The term Validity in logic Logic, from the Greek λογική is defined by the Penguin Encyclopedia to be "The formal systematic study of the principles of valid inference and correct reasoning". As a discipline, logic dates back to Aristotle, who established its fundamental place in philosophy. It became part of the classical trivium, a fundamental part of a applies to arguments In logic, an argument is a set of one or more meaningful declarative sentences known as the premises along with another meaningful declarative sentence (or "proposition") known as the conclusion. A deductive argument asserts that the truth of the conclusion is a logical consequence of the premises; an inductive argument asserts that the or statements.

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Validity of arguments

An argument In logic, an argument is a set of one or more meaningful declarative sentences known as the premises along with another meaningful declarative sentence (or "proposition") known as the conclusion. A deductive argument asserts that the truth of the conclusion is a logical consequence of the premises; an inductive argument asserts that the is valid if and only if the truth of its premises entails In logic and mathematics, entailment or logical implication is a logical relation that holds between a set T of propositions and a proposition B when every model of T is also a model of B. In symbols, the truth of its conclusion, it would be self-contradictory to affirm the premises and deny the conclusion. The corresponding conditional In logic, the corresponding conditional of an argument is a material conditional whose antecedent is the conjunction of the argument's (or derivation's) premises and whose consequent is the argument's conclusion. An argument is valid if and only if its corresponding conditional is a logical truth. It follows that an argument is valid if and only of a valid argument is a logical truth and the negation of its corresponding conditional is a contradiction. The conclusion is a logical consequence Logical consequence is a fundamental concept in logic. It is the relation that holds between a set of sentences and a sentence (proposition) when the former "entails" the latter. For example, 'Kermit is green' is said to be a logical consequence of 'All frogs are green' and 'Kermit is a frog', because it would be "self-contradictory& 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 A syllogism, or logical appeal, , (usually the categorical syllogism) is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form:

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 The form or logical form of an argument is the representation of its sentences using the formal grammar and symbolism of a logical system to display its similarity with all other arguments of the same type 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 mortal.
Socrates is mortal.
Therefore, Socrates is a man.

In this case, the conclusion does not follow inescapably from the premises: a universe is easily imagined in which ‘Socrates’ is not a man but a woman, so that in fact the above premises would be true but the conclusion false. This possibility makes the argument invalid. (Although whether or not an argument is valid does not depend on what anyone could actually imagine to be the case, this approach helps us evaluate some arguments.)

A standard view is that whether an argument is valid is a matter of the argument’s logical form The form or logical form of an argument is the representation of its sentences using the formal grammar and symbolism of a logical system to display its similarity with all other arguments of the same type. Many techniques are employed by logicians to represent an argument’s logical form. A simple example, applied to the above two 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 Q.
S is a Q.
Therefore, S is a P.

These abbreviations make plain the logical form of each respective argument. At this level, notice that we can talk about any arguments that may take on one or the other of the above two configurations, by replacing the letters P, Q and s by appropriate expressions. Of particular interest is the fact that we may exploit an argument's form to help discover whether or not the argument from which it has been obtained is or is not valid. To do this, we define an “interpretation In logic, an interpretation is a function that provides the extension of symbols and strings of symbols of an object language. For example, an intepretation function could take the predicate T and assign it the extension {a} (for "Abraham Lincoln"), signifying that under this function Abraham Lincoln is considered tall. The object” of the argument as an assignment of sets of objects to the upper-case letters in the argument form, and the assignment of a single individual member of a set to the lower-case letters of the argument form. Thus, letting P stand for the set of men, Q stand for the set of mortals, and s stand for Socrates is an interpretation of each of the above arguments. Using this terminology, we may give a formal analogue of the definition of deductive validity:

An argument is formally valid if its form is one such that for each interpretation under which the premises are all true also the conclusion is true.

As already seen, the interpretation given above does cause the second argument form to have true premises and false conclusion, hence demonstrating its invalidity.

Validity of statements

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

If no god is mortal, then no mortal is a god.

In logical form, this is:

If (No P is a Q), then (No Q is a P).

A given statement may be entailed by other statements, i.e. if the given statement must be true if the other statements are true. This means that an argument with the given statement as its conclusion and the other statements as its premises is a valid argument. The corresponding conditional In logic, the corresponding conditional of an argument is a material conditional whose antecedent is the conjunction of the argument's (or derivation's) premises and whose consequent is the argument's conclusion. An argument is valid if and only if its corresponding conditional is a logical truth. It follows that an argument is valid if and only of a valid argument is a logical truth.

Validity and Soundness

One thing we should note is that the validity of deduction is not at all affected by the truth of the premise or the truth of the conclusion. The following deduction is perfectly valid:

All fire-breathing rabbits live on Mars
All humans are fire-breathing rabbits
Therefore all humans live on Mars

The problem with the argument is that it is not sound In mathematical logic, a logical system has the soundness property if and only if its inference rules prove only formulas that are valid with respect to its semantics. In most cases, this comes down to its rules having the property of preserving truth, but this is not the case in general. The word derives from the Germanic 'Sund' as in gesund,. In order for a deductive argument to be sound, the deduction must be valid and the premise must be true.

Satisfiability and validity

Main article: Satisfiability and validity In mathematical logic, satisfiability and validity are elementary concepts concerning interpretation . A formula is satisfiable with respect to a class of interpretations if it is possible to find an interpretation that makes the formula true. A formula is valid if all such interpretations make the formula true. These notions can be relativised to

Model theory In mathematics, model theory is the study of mathematical structures such as groups, fields, graphs, or even models of set theory, using tools from mathematical logic. Model theory has close ties to algebra and universal algebra analyses 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 premisses, validate the conclusion.

Logical truths

Logical truths (including tautologies In propositional logic, a tautology is a propositional formula that is true under any possible valuation (also called a truth assignment or an interpretation) of its propositional variables. For example, the propositional formula ("A or not-A") is a tautology, because the statement is true for any valuation of A. Examples can be more) are necessarily true. One theory is that a proposition In logic and philosophy, proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or (b) the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence. Propositions in either case are intended to be truth-bearers, that is, they are either true or false such as “If p and q, then p” and the proposition “All husbands are married” are logical truths because they are true due to their inherent meanings and not because of any facts of the world. They are such that they could not be untrue.

A logical truth was considered by Ludwig Wittgenstein Described by Bertrand Russell as "the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, and dominating," Wittgenstein is considered by many to be the greatest philosopher of the 20th century. He helped inspire two of the century's principal philosophical movements: the Vienna to be a statement which is true in all possible worlds[1]. This is contrasted with synthetic claim (or fact A fact is a pragmatic truth, a statement that can, at least in theory, be checked and either confirmed or denied. Facts are often contrasted with opinions and beliefs, statements which are held to be true, but are not amenable to pragmatic confirmation or denial) which is 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.

Later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations.

See also

Logic portal Logic is the study of the principles and criteria of valid inference and demonstration

References

  1. ^ Ludwig Wittgenstein Described by Bertrand Russell as "the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, and dominating," Wittgenstein is considered by many to be the greatest philosopher of the 20th century. He helped inspire two of the century's principal philosophical movements: the Vienna, Tractatus Logico-Philosophicus Tractatus Logico-Philosophicus is the only book-length philosophical work published by the Austrian philosopher Ludwig Wittgenstein during his lifetime. He wrote it as a soldier and a prisoner of war during World War I. First published in German in 1921 as Logisch-Philosophische Abhandlung, it is now widely considered one of the most important

External links

Look up validity in Wiktionary Wiktionary is a multilingual, Web-based project to create a free content dictionary, available in over 151 languages. Unlike standard dictionaries, it is written collaboratively by volunteers, dubbed "Wiktionarians", using wiki software, allowing articles to be changed by almost anyone with access to the website, the free dictionary.
Logic Logic, from the Greek λογική is defined by the Penguin Encyclopedia to be "The formal systematic study of the principles of valid inference and correct reasoning". As a discipline, logic dates back to Aristotle, who established its fundamental place in philosophy. It became part of the classical trivium, a fundamental part of a
History and core topics
History General The history of logic is the study of the development of the science of valid inference . While many cultures have employed intricate systems of reasoning, and logical methods are evident in all human thought, an explicit analysis of the principles of reasoning was developed only in three traditions: those of China, India, and Greece. Of these, · Chinese In the history of logic, logic in China plays a particularly interesting role due to its length and relative isolation from the strong current of development of the study of logic in Europe and the Islamic world, though it may have some influence from Indian logic due to the spread of Buddhism · Greek The Organon is the name given by Aristotle's followers, the Peripatetics, to the standard collection of his six works on logic. The works are Categories, On Interpretation, Prior Analytics, Posterior Analytics, Topics and Sophistical Refutations · Indian The development of Indian logic can be said to date back to the anviksiki of Medhatithi Gautama ; the Sanskrit grammar rules of Pāṇini (c. 5th century BCE); the Vaisheshika school's analysis of atomism (c. 2nd century BCE); the analysis of inference by Gotama (c. 2nd century BCE), founder of the Nyaya school of Hindu philosophy; and the · Islamic Logic played an important role in early Islamic philosophy. Islamic law placed importance on formulating standards of argument, which gave rise to a novel approach to logic in Kalam, as seen in the method of qiyas. This approach, however, was later displaced to some extent by ideas from ancient Greek and Hellenistic philosophy with the rise of the
Core topics Reason Reason, as used in this article, refers to mental faculties that generate or affirm propositions, by activities of the mind such as judging, predicting, inferring, generalizing, and comparing · Philosophical logic Philosophical logic is the study of the more specifically philosophical aspects of logic. The term contrasts with philosophy of logic, metalogic, and mathematical logic; and since the development of mathematical logic in the late nineteenth century, it has come to include most of those topics traditionally treated by logic in general.[citation · Philosophy of logic Following the developments in Formal logic with symbolic logic in the late nineteenth century and mathematical logic in the twentieth, topics traditionally treated by logic not being part of formal logic have tended to be termed either philosophy of logic or philosophical logic if no longer simply logic · Mathematical logic Mathematical logic is a subfield of mathematics with close connections to computer science and philosophical logic. The field includes the mathematical study of logic and the applications of formal logic to other areas of mathematics. The unifying themes in mathematical logic include the study of the expressive power of formal systems and the · Metalogic Metalogic is the study of the metatheory of logic. While logic is the study of the manner in which logical systems can be used to decide the correctness of arguments, metalogic studies the properties of the logical systems themselves. According to Geoffrey Hunter, while logic concerns itself with the "truths of logic," metalogic concerns · Logic in computer science Logic in computer science describes topics where logic is applied to computer science and artificial intelligence. These include:
Key concepts and logics
Reasoning Humans have the ability to engage in reasoning about their own reasoning. Different forms of such reflection on reasoning occur in different fields. In philosophy, the study of reasoning typically focuses on what makes reasoning efficient or inefficient, appropriate or inappropriate, good or bad. Philosophers do this by either examining the form Deduction Deductive reasoning, sometimes called deductive logic, is reasoning which constructs or evaluates deductive arguments. In logic, an argument is said to be deductive when the truth of the conclusion is purported to follow necessarily or be a logical consequence of the premises and its corresponding conditional is a necessary truth. Deductive · Induction Induction or inductive reasoning, sometimes called inductive logic, is reasoning which takes us "beyond the confines of our current evidence or knowledge to conclusions about the unknown." The premises of an inductive argument indicate some degree of support for the conclusion but do not entail it; i.e. they do not ensure its truth · Abduction Abduction is a method of logical inference introduced by Charles Sanders Peirce which comes prior to induction and deduction for which the colloquial name is guessing. Abductive reasoning starts when an inquirer considers of a set of seemingly unrelated facts, armed with the hunch that they are somehow connected. The term abduction is commonly
Informal The precise nature and definition of informal logic are matters of some dispute. Ralph H. Johnson and J. Anthony Blair define informal logic as "a branch of logic whose task is to develop non-formal standards, criteria, procedures for the analysis, interpretation, evaluation, criticism and construction of argumentation." This definition Proposition In logic and philosophy, proposition refers to either the "content" or "meaning" of a meaningful declarative sentence or (b) the pattern of symbols, marks, or sounds that make up a meaningful declarative sentence. Propositions in either case are intended to be truth-bearers, that is, they are either true or false · Inference The process by which a conclusion is inferred from multiple observations is called inductive reasoning. The conclusion may be correct or incorrect, or correct, or correct to within a certain degree of accuracy, or correct in certain situations. Conclusions inferred from multiple observations may be tested by additional observations · Argument In logic, an argument is a set of one or more meaningful declarative sentences known as the premises along with another meaningful declarative sentence (or "proposition") known as the conclusion. A deductive argument asserts that the truth of the conclusion is a logical consequence of the premises; an inductive argument asserts that the · Validity · Cogency An argument is cogent if and only if the truth of the argument's premises would render the truth of the conclusion probable , and the argument's premises are, in fact, true. Cogency can be considered inductive logic's analogue to deductive logic's "soundness." As an example, consider the following · Term logic In philosophy, term logic, also known as traditional logic, is a loose name for the way of doing logic that began with Aristotle, and that was dominant until the advent of modern predicate logic in the late nineteenth century · Critical thinking Critical thinking is purposeful and reflective judgment about what to believe or what to do in response to observations, experience, verbal or written expressions, or arguments. Critical thinking might involve determining the meaning and significance of what is observed or expressed, or, concerning a given inference or argument, determining · Fallacies · Syllogism · Argumentation theory
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