophy is as well grounded as the ‘certainties’ of mathematics. Langer is aware that most people learn the techniques of mathematics as simple rules (definitions; like “minus times minus equals plus”) with minimal understanding of the concepts involved, so she deliberately adopts the powerful strategy of developing understanding first and foremost, announcing that Logic is the study of generality and especially the systematic elaboration of the concept of abstractions with the objective of the development (Langer Platonically says “discovery”) of abstract forms, contrasted with science that truly discovers particular facts about reality. Admittedly, Langer does recommend lots of practicing with logical symbol manipulation, as in arithmetic learning. When first written (1953) there were few extent textbooks on symbolic logic, even though many of its techniques were invented almost 40 years earlier. The initial chapter offers the common sense promise of understanding the powerful idea of generalized patterns that philosophers call Forms, appealing to our shared experience of some transformations of common matter in its various forms, like liquid, solid or gaseous manifestations of water. The power of the Atomic Hypothesis is that it explains the protean nature of matter to take on many different spatial structures. The ultimate limit of this analytic style of thinking was implicit in the discovery of the electron in 1897. We all have direct sensory awareness of the world but it is only our mental symbolism that enables us to create theories beyond this direct experience. These intimations of Transformations suggest the value of studying form in general. Langer cleverly builds on this intuition of common introduction to material forms to move on to abstract formalism as found in linguistics and the arts, calling the generality: “logical form” to distinguish this from physical shape.

3 Langer identifies the commonality with our intuitive awareness of structure. i.e. ‘constructed’ and to emphasize the move beyond materialism by calling the medium wherein a form is expressed its ‘content’, thus the human form may be expressed in flesh and blood or in stone or in a painting etc. Some forms may (by agreement) represent something entirely different; this is the power of speech or human languages. This is the path Langer uses to go from separate words to integrated logical ideas or “propositions” to use the technical term. Most real human languages have their own sets of structural rules, known as the Syntax of that language, which Langer claims reflects the logical form of our thoughts but this seems an over-extended claim as the massive variety of human languages attests. The shared use of a single verb in many different statements leads to the idea of the form of a sentence. Unfortunately, this shared verbal usage results in the idea of relationships while a single word as the subject of several sentences leads to a simple focus on nouns and this reflects the history of western philosophy, where relations (the generality of relationships) are much more difficult to analyze than nouns or names. When we strip away most of the features that exemplify a particular example of reality we are pursuing the strategy of abstraction: an art form that requires much skill to leave enough “flesh on the bone” to be useful but removing all that is irrelevant to the discussion. Number is one of the commonest examples of an abstraction when we repeatedly perform the activity of counting discrete real objects. The results may agree or differ in subtle ways (e.g. two more). It is only recently that we have come to appreciate the cultural differences (and difficulties) that arise with what was once thought (in the Greek tradition) as a universal mode of thinking. Tragically, it is only in learning the rules of arithmetic that most children are exposed to a process of abstraction but few teachers make this technique explicit; perhaps why so many find algebra too difficult. Science relies on intuiting physical analogies or forms. When we deal with forms abstracted from particulars (e.g. rhythms) we call them concepts. Much intellectual thinking (and certainly most philosophy following Plato) is centered on conceptual thinking. When we go backwards from the generality to examples, we talk about the interpretation (or ‘extension’) of a concept. It is truly not a coincidence that most of us develop our basic conceptual vocabulary from numerous examples that are based on our intuitive processes giving a broad and not always shared sets of examples; a common source of semantic miscommunications. It is also not a coincidence that most of us learn the standard, shared meanings of adverbs and adjectives. In order to control social thought, Aristotle (the “Sorcerer’s Apprentice”) invented the rules of the verbal game of conceptual thinking, known since as Logic. Professional thinkers, called Philosophers, have ever since expended great educational effort to preserve these rules for two millennia. All people who observe these rules are given the positive social cachet of “being rational”. Langer exposes her private motivation defending symbolic logic by claiming that mathematics is simply a branch of logic and she simply assumes that mathematics is both a science and inherently a positive contribution to humanity. Langer, like most other mathematicians, simply assumes that results of this ‘science’ are always Truths. Then, by rhetorical analogy, Langer extends the “indisputable utility” of mathematics to the natural sciences to philosophy where logic contributes to the general understanding of the world; in spite of the little appreciated failure of philosophers to agree amongst themselves in their world-views or even getting to common agreement on the perennial ‘questions’ of philosophy that have arisen since Plato. Annoyingly, Langer just trumpets the ‘Party Line’ when she writes that logic “illuminates problems that have been obscure for hundreds and even thousands of years with logic being the philosophers’ telescope.” These generalities are used as arguments without providing any detailed examples to back her confidence. As a general rule of thumb (for regular folk) be very suspicious of intellectuals who fail to provide any real examples of their vague concepts; the world really does consist of particulars and not their “universals”. Science has progressed more from improvements in technology (like microscopes) rather than the (mistaken) opposite view. We are a technologically successful society.

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LOGICAL STRUCTURE Langer jumps into her second chapter to elaborate on her introduction of structure but illustrates a key technique in philosophy to introduce a new term by inventing new words, so that parts of a structure now become ‘elements’, so that properties of “objects” may also covered by this new generality. These ‘magicians’ hide the rabbit in the hat by talking about common objects with well-recognized properties and implicitly appealing to this broad understanding – the same ‘trick’ was played by Euclid in his focus on perfect circles that we all have an approximate awareness of without there being any real examples in nature. Thus, most real objects (at our scale of reality) have the property of location so Langer can introduce the idea of relative locations (e.g. North of) for moving from things to ‘relations’ – a needed step, as this is how many animals view the real world (obviously without a compass). The elements in relations are then used to lead to the concept of a proposition as a verbal means of expressing a relation amongst several elements (or terms to use the logician’s vocabulary), while such propositions can be used to refer to a state of affairs (allowed to be equally real or imaginary). Implicitly, such propositions are to “asserted” by some (invisible) speaker. Langer admits that the verb usually performs the key functions: it ‘names’ the relation and asserts that it holds among the named elements. Here, we see the role of nouns appearing as our “mental names” for things; prepositions act as implicit verbal relations hiding the spatial, temporal or active/passive roles of the objects involved. The flexibility of Natural Language is used as a motivation to translate sets of propositions (i.e. related sentences) into abstract symbolism [incidentally, I have created a SemanticAlgebra that avoids this step by adding certain simple ‘semantic role’ “operators” while retaining common vocabulary] that makes immediate understanding difficult because all we are left with are syntactical rules for this strange symbology. This new mathematical style symbolism is the essence of symbolic logic. The richness of Natural Language is used by logicians as an excuse for replacing it with their simple signs. Langer provides an amusing example of six different meanings of the English word “IS”; these are replaced by the logical symbols: =, ∈, ⊃, ∃. These various mathematical, Greek and logical symbols are ‘explained’ to a novice reader by explicit definitions, itself variously symbolized as: ≡ , where ‘≡’ ≡ “is defined as”; then we are ‘off to the races’, so that: ‘∃’ ≡ “there exists”. One can readily see where this Game is going: to justify mathematics itself. Now, it is a psychological fact, that most short-term human memory is limited at any one time in its capacity, on average we can recall about seven ideas, so that a good short-hand notation can assist us while some symbols, like the one for zero (0) actually revolutionized European arithmetic.

TRUTH The most powerful magic ‘trick’ is brought on stage by Langer in her third chapter, where she brings in the idea of “Truth-Value” to reflect the fact that when a speaker says something about the world they may be mistaken – or worse, lying; in other words, the proposition, when understood, may be viewed as “true” or “false” if we interpret the meaning in terms of the real world. The Game of Logic is to see how far we can link such propositions together, so that when we know certain components are ‘true’ then see if the conclusions of the whole set are true or not; the final result is called its “truth-value”. It is like trying to solve a simple problem in algebra when we told the result must only be one or zero. The weakness of this game is to know how truthful are the claims that key parts are indeed true or not. As we will see, logic does very little to help with this fundamental problem although this is what the common man expects of so-called logic and rationality. Another huge problem with logic is that the time of the statements is not provided; they are supposed to be always the case; unfortunately, science itself follows in this mathematical style simplification: an equation (like E=mc2) is assumed to be always true. Experience shows us that context almost always plays an important role in life.

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5 Thus, adding one ice-cube to a hot stove-top a minute or so after a first one was placed there is supposed to result in two ice-cubes being found there soon after; since we all know that: “one plus one equals two”. The triviality of logic becomes obvious when one realizes that the key-parts of this game are played only with the logical-links for the ideas about: AND, OR, NOT. Rather than use these English words, the logicians invented an international set of symbols: ‘&’ or ‘∪’ (called conjunction), ‘∨’ (called “disjunction” = “one or the other or both”) and ‘~’ (for Truth-flip/reverse). It can be guessed that these are somehow connected to our old arithmetic friends: plus ‘+’ and minus ‘−’ [although minus is a tricky symbol that is rarely discussed properly in school, as it plays two very different roles.] Therefore, with simple, relative compass statements, we can always say about two cars (A and B) sitting on a North-South road: that one is North of the other; thus, in logical symbology: ~ (A .N. B) ∨ ~ ( B .N. A) when ‘.N.’ ≡ “North-Of”. Then again, this may be true on a single-width highway but not on a twin-lane road when they are side-by-side while here the reader must be familiar with the compass concepts. This exception illustrates the bogus assumption that all propositions are either ‘true’ or ‘false’; mathematicians rely on this key assumption, so it is called the LAW of the ‘Excluded Middle’, forcing the world into a Greek choice of Either/Or when at different times opposite results may apply; so after a successful ‘Sex-Change’ operation a person can have been both a Man and a Woman or if the operation failed, sadly, maybe neither. This rare example illustrates that both mathematics and logic rely on the LAW of Perfection; fortunately other cultures (especially Asian) readily assume both outcomes are possible. There is a disputation about one other extension associated with our causal idea of: IF / THEN called implication but we will leave this now as too contentious to be worthy of its introduction, as most readers could readily construct ambiguous examples. Here we can see the distinction between deduction, where the results of a logical-equation can be derived, whereas in most real situations we need to be told about the truthfulness of every statement, giving an inductive situation; this is the implied case in science where every experiment is expected to produce the same results (certainly within the error of accuracy expected), so we repeat to confirm this. These discussions form the preamble for Langer’s introduction in chapter IV to the idea of quantifiers in logic, which are simply the consequences of using the English words: ALL or SOME or NONE. This can be useful when the truth-values of all the elements referred to in a situation are known (how?). In practice, this means we are dealing with definitions that imply all examples always fit the definition, as in the old war-horse that: “All bachelors are unmarried”. This is simply an implicit restatement of the definition of the word ‘bachelor’ as: “An adult man, who is not married”. However, even this definition applies to most married men who are now dead, recalling how many unstated qualifiers are usually needed in reality. This is where symbolic logic smuggles in its next mathematical ‘cousin’ when it introduces algebraic-like ‘variables’; say the Greek letters x, y, z for exemplars who can remain nameless but can still play a useful role in logical productions; they resemble unidentified English pronouns. Therefore, instead of saying: “There is at least one (i.e. “some”) example of the type CAR that has the color ‘red’, we can write (with x being ‘it’): (∃x) : x is-red & x is-a CAR When ‘is-red’ ≡ “red colored”. The other “elaboration” that is made for collections (or even definitions) is: “ALL examples ...” that is written by logicians as: (∀x) : x ... . It always implies the truthfulness of all examples; so universal quantifier. These are used to rewrite the four classic logical statements of Aristotle’s basic quantified propositions: 1] (∀x) : x is A. 2] (∀x) : ~ (x is A). 3] (∃x) : x is A. 4] (∃x) : ~ (x is A). These collections are referred to by the technical term: CLASS resulting in Boole’s algebra of classes. 5

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DEFINITIONS In order to demonstrate that logic is all about word-games, it is sufficient to recall an early discussion by Langer on the distinction between concepts and conceptions. Conceptions are seen as the private understanding of a concept possessed by each individual; they reflect our unique personal history and all the associations we have developed around that concept in our private memory. Concepts are just the shared, minimal public definitions that everyone in the specific logical discussion can agree upon. These highly restricted discussions are referred to technically as the “Universe of Discourse”; Logic is only concerned with concepts discussed within agreed Discourse-Context, like a national Law-Court. Specific propositions always concern a certain subject, called an individual, and this can, using our senses, be pointed out, called an individual. Completely general propositions never mention individuals but can mention a member (or members) of a certain class: and the idea of “inclusion” or technically class-membership, simplified to our phrase: “IS-A” in contrast to the simpler “IS” as in “Water is a molecule” and “Ice is water”. Here class is an abstract idea (logical construction), exemplified as a collection of everything to which a specific concept applies. For finite collections, a given group of objects of the same type are examples of the concept, called an “extension” but its verbal definition is simply called its “intension”. Logicians prefer working with intensions, using the notation: ‘∈’ ≡ “is a member of”. Ordinary people prefer to work with extensions. When the number of extensions in a ‘target’ collection is small (say less than 7) most people can confirm claims about the set as obvious but logicians have to prove these claims deductively, justifying their effort since their ‘proof’ applies to any number of examples, even when too many to count to confirm. Aristotle was enamored of his idea of class as it allowed him to construct classification hierarchies by joining together groups or dividing classes into sub-classes by applying a differentia definition. Again, this technique only works when membership does not change over time. This can cause problems as when citizenship is assigned for legal reasons and not simply because of the accident of one’s ancestors. These ideas led to one of the most famous arguments in philosophy: “Socrates is a human. All humans are mortal. Therefore Socrates is mortal”. Here, human is a class with the assumption that as humans are animals and all animals die then this must apply to poor old Socrates. But, again Time raises its lovely head and the mortality assumption can only apply to the past as we have no idea whether a technique will be invented for immortality. As Socrates certainly died a long time ago, we are confusing logic with historical fact. Check out another example of ludicrous and long-winded logic by examining the propositional form needed to express the idea that there is only one example ever of the concept ‘A’ (e.g. Virginal-Birth) becomes, when translated back into English: “There is at least one something x, such that for another something y where both are examples of A then y must be identical with x.” (∃x) (∀y) : (x ∈ A) & (y ∈ A) & (y = x) Another bizarre construct is the notion of a null class with zero examples, all of which have the value false; so “all null classes are identical” [sorry, go figure – it’s just logical.] All of this symbolic ‘machinery’ was constructed to expand Aristotle’s classification scheme that was based on a single subject and one property (or ‘quality’), known by the grammarians as a ‘predicate’; whereas propositions specified relations between two or more exemplars. Traditional logic began with predicate-logic but Langer only brings it up after completing most of her text because she sees them as too particular and she always prefers generalities. Unfortunately, logicians have taken too much of a short-cut: for example the predicate ‘is-white’ as in a statement: “Swans are-white” derives from a fuller statement, like “Swans have feathers and the color of the feathers are white.” The problem is that the only connection among elements which can be established by predicative propositions is common membership in a class (this was Aristotle’s singular objective). Predicates often just define a class. 6

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THEOREMS and VALIDITY Just like manipulating numbers in arithmetic only results in other numbers, so there can be no new knowledge generated by the propositional calculus; as Langer admits: “a proposition cannot be deduced from anything but another proposition.” But, even then, like geometry, logic is no stronger than the initial assumptions (grandly called “axioms”) whose ‘truth’ is vouched for when we accept it intuitively, demonstrating that ultimately rationality is grounded in experience. Furthermore, Langer says: “the self-evidence of any proposition is a very questionable affair. The power to doubt familiar propositions is relative to one’s logical imagination (?) and to the force of verbal and mental habits.” – this explains the efforts made to ‘educate’ society’s intellectual elites when young; she also warns, using the history of science: “What is intuitively accepted by all people may turn out to be false.” This leads to the trumpeted feature of ‘logical proofs’ or ‘theorems’ where resulting propositions are seen as ‘proved’ as long as the prior propositions are “granted” or previously proved. Although logic makes many claims that are concerned about truth and falsity the reality is that these are two complementary values and almost nothing to do with the concept of Truth that most people work with: implicitly a statement that “accurately reflects concrete reality”. As Langer confesses halfway into her book: “For the truth of postulates there is no logical guarantee. There is no difference between a proposition that expresses an actual state of affairs in the world and one that expresses merely a conceivable state; no formal distinction between factual and fictional premises.” This astonishing view shows why logicians are not called as legal experts in criminal trials. This distinction is called by these Masters of Rationality ‘logical certainty’ or “validity”. These admissions mean that ordinary people must be extremely skeptical when confronted by clever professors pushing their own views under the disguise that “it’s just logical”. Langer uses the following example to demonstrate validity: 1) Napoleon discovered the Americas. 2) Napoleon died before 1500 AD. Langer claims that together these two propositions imply that: 3) The Americas were discovered before 1500 AD; because the third proposition from the prior two “is perfectly valid, as there is nothing inconsistent about them. Langer elaborates on this vital point and warns against a danger symbol, the word “therefore”. Validity is all. The reader is here warned that chapter IX contains page after page of trivial and boring logic theorems. The omission of TIME certainly leads to simplicity in the timeless logical “equations” but they are only ‘valid’ for some foundational assumptions of what are the “objects of thought” symbolized by these algebraic letters, so a proposition about two distinct classes assumes that the equation can be written in any order (“commutation”), like the addition of simple ‘things’ [such as pebbles], as in: A + B = B + A. However, if the ‘objects’ are actually events or actions then, for example: “eating must precede dying” when time is being considered, so symbolizing these events as E(t1) and D(t2) then: E(t1) < D(t2) for time t1 before time t2; but not the opposite times. The irony of logic, present since its invention by Aristotle and continued through the vanity project of Logistics, is that has always tried to understand the nature of relationships (best represented in natural language by verbs) but have persistently focused on the singular idea appearing as nouns, with the only relation covered being that of membership in a conceptual class or simple inclusion in its geometric analogy exhibited in two-dimensional Venn Diagrams of overlapping circles. Few will be convinced by Langer that this huge effort is justified because it only shows the (claimed) pattern of classrelationships such as conjunction (AND) and disjunction (OR) alternatively total/partial inclusions or even over-lapping when presented diagrammatically.

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MATERIAL IMPLICATION In order to bypass the problem of knowing if the initial propositions are true in developing a logical proof then the ‘tricky’ function of logical implication was invented. The connective, symbolized by the odd character ‘⊃’ to indicate that IF the left-hand proposition (s) are true, then the right-hand proposition must be true, such IF jack was born in December THEN his birthday must be in the winter; this is translated into the formula: (Jack born-in December) ⊃ (Jack’s birthday is in the winter). This is viewed as central to logical proofs where conclusions are conditionally implied by their premises. Unfortunately, logicians do not care about meaning, so they have derived the bizarre result that the base rule: P ⊃ Q is equivalent to the formula: ~P ∨ Q ; i.e. P is false or Q is true. This is called in the logicgame “material implication” as it is very different than IF P THEN Q because even when P is false Q may be true. These result in the two infamous “paradoxes” of symbolic logic: 1) P ⊃ Q ⊃ P in words: a true proposition is implied by ANY proposition. 2) P ⊃ P ⊃ Q in normal language: a false proposition implies ANY proposition. The logicians break the link to standard language by changing the result to: “materially implies”. But users of so-called Truth-Tables have been warned because “material implication” will be slipped in some place because as Langer concludes: “It is the most prominent operation in logical calculus because it permits inference.” Indeed, she further admits that: “the laws of logic must be accepted before any logical calculus can be used as they only exhibit the formal properties of those laws (?), which we do, in fact, accept as valid (?) in reasoning.” [my challenging question marks.]

LOGISTICS Only the masochistic will be bedazzled by Langer’s final quarter of the book, written deliberately to praise her Ph.D. supervisor (Whitehead) although she leads into it by admitting that the difficulties of the propositional calculus had to be modified so that it could become the basis (called Logistics) for “the highest classic of symbolic logic” the ‘Big Book’; or as she falsely claims for the foundations of all mathematical reasoning but it only covers arithmetic and its symbolic forms (algebra); it does NOT cover geometry nor its “proofs”. Two further significant admissions emerge here: “The only way to pass from class-concepts to truth-value concepts is by a change of interpretations for the whole calculus.” “It becomes impossible to distinguish what we are saying from what we are talking about.” In other to avoid self-contradictions, such as Russell’s or the Liar paradox, they dropped generalities by limiting an Elementary Proposition to one which takes only individuals (things, persons, etc.) for its terms and specifically it is not about propositions. Russell re-introduced Frege’s new (unprintable) assertion symbol that is to be taken as unchallengeable (but not proven). Incredibly, there is no formal expression of Truth as this is substituted by the informal postulate: “Anything implied by a true proposition is true.” One can always spot a real mathematician (like Langer) when they claim that mathematics is the most developed and elaborate system of knowledge : a true science, “the ideal of logical thought.” While the more thoughtful mathematicians admitted that ordinary people’s view of arithmetic as simply the result of the acts of counting as valid but they wanted a solid, logical foundation to justify their own imaginative extensions, like irrationals, imaginary and transfinite numbers; as ‘real’ not fictions. Russell turned to the recent writings of Peano who had defined arithmetic around the key idea that “every number has a unique successor-number”. This approach sneakily smuggles in the endless (or infinity) concept, the clue that the theologians have returned.

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CONCLUSION Contrary to Langer’s own objective (explicitly described in her preface) logic does NOT help in any understanding language meaning (Semantics); this is reinforced in her final conclusions where logic is divorced from any metaphysical views or human psychology. An independent study of the long history of philosophy will show the strong connections to theology (Scholasticism) and the minimal progress in understanding the world by this enduring collection of wordsmiths: no REAL problems are ever solved. Like any proud mathematician, Langer describes logic as the steady progression from the concrete to the abstract, from contents with certain forms to those empty forms without their contents, from instances to universal kinds, from examples to concepts. Mathematicians seem to have been long hypnotized by the power of combining the techniques of generalization and abstraction. Unfortunately, they appear to have convinced many scientists that these are the keys to progress in all science but only simple physics and chemistry is amenable to this view, where the material objects are small enough to be so simple that every example is effectively identical – this is not the case for organic life forms with their inherent differences and vital uniqueness, presenting intrinsic challenges to medical science and pharmaceutical companies who always focus on small molecules. The urge towards excessive generality, to justify the centrality of the concept of concept in philosophy, is just simple mathematics (the binary algebra of George Boole or the ‘Class-Calculus’) as a forest of words to disguise the trail being laid to the huge effort of Logistics, creating the circular argument of mathematics being used to justify mathematics itself by trivial symbolic logic. But six year old children have more common sense about the action of counting as the foundation of arithmetic than generations of the cleverest professors appealing to the rigors of their mathematics or logic. However, Langer remained convinced that all this effort was justified as the Laws of Deductive Reason and the structure of logic itself, even though many results are trivial, such as: P = 1 ∨ P = 0, where {1,0} are the Boolean variables for true and false; so this formula only means: Any proposition is either true or false. As a theoretical physicist, I cannot recall a single major advance in the last 300 years that was brought about by improvements in conceptual thinking or symbolic logic: contrary to Langer’s many claims otherwise. Progress appears to be driven by new experimental discoveries that stimulate creative leaps of the imagination. Ironically, most people benefit from technological innovations, not science theory. It is not true that the study of symbolic logic helps prepare students for understanding (i.e. ontological) semantics the epistemological problems that have arisen in the contemporary philosophy of nature, as the centennial anniversary of the semantic confusions of the interpretation of the mathematical theory known as quantum mechanics develops from a Positivistic (logical) perspective demonstrates. The huge effort Russell put into writing his Principia must be seen as little more than a personal vanity project because within twelve years, Kurt Gödel proved that all this was ‘incomplete’ (“irrelevant”?). Few non-mathematicians have probably read all of this review, suspecting there is far too much mathematics even here. The brave readers, who have persisted, must realize that the reviewed book goes much further with this strategy than its review. Acres of words are used to try to cover up all the mathematical ideas that are smuggled into the naive readers’ minds.

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