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ArgumentForms

ArgumentForms

In logic, the argument form or test form of an argument results from replacing the different words, or sentences, that make up the argument with letters, along the lines of algebra; the letters represent logical variables. The sentence forms which classify argument forms of common arguments important are studied in college logic. Here is an example of an argument: A All humans are mortal. Socrates is human. Therefore, Socrates is mortal. We can rewrite argument A by putting each sentence on its own line: B :All humans are mortal. :Socrates is human. :Therefore, Socrates is mortal. To demonstrate the important notion of the form of an argument, substitute letters for similar items throughout B: C :All S are P. :a is S. :Therefore, a is P. All we have done in C is to put 'S' for 'human' and 'humans', 'P' for 'mortal', and a for 'Socrates'; what results, C, is the form of the original argument in A. So argument form C is the form of argument A. Moreover, each individual sentence of C is the sentence form of its respective sentence in A. Attention is given to argument and sentence form, because form is what makes an argument valid or cogent. Some examples of valid arguments forms are modus ponens, modus tollens, and the disjunctive syllogism. Two invalid argument forms are affirming the consequent and denying the antecedent.

Logical argument

An argument is an attempt to demonstrate the truth of an assertion called a conclusion, based on the truth of a set of assertions called premises. The process of demonstration of deductive (see also deduction) and inductive reasoning shapes the argument, and presumes some kind of communication, which could be part of a written text, a speech or a conversation. In ordinary, philosophical and scientific argumentation abductive arguments and arguments by analogy are also commonly used. Arguments can be valid or invalid, although how arguments are determined to be in either of these two categories can often itself be an object of much discussion. Informally one should expect that a valid argument should be compelling in the sense that it is capable of convincing someone about the truth of the conclusion. However, such a criterion for validity is inadequate or even misleading since it depends more on the skill of the person constructing the argument to manipulate the person who is being convinced and less on the argument itself. Less subjective criteria for validity of arguments are often clearly desirable, and in some cases we should even expect an argument to be rigorous, that is, to adhere to precise rules of validity. This is the case for arguments used in mathematical proofs. Note that a rigorous proof does not have to be a formal proof. In ordinary language, people refer to the logic of an argument or use terminology that suggests that an argument is based on inference rules of formal logic. Though arguments do use inferences that are indisputably purely logical (such as syllogisms), other kinds of inferences are almost always used in practical arguments. For example, arguments commonly deal with causality, probability and statistics or even specialized areas such as economics. In these cases, logic refers to the structure of the argument rather than to principles of pure logic that might be used in it.

Argument validity

In evaluating an argument, we consider separately the truth of the premises and the validity of the logical relationships between the premises, any intermediate assertions and the conclusion. The main logical property of an argument that is of concern to us here is whether it is truth preserving, that is if the premises are true, then so is the conclusion. We will usually abbreviate this property by saying simply that argument is valid. If the argument is valid, the premises together entail or imply the conclusion. The ways in which arguments go wrong tend to fall into certain patterns, called logical fallacies. Validity is a semantic characteristic of arguments; independently of this property, and more controversially, arguments should also be scrutinizable, in the sense that the argument be open to public examination and systematic in the sense that the structural components of the argument have public legitimacy.

The mathematical paradigm

In mathematics, an argument can be formalized using symbolic logic. In that case, an argument is seen as an ordered list of statements, each one of which is either one of the premises or derivable from the combination of some subset of the preceding statements and one or more axioms using rules of inference. The last statement in the list is the conclusion. Most arguments used in mathematical proof are rigorous, but not formal. In fact, strictly formal proofs of all but the most trivial assertions are extremely hard to construct and hard to understand without some assistance from a computer. One of the goals of automated theorem proving is to design computer programs to produce and check formal proofs. A study of formal systems of mathematics together with semantic questions such as completeness and validity is often called metamathematics. Of particular note in this direction are the Gödel's incompleteness theorems for first order theories of arithmetic. The prevalent belief among mathematical authors is that valid arguments in mathematics are those that can be recognized as being in principle formalizable in the encompassing formal theory. It follows that the theory of valid arguments in mathematics is reducible to the theory of valid inferences in formal mathematical theories. A theory of validity of formal mathematical theories posits two distinct elements: syntax which gives the rules for when a formula is correctly constructed and semantics which is essentially a function from formulas to truth values. An expression is said to be valid if the semantic function assigns the value true to it. A rule of inference is valid if and only if it is validity-preserving. An argument is valid if and only if it utilizes valid rules of inference. Note that in the case of mathematical semantics, both the syntax and semantics are mathematical objects. In general usage, however, arguments are rarely formal or even have the rigor of mathematical proofs.

Theories of arguments

Theories of arguments are closely related to theories of informal logic. Ideally, a theory of argument should provide some mechanism for explaining validity of arguments. One natural approach would follow the mathematical paradigm and attempt to define validity in terms of semantics of the assertions in the argument. Though such an approach is appealing in its simplicity, the obstacles to proceeding this way are very difficult for anything other than purely logical arguments. Among other problems, we need to interpret not only entire sentences, but also components of sentences, for example noun phrases such as The present value of government revenue for the next twelve years. One major difficulty of pursuing this approach is that determining an appropriate semantic domain is not an easy task, raising numerous thorny ontological issues. It also raises the discouraging prospect of having to work out acceptable semantic theories before being able to say anything useful about understanding and evaluating arguments. For this reason the purely semantic approach is usually replaced with other approaches that are more easily applicable to practical discourse. For arguments regarding topics such as probability, economics or physics, some of the semantic problems can be conveniently shoved under the rug if we can avail ourselves of an model of the phenomenon under discussion. In this case, we can establish a limited semantic interpretation using the terms of the model and the validity of the argument is reduced to that of the abstract model. This kind of reduction is used in the natural sciences generally, and would be particularly helpful in arguing about social issues if the parties can agree on a model. Unfortunately, this prior reduction seldom occurs, with the result that arguments about social policy rarely have a satisfactory resolution. Another approach is to develop a theory of argument pragmatics, at least in certain cases where argument and social interaction are closely related. This is most useful when the goal of logical argument is to establish a mutually satisfactory resolution of a difference of opinion between individuals.

Argumentative dialogue

Arguments as discussed in the preceding paragraphs are static, such as one might find in a textbook or research article. They serve as a published record of justification for an assertion. Arguments can also be interactive, in which the proposer and the interlocutor have a more symmetrical relationship. The premises are discussed, as well the validity of the intermediate inferences. For example, consider the following exchange, illustrated by the No true Scotsman fallacy: : Argument: "No Scotsman puts sugar on his porridge." : Reply: "But my friend Angus likes sugar with his porridge." : Rebuttal: "Ah yes, but no true Scotsman puts sugar on his porridge." In this dialogue, the proposer first offers a premise, the premise is challenged by the interlocutor, and finally the proposer offers a modification of the premise. This exchange could be part of a larger discussion, for example a murder trial, in which the defendant is a Scotsman, and it had been established earlier that the murderer was eating sugared porridge when he or she committed the murder. In argumentative dialogue, the rules of interaction may be negotiated by the parties to the dialogue, although in many cases the rules are already determined by social mores. In the most symmetrical case, argumentative dialogue can be regarded as a process of discovery more than one of justification of a conclusion. Ideally, the goal of argumentative dialogue is for participants to arrive jointly at a conclusion by mutually accepted inferences. In some cases however, the validity of the conclusion is secondary. For example; emotional outlet, scoring points with an audience, wearing down an opponent or lowering the sale price of an item may instead be the actual goals of the dialogue. Walton distinguishes several types of argumentative dialogue which illustrate these various goals:
- Personal quarrel.
- Forensic debate.
- Persuasion dialogue.
- Bargaining dialogue.
- Action seeking dialogue.
- Educational dialogue. Van Eemeren and Grootendorst identify various stages of argumentative dialogue. These stages can be regarded as an argument protocol. In a somewhat loose interpretation, the stages are as follows:
- Confrontation: Presentation of the problem, such as a debate question or a political disagreement
- Opening: Agreement on rules, such as for example, how evidence is to be presented, which sources of facts are to be used, how to handle divergent interpretations, determination of closing conditions.
- Argumentation: Application of logical principles according to the agreed-upon rules
- Closing: This occurs when the termination conditions are met. Among these could be for example, a time limitation or the determination of an arbiter. Van Eemeren and Grootendorst provide a detailed list of rules that must be applied at each stage of the protocol. Moreover, in the account of argumentation given by these authors, there are specified roles of protagonist and antagonist in the protocol which are determined by the conditions which set up the need for argument. Many cases of argument are highly unsymmetrical, although in some sense they are dialogues. A particularly important case of this is political argument. Much of the recent work on argument theory has considered argumentation as an integral part of language and perhaps the most important function of language (Grice, Searle, Austin, Popper). This tendency has removed argumentation theory away from the realm of pure formal logic. One of the original contributors to this trend is the philosopher Chaim Perelman, who together with Lucie Olbrechts-Tyteca, introduced the French term La nouvelle rhetorique in 1958 to describe an approach to argument which is not reduced to application of formal rules of inference. Perelman's view of argumentation is much closer to a juridical one, in which rules for presenting evidence and rebuttals play an important role. Though this would apparently invalidate semantic concepts of truth, this approach seems useful in situations in which the possibility of reasoning within some commonly accepted model does not exist or this possibility has broken down because of ideological conflict. Retaining the notion enunciated in the introduction to this article that logic usually refers to the structure of argument, we can regard the logic of rhetoric as a set of protocols for argumentation.

Other theories

In recent decades one of the more influential discussions of philosophical arguments is that by Nicholas Rescher in his book The Strife of Systems. Rescher models philosophical problems on what he calls aporia or an aporetic cluster: a set of statements, each of which has initial plausibility but which are jointly inconsistent. The only way to solve the problem, then, is to reject one of the statements. If this is correct, it constrains how philosophical arguments are formulated.

References

- Rober Audi, Epistemology, Routledge, 1998. Particularly relevant is Chapter 6, which explores the relationship between knowledge, inference and argument.
- J. L. Austin How to Do things with Words, Oxford University Press, 1976.
- H. P. Grice, Logic and Conversation in The Logic of Grammar, Dickenson, 1975.
- R. A. DeMillo, R. J. Lipton and A. J. Perlis, Social Processes and Proofs of Theorems and Programs, Communications of the ACM, Vol. 22, No. 5, 1979. A classic article on the social process of acceptance of proofs in mathematics.
- Yu Manin, A Course in Mathematical Logic, Springer Verlag, 1977. A mathematical view of logic. This book is different from most books on mathematical logic in that it emphasizes the mathematics of logic, as opposed to the formal structure of logic.
- Ch. Perelman and L Olbrechts-Tyteca, The New Rhetoric, Notre Dame, 1970. This classic was originally published in French in 1958.
- Henri Poincaré, Science and Hypothesis, Dover Publications, 1952
- Frans van Eemeren and Rob Grootendorst, Speech Acts in Argumentative Discussions, Foris Publications, 1984.
- K. R. Popper Objective Knowledge; An Evolutionary Approach, Oxford: Clarendon Press, 1972.
- L. Stebbing, A Modern Introdcution to Logic, Methuen and Co., 1948. An account of logic that covers the classic topics of logic and argument while carefully considering modern developments in logic.
- Douglas Walton, Informal Logic: A Handbook for Critical Argumentation, Cambridge, 1998
- Carlos Chesñevar, Ana Maguitman and Ronald Loui, Logical Models of Argument, ACM Computing Surveys, vol. 32, num. 4, pp.337-383, 2000.

Algebra

:This article is about the branch of mathematics. For other uses of the term see algebra (disambiguation). Algebra is a branch of mathematics which studies structure and quantity. It may be roughly characterized as a generalization and abstraction of arithmetic, in which operations are performed on symbols rather than numbers. It includes elementary algebra, taught to high school students, as well as abstract algebra which covers such structures as groups, rings and fields. Along with geometry and analysis, it is one of the three main branches of mathematics. The study of Algebra is the cause for some debate as the level taught to High School students is rarely applicable in the real world.

History

The origins of algebra can be traced to the cultures of the ancient Egyptians and Babylonians who used an early type of algebra to solve linear, quadratic, and indeterminate equations more than 3,000 years ago.
- Circa 300 BC: Greek mathematician Euclid, who taught and died at Alexandria in Egypt, in Book 2 of his Elements addresses quadratic equations, although in a strictly geometrical fashion.
- Circa 100 BC: algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu, The Nine Chapters of Mathematical Art.
- Circa 150 AD: Greek mathematician Hero of Alexandria treats algebraic equations in three volumes of mathematics.
- Circa 200 AD: Greek mathematician Diophantus, often referred to as the "father of algebra", writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers.
- 476 AD Indian mathematician, Aryabhata obtains whole number solutions to linear equations by a method equivalent to modern one. Bhaskara II (1114 AD), who wrote the text Bijaganita (algebra), was the first to recognize that a positive number has two square roots. The Hindus recognized that quadratic equations have two roots, and included negative as well as irrational roots. They treated indeterminate quadratic equations.
- 820 AD The word algebra is derived from the name of the treatise first written by Persian mathematician Khwarizmi titled: Al-Jabr wa-al-Muqabilah meaning The book of summary concerning calculating by transposition and reduction. The word al-jabr means "reunion".
- 1202 AD Algebra was introduced to Europe largely through the work of Leonardo Fibonacci of Pisa in his work Liber Abaci .

Classification

Algebra may be roughly divided into the following categories:
- elementary algebra, in which the properties of operations on the real number system are recorded using symbols as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied (note that this usually includes the subject matter of courses called intermediate algebra and college algebra);
- abstract algebra, sometimes also called modern algebra, in which algebraic structures such as groups, rings and fields are axiomatically defined and investigated;
- linear algebra, in which the specific properties of vector spaces are studied (including matrices);
- universal algebra, in which properties common to all algebraic structures are studied. In advanced studies, axiomatic algebraic systems like groups, rings, fields, and algebras over a field are investigated in the presence of a natural geometric structure (a topology) which is compatible with the algebraic structure. The list includes:
- Normed linear spaces
- Banach spaces
- Hilbert spaces
- Banach algebras
- Normed algebras
- Topological algebras
- Topological groups

Algebras

The word algebra is also used for various algebraic structures:
- algebra over a field
- algebra over a set
- Boolean algebra
- sigma-algebra
- F-algebra and F-coalgebra in category theory

References

- Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
- Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
- George Gheverghese Joseph, The Crest of the Peacock : The Non-European Roots of Mathematics (Princeton University Press, 2000).

External links

- [http://www.mathleague.com/help/algebra/algebra.htm Explanation of Basic Topics]
- [http://www.sparknotes.com/math/#algebra1 Sparknotes' Review of Algebra I and II]
- [http://www.jamesbrennan.org/algebra/ Understanding Algebra.] An online algebra text by James W. Brennan. Category:Algebra Category:Arabic words ko:대수학 ms:Algebra ja:代数学 simple:Algebra

Variable

In computer science and mathematics, a variable is a symbol denoting a quantity or symbolic representation. In mathematics, a variable often represents an unknown quantity; in computer science, it represents a place where a quantity can be stored. Variables are often contrasted with constants, which are known and unchanging. In other scientific fields such as biology, chemistry, and physics, the word variable is used to refer to a measurable factor, characteristic, or attribute of an individual or a system. In a scientific experiment, so called "independent variables" are factors that can be altered or chosen by the scientist. For example, temperature is a common environmental factor that can be controlled in laboratory experiments. "Dependent variables" or "response variables" are those that are measured and collected as data. An independent variable is presumed to affect a dependent one. In social sciences, variable is a logical set of attributes. A variable such as "gender" would have two attributes: male and female.

General overview

Variables are used in open sentences. For instance, in the formula: x + 1 = 5, x is a variable which represents an "unknown" number. In mathematics, variables are usually represented by letters of the Roman alphabet, but are also represented by letters of other alphabets; as well as various other symbols. In computer programming, variables are usually represented by either single letters or alphanumeric strings.

Why are variables useful?

Variables are useful in mathematics and computer programming because they allow instructions to be specified in a general way. If one were forced to use actual values, then the instructions would only apply in a more narrow, and specific set of situations. For example: specify a mathematical definition for finding the square of ANY number: square(x) = x · x. Now, all we need to do to find the square of a number is replace x with any number we want.
- square(x) = x · x = y
- square(1) = 1 · 1 = 1
- square(2) = 2 · 2 = 4
- square(3) = 3 · 3 = 9 etc... In the above example, the variable x is a "placeholder" for ANY number. One important thing we are assuming is that the value of each occurrence of x is the same -- that x does not get a new value between the first x and the second x. In computer programming languages without referential transparency, such changes can occur.

Computer programming

In programming languages, a variable can be thought of as a place to store a value in computer memory. In general, a variable binds an object to a name so that the object could be accessed later, much like a person has a name and people could refer to him by that name. This is analogous to the use of variables in the mathematics and variables in computer programming work usually in the similar manner. Put in another way, an object could exist without it being bound to a certain variable. Typically, the name of a variable is bound to a particular address of some bytes on the memory, and any operations on the variable would manipulate that block. This is called name binding. If the space is way too large or its size is unknown beforehand, the use of referencing is more common, in which a value is not directly stored in the variable but a location information for it is. Importation questions about variables are twofold: its life-time and scope. For space efficiency, a memory space needed for a variable is allocated when first used and freed if no longer needed. The scope helps determine the life-time of variables. Usually, a variable is set to reside in some scope in program code, and entrance and leave of the scope coincides with the beginning and ending of a variable life, respectively. Put in conceptual terms, a variable is visible in its scope, and computers could assume the variable is needed only when it is visible. In this way, however, unused variables might be given a space, which is going to be never used. Because of this, a compiler often warns programmers when a variable is declared but not used at all. While a variable stores simple data like integers and literal strings, some languages allow a variable to store datatype as well. They enable parametric polymorphic functions to be written. They operate like variables, in that they can represent any type. For example, with the function length -- to determine the length of a list, it is only necessary to know the amount of elements in the list -- the type of the elements does not count, so the type signature can be represented with a type variable and thus is parametric polymorphic. Variables could be either mutable or immutable. Mutable variables could be thought of ones having l-value while immutable ones having r-value. One characteristic of functional programming is that a variable is immutable. Because immutable variables are semantically the same as constants given a name or constant functions, when one talks about variables, they usually mean mutable variables. See name for naming rules and convention of variable names. In C++ (not in C), "mutable" is a keyword to allow a mutable member to be modified by a const member function. ---- In programming languages, a variable can be thought of as a place to store a value in computer memory. More precisely, a variable associates a name (sometimes called an identifier) with the location of the value; the value in turn is stored as a data object in this location. The specifics of variable allocation and the representation of values vary widely, both among languages and among implementations of any given language.

Constant

A constant is similar, but does not change its value during program execution, and is the same when running the program again. The value is specified only once but the constant can be referenced multiple times in the program. Changing the value is done by a simple change of the program. Using a constant should be distinguished from specifying the value multiple times in the program. The latter would make a change of the value cumbersome. Even if the value is used in one program location only, advantages over just putting the value there are that the name of the constant can be informative about its meaning, and the value is specified at a more convenient standard location in the program, such as at the beginning. Currently, programming languages provide one of 2 kinds of constant variables:
- Static constant: or Manifest constant the declaration of a static constant assigns one fixed value to a name, which will be known before the program starts. Visual Basic uses this style. Example of static constant: : CONST a = 60
- Dynamic constant: The dynamic constant can be assigned with an expression, possibly involving non-constant operands. The value of these constants may rely on variables defined while a computer program is running, so unlike static constants, the values that dynamic constants will take on cannot be determined at compile time. Java uses the dynamic constant style. : final int a = b + 20;

Variables names

Variables are denoted by identifiers. In some languages, specific characters are prepended or appended to variable identifiers to indicate the variable's type. For example:
- in BASIC, the suffix $ on a variable name indicates that its value is a string;
- in Perl, the prefixes $, @, %, and & indicate scalar, array, hash, and subroutine variables, respectively. The identifier article describes cases where the actual identifier itself, rather than a prefix or suffix, is interpreted by the compiler or interpreter. See also:
- namespaces

Scope and extent

The scope and extent (or lifetime) of a variable describe where in the program's text it may be used, and when in the program's execution it has a value. In most languages, variables can have different scopes. The scope of a variable is the portion of the program code for which the variable's name has meaning. For instance, a variable with lexical scope is meaningful only within a certain block of statements or subroutine. A global variable, or one with indefinite scope, may be referred to anywhere in the program. When a variable has gone out of scope, it is erroneous or meaningless to refer to it. Lexical analysis of a program can determine whether variables are used out of scope. Likewise, the bindings of variables to values can have different extent. The extent of a binding is the length of time -- part of the course of the program's execution -- during which the variable continues to refer to the same value or place. A running program may enter and leave a given extent many times, as in the case of a closure. A variable can be unbound, meaning that it is in scope but has never been given a value, or its value has been destroyed; in many languages, it is an error to try to use the value of an unbound variable, or may yield unpredictable results. In other words, scope is a lexical fact, but extent a runtime (dynamic) fact. If a variable name is out of scope, then it is an error for that name to be used in the program code. In compiled languages, this error can be detected statically at compile-time. If a variable is out of extent, its value cannot be referred to (since it doesn't have one; it is unbound) but it may be given a value, which gives it a new extent. When a variable binding extends (in time) as the program's execution passes out of the variable's scope, this is not a bug. It is allowed for Lisp closures or C static variables, for example: when execution passes back into the variable's scope, the variable may be referred to again. But when a variable's extent ends, it becomes unbound -- if it is still in scope, referring to it is an error (or, in C, results in undefined behavior). It is considered good programming practice to make the scope of variables as narrow as feasible so that different parts of a program do not accidentally interact with each other by modifying each other's variables. This also prevents action at a distance. Common techniques for doing so are to have different sections of your program use different namespaces, or else make individual variables private through either dynamic variable scoping or lexical variable scoping. Many programming languages employ a reserved value (often named null or nil) to indicate an invalid or uninitialized variable.

Memory allocation

Bound variables have values. A value, however, is an abstraction, an idea; in implementation, a value is represented by some data object, which is stored somewhere in computer memory. The program, or the runtime environment, must set aside memory for each data object and, since memory is finite, ensure that this memory is yielded for re-use when the object is no longer needed to represent some variable's value. The handling of memory for variables is highly dependent on the programming language environment. Many language implementations handle the simplest cases easily by distinguishing those variables whose extent lasts no longer than a single function call. Space for these local variables are allocated on the execution stack, where their memory is automatically reclaimed when the function returns. Space for other objects must be allocated on the heap, or pool of unused memory. These must be reclaimed specially when the objects are no longer needed. In a garbage-collected (gc) language such as Java or Lisp, the runtime environment automatically "reaps" objects when it can be proven that no extant variable refers to them. In a non-gc language such as C, it is up to the program (and thus the programmer) to explicitly allocate memory; and in turn to state when memory can be reclaimed, by explicitly freeing it. Failure to do so leads to memory leaks, in which the heap is depleted over the program's run. If the program runs long enough, it will exhaust available memory and fail. Memory allocation goes beyond single variables. A variable may refer to a data structure created dynamically, where many structure components are not directly named by variables, but are reachable from a variable by traversing the structure. For this reason, garbage collectors (and programs in languages which lack them) must deal with the case where a portion of the memory reachable from a variable needs to be reclaimed.

Typed and untyped variables

In statically-typed languages such as Java or ML, a variable also has type, meaning that only values of a given sort can be stored in it. In dynamically-typed languages such as Python or Lisp, it is values and not variables which carry type. See type system. Typing of variables also allows polymorphisms to be resolved at compile time.

Parameters

The arguments or formal parameters of functions are also referred to as variables. For instance, in these equivalent functions in Python and Lisp def addtwo(x): return x + 2 (defun addtwo (x) (+ x 2)) the variable named x is an argument. It is given a value when the function is called. In most languages, function arguments have local scope; this specific variable named x can only be referred to within the addtwo function, though of course other functions can also have variables called x.

External link

- [http://www.legislation.hmso.gov.uk/acts/acts2003/30042--b.htm Example of the use of variables (persons A and B) in a law] (NB: this comes from the Sexual Offences Act.) Category:Algebra Category:Elementary mathematics ja:変数

Modus ponens

In Logic, Modus ponens (Latin: mode that affirms) is a valid, simple argument form (often abbreviated to MP): :If P, then Q. :P. :Therefore, Q. or in logical operator notation: :P → Q :P :⊢ Q where ⊢ represents the logical assertion. or may also be written: :P P → Q : Q In Metalogics the modus ponens is the cut-rule. The cut-elimination theorem says that the cut is valid (admissible rule) in some logical calculus (sequent calculus). The argument form has two premises. The first premise is the "if-then" or conditional claim, namely that P implies Q. The second premise is that P, the antecedent of the conditional claim, is true. From these two premises it can be logically concluded that Q, the consequent of the conditional claim, must be true as well. Here is an example of an argument that fits the form modus ponens: :If democracy is the best system of government, then everyone should vote. :Democracy is the best system of government. :Therefore, everyone should vote. The fact that the argument is valid cannot assure us that any of the statements in the argument are true; the validity of modus ponens tells us that the conclusion must be true if all the premises are true. It is wise to recall that a valid argument within which one or more of the premises are not true is called an unsound argument, whereas if all the premises are true, then the argument is sound. In most logical systems, Modus ponens is considered to be valid. However, the instances of its use may be either sound or unsound. :If the argument is modus ponens and its premises are true, then it is sound. :The premises are true. :Therefore, it is a sound argument. A propositional argument using modus ponens is said to be deductive. Modus ponens can also be referred to as affirming the antecedent or "Law of Detachment". For an amusing dialog that problematizes modus ponens, see Lewis Carroll's "What the Tortoise Said to Achilles."

Disjunctive syllogism

A disjunctive syllogism, also known as modus tollendo ponens (literally: mode which, by denying, affirms) is a valid, simple argument form: : P or Q : Not P : Therefore, Q In logical operator notation: :

p \lor q,

:¬

p \quad

\vdash q

where

\vdash

represents the logical assertion. Roughly, we are told that it has to be one or the other that is true; then we are told that it is not the one that is true; so we infer that it has to be the other that is true. The reason this is called "disjunctive syllogism" is that, first, it is a syllogism--a three-step argument--and second, it contains a disjunction, which means simply an "or" statement. "Either P or Q" is a disjunction; P and Q are called the statement's disjuncts. Here is an example: :Either I will choose soup or I will choose salad. :I will not choose soup. :Therefore, I will choose salad. Here is another example: :Either the Browns win or the Bengals win. :The Browns do not win. :Therefore, the Bengals win.

Inclusive versus exclusive disjunction

It should be noted with importance that there are two kinds of logical disjunction:
- inclusive means "and/or" where at least one term must be true or they can both be true.
- exclusive ("xor") means one must be true and the other must be false. Both terms cannot be true and both cannot be false. The popular English language concept of or is often ambiguous between these two meanings, but the difference is pivotal in evaluating disjunctive arguments. This argument: :Either P or Q. :Not P. :Therefore, Q. is valid and indifferent between both meanings. However, only in the exclusive meaning is the following form valid: :Either P or Q (exclusive). :P. :Therefore, not Q. With the inclusive meaning you could draw no conclusion from the first two premises of that argument. See affirming a disjunct.

Related argument forms

Unlike modus ponendo ponens and modus tollendo tollens, with which it should not be confused, modus tollendo ponens is often not made an explicit rule or axiom of logical systems, as the above arguments can be proven with a (slightly devious) combination of reductio ad absurdum and disjunction elimination. Modus tollendo ponens should also not be confused with modus ponendo tollens. Other forms of syllogism:
- hypothetical syllogism
- categorical syllogism

External link

- [http://logik.phl.univie.ac.at/~chris/beispielskriptum/node7.html Proof of MTP] Category:Rules of inference

Affirming the consequent

Affirming the consequent is a logical fallacy in the form of a hypothetical proposition. The fallacy of affirming the consequent occurs when a hypothetical proposition comprising an antecedent and a consequent asserts that the truthhood of the consequent implies the truthhood of the antecedent. This is fallacious because it assumes a bidirectionality when it does not necessarily exist. This fallacy has the following argument form: :If P, then Q. :Q. :Therefore, P. This logical error is called the fallacy of affirming the consequent because it is mistakenly concluded from the second premise that the affirmation of the consequent entails the truthhood of the antecedent. One way to demonstrate the invalidity is to use a counterexample. Here is an argument that is obviously incorrect: :If Stephen King wrote the Bible (P), then Stephen King is a good writer (Q). :Stephen King is a good writer (Q). :Therefore, Stephen King wrote the Bible (P). The previous argument was obviously incorrect, but the next argument may be more deceiving: :If someone is human (P), then she is mortal (Q). :Anna is mortal (Q). :Therefore Anna is human (P). But in fact Anna can be a cat; very much a mortal, but not a human one. However, be aware that a similar argument form is valid in which the first premise asserts "if and only if" rather than "if".

Analytic proposition

In philosophy, an analytic statement, or analytic proposition, is one such that its truth can be determined (solely) through analysis of its meaning. Loosely defined, an analytic proposition is a proposition the negation of which is self-contradictory, or a proposition that is true in every conceivable world, or a proposition that is true by definition. For example, All white cats are white is not only true, but also necessarily true — since a negation of it — "Not all white cats are white" is self-contradictory. The statement :Vegetarians don't eat meat is true by virtue of the meanings of its words, and it doesn't make sense to think of going out and studying the behaviour of vegetarians to see whether it's true or not. Statements that aren't analytic — that is, whose truth or falsity cannot be established by reflecting on their meaning — are termed synthetic; see synthetic proposition. There is no single, generally accepted, precise definition for analytic proposition, but philosophers have proposed a small number of closely related definitions, some of which are presented in the remainder of this article. The term was first defined by Immanuel Kant (1724–1804): :Either the predicate B belongs to subject A, as something which is contained (though covertly) in the conception of A; or the predicate B lies entirely out of the conception of A, although it stands in connection with it. In the first instance, I term the judgement analytical, in the second, synthetical. Analytical judgements (affirmative) are therefore those in which the connection of the predicate with the subject is cogitated through identity; those in which this connection is cogitated without identity, are called synthetical judgments. --(From the Introduction to The Critique of Pure Reason.) This definition is narrower than definitions currently in use. Later philosophers pointed out that if Kant’s definition is accepted, some propositions that are true by definition are not analytic. For example, 'A is A' is analytic by Kant’s definition. But an equally obvious 'If A, then A' is not analytic since it is not framed in the subject-predicate form. As a result, the definition of analytic proposition was expanded to include statements that are not in subject-predicate form. Two principle definitions for 'analytic proposition' have since been advanced: # An analytic proposition is one the negation of which is self-contradictory. If you deny a true analytic proposition, you always get a self-contradictory proposition.

# An analytic proposition is a proposition the truth of which can be determined solely through the analysis of the meaning of its words. Nothing in the world apart from language needs to be examined. :For example, "All bachelors are unmarried" is true if we take "bachelor" to mean "unmarried man" – and its negation is self-contradictory – so it is an analytic proposition. Its truth is apparent through the definition of its words. :But if by "bachelors" we mean "individuals who have received a certain kind of academic degree" then we have a statement that may or may not be true, but certainly one that can be negated with no contradiction. In other words, in this case we no longer have an analytic proposition, but a synthetic one. Analytic propositions need not be trivial tautologies like "All white cats are cats". Complex mathematical and geometrical theorems are analytic propositions, since a denial of such theorems leads to a contradiction. However, in the case of mathematical and geometric theorems, the statement that analytic propositions are true in any conceivable world breaks down. For example, the theorems of Euclidean geometry are analytic – but only if the axioms of Euclidean geometry are assumed. In other words, these theorems are analytic within a specific deductive system rather than "any conceivable world".

Analytic propositions and a priori knowledge

Analytic propositions and a priori knowledge are related, though not the same. Analytic propositions are propositions of a certain kind. A priori knowledge is knowledge that can be acquired without experience of the world. So knowledge of analytic propositions is commonly held to be a priori knowledge. Whether other kinds of a priori knowledge can exist is a matter of considerable debate within philosophy (see synthetic proposition).

References

- [http://plato.stanford.edu/entries/analytic-synthetic/ Stanford Encyclopedia of Philosophy entry] Category:Philosophical terminology Category:Philosophy of language

Category:Logic

Logic, in its purest form, is the reasoning used to take a set of assumptions and reach a conclusion. More specifically, logic is the study of prescriptive systems of reasoning, that is, systems proposed as guides for how people (as well, perhaps, as other intelligent beings/machines/systems) ought to reason. Logic says which forms of inference are valid and which are not. Traditionally, logic is studied as a branch of philosophy, but it can also be considered a branch of mathematics and computer science. How people actually reason is usually studied under other headings, including cognitive psychology. :See table of logic symbols for explanations of symbols used in logic. Category:Abstraction Category:Branches of philosophy Category:Interdisciplinary fields ja:Category:論理学

Ozzano Monferrato

Ozzano Monferrato to miejscowość i gmina we Włoszech, w regionie Piemont, w prowincji Alessandria. Wg danych na rok 2004 gminę zamieszkuje 1 567 osób, 104,5 os./km². Źródło danych: [http://www.istat.it Istituto Nazionale di Statistica] Kategoria:Miejscowości Włoch Kategoria:Prowincja Alessandria

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