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Saturday, January 25, 2020

Logical consequence

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Logical_consequence

Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusion is entailed by the premises, because the conclusion is the consequence of the premises. The philosophical analysis of logical consequence involves the questions: In what sense does a conclusion follow from its premises? and What does it mean for a conclusion to be a consequence of premises? All of philosophical logic is meant to provide accounts of the nature of logical consequence and the nature of logical truth.

Logical consequence is necessary and formal, by way of examples that explain with formal proof and models of interpretation. A sentence is said to be a logical consequence of a set of sentences, for a given language, if and only if, using only logic (i.e. without regard to any personal interpretations of the sentences) the sentence must be true if every sentence in the set is true.

Logicians make precise accounts of logical consequence regarding a given language , either by constructing a deductive system for or by formal intended semantics for language . The Polish logician Alfred Tarski identified three features of an adequate characterization of entailment: (1) The logical consequence relation relies on the logical form of the sentences, (2) The relation is a priori, i.e. it can be determined with or without regard to empirical evidence (sense experience), and (3) The logical consequence relation has a modal component.

Formal accounts

The most widely prevailing view on how to best account for logical consequence is to appeal to formality. This is to say that whether statements follow from one another logically depends on the structure or logical form of the statements without regard to the contents of that form. 

Syntactic accounts of logical consequence rely on schemes using inference rules. For instance, we can express the logical form of a valid argument as:
All X are Y
All Y are Z
Therefore, all X are Z.
This argument is formally valid, because every instance of arguments constructed using this scheme is valid. 

This is in contrast to an argument like "Fred is Mike's brother's son. Therefore Fred is Mike's nephew." Since this argument depends on the meanings of the words "brother", "son", and "nephew", the statement "Fred is Mike's nephew" is a so-called material consequence of "Fred is Mike's brother's son," not a formal consequence. A formal consequence must be true in all cases, however this is an incomplete definition of formal consequence, since even the argument "P is Q's brother's son, therefore P is Q's nephew" is valid in all cases, but is not a formal argument.

A priori property of logical consequence

If you know that follows logically from , then no information about the possible interpretations of or will affect that knowledge. Our knowledge that is a logical consequence of cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, so they must be knowable a priori. However, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a priori property of logical consequence is considered to be independent of formality.

Proofs and models

The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of proofs and via models. The study of the syntactic consequence (of a logic) is called (its) proof theory whereas the study of (its) semantic consequence is called (its) model theory.

Syntactic consequence

A formula is a syntactic consequence within some formal system of a set of formulas if there is a formal proof in of from the set .
Syntactic consequence does not depend on any interpretation of the formal system.

Semantic consequence

A formula is a semantic consequence within some formal system of a set of statements
if and only if there is no model in which all members of are true and is false. Or, in other words, the set of the interpretations that make all members of true is a subset of the set of the interpretations that make true. 

Modal accounts

Modal accounts of logical consequence are variations on the following basic idea:
is true if and only if it is necessary that if all of the elements of are true, then is true.
Alternatively (and, most would say, equivalently):
is true if and only if it is impossible for all of the elements of to be true and false.
Such accounts are called "modal" because they appeal to the modal notions of logical necessity and logical possibility. 'It is necessary that' is often expressed as a universal quantifier over possible worlds, so that the accounts above translate as:
is true if and only if there is no possible world at which all of the elements of are true and is false (untrue).
Consider the modal account in terms of the argument given as an example above:
All frogs are green.
Kermit is a frog.
Therefore, Kermit is green.
The conclusion is a logical consequence of the premises because we can't imagine a possible world where (a) all frogs are green; (b) Kermit is a frog; and (c) Kermit is not green. 

Modal-formal accounts

Modal-formal accounts of logical consequence combine the modal and formal accounts above, yielding variations on the following basic idea:
if and only if it is impossible for an argument with the same logical form as / to have true premises and a false conclusion.
 

Warrant-based accounts

The accounts considered above are all "truth-preservational," in that they all assume that the characteristic feature of a good inference is that it never allows one to move from true premises to an untrue conclusion. As an alternative, some have proposed "warrant-preservational" accounts, according to which the characteristic feature of a good inference is that it never allows one to move from justifiably assertible premises to a conclusion that is not justifiably assertible. This is (roughly) the account favored by intuitionists such as Michael Dummett

Non-monotonic logical consequence

The accounts discussed above all yield monotonic consequence relations, i.e. ones such that if is a consequence of , then is a consequence of any superset of . It is also possible to specify non-monotonic consequence relations to capture the idea that, e.g., 'Tweety can fly' is a logical consequence of
{Birds can typically fly, Tweety is a bird}
but not of
{Birds can typically fly, Tweety is a bird, Tweety is a penguin}.

Validity (logic)

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Validity_(logic)

In logic, more precisely in deductive reasoning, an argument is valid if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. Valid arguments must be clearly expressed by means of sentences called well-formed formulas (also called wffs or formulas). The validity of an argument can be tested, proved or disproved, and depends on its logical form.

Arguments

Argument terminology used in logic

In logic, an argument is a set of statements expressing the premises (whatever consists of empirical evidences and axiomatic truths) and an evidence-based conclusion.
 
An argument is valid if and only if it would be contradictory for the conclusion to be false if all of the premises are true. Validity doesn't require the truth of the premises, instead it merely necessitates that conclusion follows from the formers without violating the correctness of the logical form. If also the premises of a valid argument are proven true, this is said to be sound.

The corresponding conditional of a valid argument is a logical truth and the negation of its corresponding conditional is a contradiction. The conclusion is a logical consequence of its premises.
An argument that is not valid is said to be "invalid".

An example of a valid argument is given by the following well-known syllogism:
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.
What makes this a valid argument is not that it has true premises and a true conclusion, but the logical necessity of the conclusion, given the two premises. The argument would be just as valid were the premises and conclusion false. The following argument is of the same logical form but with false premises and a false conclusion, and it is equally valid:
All cups are green.
Socrates is a cup.
Therefore, Socrates is green.
No matter how the universe might be constructed, it could never be the case that these arguments should turn out to have simultaneously true premises but a false conclusion. The above arguments may be contrasted with the following invalid one:
All men are immortal.
Socrates is a man.
Therefore, Socrates is mortal.
In this case, the conclusion contradicts the deductive logic of the preceding premises, rather than deriving from it. Therefore, the argument is logically 'invalid', even though the conclusion could be considered 'true' in general terms. The premise 'All men are immortal' would likewise be deemed false outside of the framework of classical logic. However, within that system 'true' and 'false' essentially function more like mathematical states such as binary 1s and 0s than the philosophical concepts normally associated with those terms.

A standard view is that whether an argument is valid is a matter of the argument's logical form. Many techniques are employed by logicians to represent an argument's logical form. A simple example, applied to two of the above illustrations, is the following: Let the letters 'P', 'Q', and 'S' stand, respectively, for the set of men, the set of mortals, and Socrates. Using these symbols, the first argument may be abbreviated as:
All P are Q.
S is a P.
Therefore, S is a Q.
Similarly, the second argument becomes:
All P are not Q.
S is a P.
Therefore, S is a Q.
An argument is termed formally valid if it has structural self-consistency, i.e. if when the operands between premises are all true, the derived conclusion is always also true. In the third example, the initial premises cannot logically result in the conclusion and is therefore categorized as an invalid argument. 

Valid formula

A formula of a formal language is a valid formula if and only if it is true under every possible interpretation of the language. In propositional logic, they are tautologies

Statements

A statement can be called valid, i.e. logical truth, if it is true in all interpretations.

Soundness

Validity of deduction is not affected by the truth of the premise or the truth of the conclusion. The following deduction is perfectly valid:
All animals live on Mars.
All humans are animals.
Therefore, all humans live on Mars.
The problem with the argument is that it is not sound. In order for a deductive argument to be sound, the deduction must be valid and all the premises true.

Satisfiability

Model theory analyzes formulae with respect to particular classes of interpretation in suitable mathematical structures. On this reading, formula is valid if all such interpretations make it true. An inference is valid if all interpretations that validate the premises validate the conclusion. This is known as semantic validity.

Preservation

In truth-preserving validity, the interpretation under which all variables are assigned a truth value of 'true' produces a truth value of 'true'.

In a false-preserving validity, the interpretation under which all variables are assigned a truth value of 'false' produces a truth value of 'false'.
Preservation properties Logical connective sentences
True and false preserving: Proposition  • Logical conjunction (AND, )  • Logical disjunction (OR, )
True preserving only: Tautology ( )  • Biconditional (XNOR, )  • Implication ( )  • Converse implication ( )
False preserving only: Contradiction ( ) • Exclusive disjunction (XOR, )  • Nonimplication ( )  • Converse nonimplication ( )
Non-preserving: Negation ( )  • Alternative denial (NAND, ) • Joint denial (NOR, )

Logical truth

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Logical_truth
 
Logical truth is one of the most fundamental concepts in logic, and there are different theories on its nature. A logical truth is a statement which is true, and remains true under all reinterpretations of its components other than its logical constants. It is a type of analytic statement. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, as well as logical consequence.

Logical truths (including tautologies) are truths which are considered to be necessarily true. This is to say that they are considered to be such that they could not be untrue and no situation could arise which would cause us to reject a logical truth. It must be true in every sense of intuition, practices, and bodies of beliefs. However, it is not universally agreed that there are any statements which are necessarily true.

A logical truth is considered by some philosophers to be a statement which is true in all possible worlds. This is contrasted with facts (which may also be referred to as contingent claims or synthetic claims) which are true in this world, as it has historically unfolded, but which is not true in at least one possible world, as it might have unfolded. The proposition "If p and q, then p" and the proposition "All married people are married" are logical truths because they are true due to their inherent structure and not because of any facts of the world. Later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations.

The existence of logical truths has been put forward by rationalist philosophers as an objection to empiricism because they hold that it is impossible to account for our knowledge of logical truths on empiricist grounds. Empiricists commonly respond to this objection by arguing that logical truths (which they usually deem to be mere tautologies), are analytic and thus do not purport to describe the world.

Logical truths and analytic truths

Logical truths, being analytic statements, do not contain any information about any matters of fact. Other than logical truths, there is also a second class of analytic statements, typified by "no bachelor is married". The characteristic of such a statement is that it can be turned into a logical truth by substituting synonyms for synonyms salva veritate. "No bachelor is married" can be turned into "no unmarried man is married" by substituting "unmarried man" for its synonym "bachelor".

In his essay Two Dogmas of Empiricism, the philosopher W. V. O. Quine called into question the distinction between analytic and synthetic statements. It was this second class of analytic statements that caused him to note that the concept of analyticity itself stands in need of clarification, because it seems to depend on the concept of synonymy, which stands in need of clarification. In his conclusion, Quine rejects that logical truths are necessary truths. Instead he posits that the truth-value of any statement can be changed, including logical truths, given a re-evaluation of the truth-values of every other statement in one's complete theory.

Truth values and tautologies

Considering different interpretations of the same statement leads to the notion of truth value. The simplest approach to truth values means that the statement may be "true" in one case, but "false" in another. In one sense of the term tautology, it is any type of formula or proposition which turns out to be true under any possible interpretation of its terms (may also be called a valuation or assignment depending upon the context). This is synonymous to logical truth.

However, the term tautology is also commonly used to refer to what could more specifically be called truth-functional tautologies. Whereas a tautology or logical truth is true solely because of the logical terms it contains in general (e.g. "every", "some", and "is"), a truth-functional tautology is true because of the logical terms it contains which are logical connectives (e.g. "or", "and", and "nor"). Not all logical truths are tautologies of such a kind. 

Logical truth and logical constants

Logical constants, including logical connectives and quantifiers, can all be reduced conceptually to logical truth. For instance, two statements or more are logically incompatible if, and only if their conjunction is logically false. One statement logically implies another when it is logically incompatible with the negation of the other. A statement is logically true if, and only if its opposite is logically false. The opposite statements must contradict one another. In this way all logical connectives can be expressed in terms of preserving logical truth. The logical form of a sentence is determined by its semantic or syntactic structure and by the placement of logical constants. Logical constants determine whether a statement is a logical truth when they are combined with a language that limits its meaning. Therefore, until it is determined how to make a distinction between all logical constants regardless of their language, it is impossible to know the complete truth of a statement or argument.

Logical truth and rules of inference

The concept of logical truth is closely connected to the concept of a rule of inference.

Logical truth and logical positivism

Logical positivism was a movement in the early 20th century that tried to reduce the reasoning processes of science to pure logic. Among other things, the logical positivists claimed that any proposition that is not empirically verifiable is neither true nor false, but nonsense. This movement faded out due to various problems with their approach among which was a growing understanding that science does not work in the way that the positivists described. Another problem was that one of the favorite slogans of the movement: "any proposition that is not empirically verifiable is nonsense" was itself not empirically verifiable, and therefore, by its own terms, nonsense.

Non-classical logics

Non-classical logic is the name given to formal systems which differ in a significant way from standard logical systems such as propositional and predicate logic. There are several ways in which this is done, including by way of extensions, deviations, and variations. The aim of these departures is to make it possible to construct different models of logical consequence and logical truth.

Science in newly industrialized countries

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Science_in_newly_industrialized_countries

Scientific research is concentrated in the developed world, with only a marginal contribution from the rest of the world. Most Nobel Laureates are either from United States, Europe, or Japan. Many newly industrialized countries have been trying to establish scientific institutions, but with limited success. There is an insufficient dedicated, inspired and motivated labor pool for science and insufficient investment in science education.

The limited success of Newly Industrialized Countries

The reason that there have been so few scientists, who have made their mark globally, from most NIC's (Newly Industrialized Countries) is partly historical and partly social  A true scientist is nurtured from the school upwards to scientific establishments. Only if there are inspired and dedicated school science teachers in abundance, there will be a sufficient number of inspired students who would like to take science as a career option and who may one day become a successful scientist. 

The common thread

A common thread can indeed be discerned in the state of science in many NICs. Thus although, most of the science establishments in the major NICs can be said to be doing fairly well, none of them have been as successful as the developed countries.

After the Second World War, a small technical elite arose in developing countries such as India, Pakistan, Brazil, and Iraq who had been educated as scientists in the industrialized world. They spearheaded the development of science in these countries, presuming that by pushing for Manhattan project-type enterprises in nuclear power, electronics, pharmaceuticals, or space exploration they could leapfrog the dismally low level of development of science establishments in their countries. India, for example, started a nuclear energy program that mobilized thousands of technicians and cost hundreds of millions of dollars but had limited success. Though China, North Korea, India and Pakistan have been successful in deploying nuclear weapons and some of them e.g. China and India have launched fairly successful space programs, (for example, Chandrayaan I (Sanskrit चंद्रयान-1), which literally means "Moon Craft," is an unmanned lunar mission by the Indian Space Research Organisation and it hopes to land a motorised rover on the moon in 2010 or 2011 as a part of its second Chandrayaan mission; Chang'e I, China's moon probing project is proceeding in full swing in a well-organized way), the fact remains that most of the scientists responsible for these deeds had received their terminal education from some institution or university in US or Europe. In addition there have been hardly any Nobel laureates in science who have conducted the path-breaking research in a native science establishment. 

Science in Brazil

Brazilian science effectively began in the 19th century, until then, Brazil was a poor colony, without universities, printing presses, libraries, museums, etc. This was perhaps a deliberate policy of the Portuguese colonial power, because they feared that the appearance of educated Brazilian classes would boost nationalism and aspirations toward political independence.

The first attempts of having a Brazilian science establishment were made around 1783, with the expedition of Portuguese naturalist Alexandre Rodrigues, who was sent by Portugal's prime minister, the Marquis of Pombal, to explore and identify Brazilian fauna, flora and geology. His collections, however, were lost to the French, when Napoleon invaded, and were transported to Paris by Étienne Geoffroy Saint-Hilaire. In 1772, the first learned society, the Sociedade Scientifica, was founded in Rio de Janeiro, but lasted only until 1794. Also, in 1797, the first botanic institute was founded in Salvador, Bahia. In the second and third decades of the twentieth century, the main universities in Brazil were organised from a set of existing medical, engineering and law schools. The University of Brazil dates from 1927, the University of São Paulo - today the largest in the Country - dates from 1934.

Today, Brazil has a well-developed organization of science and technology. Basic research in science is largely carried out in public universities and research centers and institutes, and some in private institutions, particularly in non-profit non-governmental organizations. More than 90% of funding for basic research comes from governmental sources.

Applied research, technology and engineering is also largely carried out in the university and research centers system, contrary-wise to more developed countries such as the United States, South Korea, Germany, Japan, etc. A significant trend is emerging lately. Companies such as Motorola, Samsung, Nokia and IBM have established large R&D&I centers in Brazil. One of the incentive factors for this, besides the relatively lower cost and high sophistication and skills of Brazilian technical manpower, has been the so-called Informatics Law, which exempts from certain taxes up to 5% of the gross revenue of high technology manufacturing companies in the fields of telecommunications, computers, digital electronics, etc. The Law has attracted annually more than 1,5 billion dollars of investment in Brazilian R&D&I. Multinational companies have also discovered that some products and technologies designed and developed by Brazilians are significantly competitive and are appreciated by other countries, such as automobiles, aircraft, software, fiber optics, electric appliances, and so on.

The challenges Brazilian science faces today are: to expand the system with quality, supporting the installed competence; transfer knowledge from the research sector to industry; embark on government action in strategic areas; enhance the assessment of existing programmes and commence innovative projects in areas of relevance for the Country. Furthermore, scientific dissemination plays a fundamental role in transforming the perception of the public at large of the importance of science in modern life. The government has undertaken to meet these challenges using institutional base and the operation of existing qualified scientists.

Science in China

  1. A question that has been intriguing many historians studying China is the fact that China did not develop a scientific revolution and Chinese technology fell behind that of Europe. Many hypotheses have been proposed ranging from the cultural to the political and economic. Nathan Sivin has argued that China indeed had a scientific revolution in the 17th century and that we are still far from understanding the scientific revolutions of the West and China in all their political, economic and social ramifications. Some like John K. Fairbank are of the opinion that the Chinese political system was hostile to scientific progress.
Needham argued, and most scholars agreed, that cultural factors prevented these Chinese achievements from developing into what could be called "science". It was the religious and philosophical framework of the Chinese intellectuals which made them unable to believe in the ideas of laws of nature. More recent historians have questioned political and cultural explanations and have focused more on economic causes. Mark Elvin's high level equilibrium trap is one well-known example of this line of thought, as well as Kenneth Pomeranz' argument that resources from the New World made the crucial difference between European and Chinese development. 

Thus, it was not that there was no order in nature for the Chinese, but rather that it was not an order ordained by a rational personal being, and hence there was no conviction that rational personal beings would be able to spell out in their lesser earthly languages the divine code of laws which he had decreed aforetime. The Taoists, indeed, would have scorned such an idea as being too naive for the subtlety and complexity of the universe as they intuited it. Similar grounds have been found for questioning much of the philosophy behind traditional Chinese medicine, which, derived mainly from Taoist philosophy, reflects the classical Chinese belief that individual human experiences express causative principles effective in the environment at all scales. Because its theory predates use of the scientific method, it has received various criticisms based on scientific thinking. Even though there are physically verifiable anatomical or histological bases for the existence of acupuncture points or meridians, for instance skin conductance measurements show increases at the predicted points.

Today, science and technology establishment in the People's Republic of China is growing rapidly. Even as many Chinese scientists debate what institutional arrangements will be best for Chinese science, reforms of the Chinese Academy of Sciences continue. The average age of researchers at the Chinese Academy of Sciences has dropped by nearly ten years between 1991 and 2003. However, many of them are educated in the United States and other foreign countries.

Chinese university undergraduate and graduate enrollments more than doubled from 1995 to 2005. The universities now have more cited PRC papers than CAS in the Science Citation Index. Some Chinese scientists say CAS is still ahead on overall quality of scientific work but that lead will only last five to ten years. 

Several Chinese immigrants to the United States have also been awarded the Nobel Prize, including:, Samuel C. C. Ting, Chen Ning Yang, Tsung-Dao Lee, Yuan T. Lee, Daniel C. Tsui, and Gao Xingjian. Other overseas ethnic Chinese that have achieved success in sciences include Fields Medal recipient Shing-Tung Yau and Terence Tao, and Turing Award recipient Andrew Yao. Tsien Hsue-shen was a prominent scientist at NASA's Jet Propulsion Laboratory, while Chien-Shiung Wu contributed to the Manhattan Project (some argue she never received the Nobel Prize unlike her colleagues Tsung-Dao Lee and Chen Ning Yang due to sexism by the selection committee). Others include Charles K. Kao, a pioneer in fiber optics technology, and Dr. David Ho, one of the first scientists to propose that AIDS was caused by a virus, thus subsequently developing combination antiretroviral therapy to combat it. Dr. Ho was named TIME magazine's 1996 Man of the Year. In 2015, Tu Youyou, a pharmaceutical chemist, became the first native Chinese scientist, born and educated and carried out research exclusively in the People's Republic of China, to receive the Nobel Prize in natural sciences.

Science in India

The earliest applications of science in India took place in the context of medicine, metallurgy, construction technology (such as ship building, manufacture of cement and paints) and in textile production and dyeing. But in the process of understanding chemical processes, led to some theories about physical processes and the forces of nature that are today studied as specific topics within the fields of chemistry and physics.

Many mathematical concepts today were contributed by Indian mathematicians like Aryabhata.

There was really no place for scientists in the Indian caste system. Thus while there were/are castes for the learned brahmins, the warriors kshatriyas, the traders vaishyas and the menial workers shudras, maybe even the bureaucrats (the kayasths) there was/is hardly any formal place in the social hierarchy for a people who discover new knowledge or invent new devices based on the recently discovered knowledge, even though scientific temper has always been in India, in the form of logic, reasoning and method of acquiring knowledge. Its therefore no wonder that some Indians quickly learned to value science, especially those belonging to the privileged Brahmin caste during the British colonial rule that lasted over two centuries. Some Indians did succeed to achieve notable success and fame, examples include Satyendra Nath Bose, Meghnad Saha, Jagdish Chandra Bose and C. V. Raman even though they belonged to different castes. The science communication had begun with publication of a scientific journal, Asiatick Researches in 1788. Thereafter, the science communication in India has evolved in many facets. Following this, there has been a continuing development in the formation of scientific institutions and publication of scientific literature. Subsequently, scientific publications also started appearing in Indian languages by the end of eighteenth century. The publication of ancient scientific literature and textbooks at mass scale started in the beginning of nineteenth century. The scientific and technical terms, however, had been a great difficulty for a long time for popular science writing.

Ecological civilization

From Wikipedia, the free encyclopedia https://en.wikipedia.org/wiki/Eco...