1.
Formal logic
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Logic, originally meaning the word or what is spoken, is generally held to consist of the systematic study of the form of arguments. A valid argument is one where there is a relation of logical support between the assumptions of the argument and its conclusion. Historically, logic has been studied in philosophy and mathematics, and recently logic has been studied in science, linguistics, psychology. The concept of form is central to logic. The validity of an argument is determined by its logical form, traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic. Informal logic is the study of natural language arguments, the study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as an application of a wholly abstract rule, that is. The works of Aristotle contain the earliest known study of logic. Modern formal logic follows and expands on Aristotle, in many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language, Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is divided into two main branches, propositional logic and predicate logic. Mathematical logic is an extension of logic into other areas, in particular to the study of model theory, proof theory, set theory. Logic is generally considered formal when it analyzes and represents the form of any valid argument type, the form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. Simply put, formalising simply means translating English sentences into the language of logic and this is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a variety of form. Second, certain parts of the sentence must be replaced with schematic letters, thus, for example, the expression all Ps are Qs shows the logical form common to the sentences all men are mortals, all cats are carnivores, all Greeks are philosophers, and so on. The schema can further be condensed into the formula A, where the letter A indicates the judgement all - are -, the importance of form was recognised from ancient times
2.
Symbolic logic
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Mathematical logic is a subfield of mathematics exploring the applications of formal logic to mathematics. It bears close connections to metamathematics, the foundations of mathematics, the unifying themes in mathematical logic include the study of the expressive power of formal systems and the deductive power of formal proof systems. Mathematical logic is divided into the fields of set theory, model theory, recursion theory. These areas share basic results on logic, particularly first-order logic, in computer science mathematical logic encompasses additional topics not detailed in this article, see Logic in computer science for those. Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics and this study began in the late 19th century with the development of axiomatic frameworks for geometry, arithmetic, and analysis. In the early 20th century it was shaped by David Hilberts program to prove the consistency of foundational theories, results of Kurt Gödel, Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, the Handbook of Mathematical Logic makes a rough division of contemporary mathematical logic into four areas, set theory model theory recursion theory, and proof theory and constructive mathematics. Each area has a focus, although many techniques and results are shared among multiple areas. The borderlines amongst these fields, and the lines separating mathematical logic, Gödels incompleteness theorem marks not only a milestone in recursion theory and proof theory, but has also led to Löbs theorem in modal logic. The method of forcing is employed in set theory, model theory and these foundations use toposes, which resemble generalized models of set theory that may employ classical or nonclassical logic. Mathematical logic emerged in the century as a subfield of mathematics independent of the traditional study of logic. Before this emergence, logic was studied with rhetoric, through the syllogism, the first half of the 20th century saw an explosion of fundamental results, accompanied by vigorous debate over the foundations of mathematics. Theories of logic were developed in many cultures in history, including China, India, Greece, in the middle of the nineteenth century, George Boole and then Augustus De Morgan presented systematic mathematical treatments of logic. Their work, building on work by such as George Peacock. Charles Sanders Peirce built upon the work of Boole to develop a system for relations and quantifiers. Gottlob Frege presented an independent development of logic with quantifiers in his Begriffsschrift, published in 1879, Freges work remained obscure, however, until Bertrand Russell began to promote it near the turn of the century. The two-dimensional notation Frege developed was never adopted and is unused in contemporary texts. From 1890 to 1905, Ernst Schröder published Vorlesungen über die Algebra der Logik in three volumes
3.
Logic
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Logic, originally meaning the word or what is spoken, is generally held to consist of the systematic study of the form of arguments. A valid argument is one where there is a relation of logical support between the assumptions of the argument and its conclusion. Historically, logic has been studied in philosophy and mathematics, and recently logic has been studied in science, linguistics, psychology. The concept of form is central to logic. The validity of an argument is determined by its logical form, traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic. Informal logic is the study of natural language arguments, the study of fallacies is an important branch of informal logic. Since much informal argument is not strictly speaking deductive, on some conceptions of logic, formal logic is the study of inference with purely formal content. An inference possesses a purely formal content if it can be expressed as an application of a wholly abstract rule, that is. The works of Aristotle contain the earliest known study of logic. Modern formal logic follows and expands on Aristotle, in many definitions of logic, logical inference and inference with purely formal content are the same. This does not render the notion of informal logic vacuous, because no formal logic captures all of the nuances of natural language, Symbolic logic is the study of symbolic abstractions that capture the formal features of logical inference. Symbolic logic is divided into two main branches, propositional logic and predicate logic. Mathematical logic is an extension of logic into other areas, in particular to the study of model theory, proof theory, set theory. Logic is generally considered formal when it analyzes and represents the form of any valid argument type, the form of an argument is displayed by representing its sentences in the formal grammar and symbolism of a logical language to make its content usable in formal inference. Simply put, formalising simply means translating English sentences into the language of logic and this is called showing the logical form of the argument. It is necessary because indicative sentences of ordinary language show a variety of form. Second, certain parts of the sentence must be replaced with schematic letters, thus, for example, the expression all Ps are Qs shows the logical form common to the sentences all men are mortals, all cats are carnivores, all Greeks are philosophers, and so on. The schema can further be condensed into the formula A, where the letter A indicates the judgement all - are -, the importance of form was recognised from ancient times
4.
Reference
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Reference is a relation between objects in which one object designates, or acts as a means by which to connect to or link to, another object. The first object in this relation is said to refer to the second object, the second object, the one to which the first object refers, is called the referent of the first object. In some cases, methods are used that intentionally hide the reference from some observers, References feature in many spheres of human activity and knowledge, and the term adopts shades of meaning particular to the contexts in which it is used. Some of them are described in the sections below, the word reference is derived from Middle English referren, from Middle French référer, from Latin referre, to carry back, formed from the prefix re- and ferre, to bear. A number of words derive from the root, including refer, referee, referential, referent. The verb refer and its derivatives may carry the sense of link to or connect to, another sense is consult, this is reflected in such expressions as reference work, reference desk, job reference, etc. In semantics, reference is generally construed as the relationships between nouns or pronouns and objects that are named by them, hence, the word John refers to the person John. The word it refers to some previously specified object, the object referred to is called the referent of the word. Sometimes the word-object relation is called denotation, the word denotes the object, the converse relation, the relation from object to word, is called exemplification, the object exemplifies what the word denotes. In syntactic analysis, if a word refers to a previous word and this problem led Frege to distinguish between the sense and reference of a word. Some cases seem to be too complicated to be classified within this framework, words can often be meaningful without having a concrete here-and-now referent. Fictional and mythological names such as Bo-Peep and Hercules illustrate this possibility, sign links with absent referents also allow for discussing abstract ideas as well as people and events of the past and future. For those who argue that one cannot directly experience the divine, additionally, certain sects of Judaism and other religions consider it sinful to write, discard, or deface the name of the divine. To avoid this problem, the signifier G-d is sometimes used, the very concept of the linguistic sign is the combination of content and expression, the former of which may refer entities in the world or refer more abstract concepts, e. g. thought. Certain parts of speech exist only to reference, namely anaphora such as pronouns. The subset of reflexives expresses co-reference of two participants in a sentence and these could be the agent and patient, as in The man washed himself, the theme and recipient, as in I showed Mary to herself, or various other possible combinations. In computer science, references are data types that refer to an object elsewhere in memory and are used to construct a variety of data structures. Generally, a reference is a value that enables a program to access the particular data item
5.
Existence
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Existence is commonly held to be that which objectively persists independent of ones presence. Ontology is the study of the nature of being, existence or reality in general, as well as of the basic categories of being. In mathematics, existence is asserted by a quantifier, the existential quantifier, the properties of the existential quantifier are established by axioms. The word existence comes from the Latin word exsistere meaning to appear, to arise, to become, or to be, but literally, it means to stand out. In Aristotles Metaphysics, there are four causes of existence or change in nature, the cause, the formal cause, the efficient cause. The Neo-Platonists and some early Christian philosophers argued about whether existence had any reality except in the mind of God, Some taught that existence was a snare and a delusion, that the world, the flesh, and the devil existed only to tempt weak humankind away from God. The medieval philosopher Thomas Aquinas argued that God is pure being, the early modern treatment of the subject derives from Antoine Arnauld and Pierre Nicoles Logic, or The Art of Thinking, better known as the Port-Royal Logic, first published in 1662. This is called affirming or denying, and in general judging and this judgment is also called a proposition, and it is easy to see that it must have two terms. One term, of which one affirms or denies something, is called the subject, the other term, the two terms are joined by the verb is. Thus every proposition has three components, the two terms, and the copula that connects or separates them, even when the proposition has only two words, the three terms are still there. For example, God loves humanity, really means God is a lover of humanity and this theory of judgment dominated logic for centuries, but it has some obvious difficulties, it only considers proposition of the form All A are B. a form logicians call universal. It does not allow propositions of the form Some A are B, if neither A nor B includes the idea of existence, then some A are B simply adjoins A to B. Arnaulds theory was current until the middle of the nineteenth century, david Hume argued that the claim that a thing exists, when added to our notion of a thing, does not add anything to the concept. For example, if we form a notion of Moses. Kant also argued that existence is not a predicate. Schopenhauer claimed that “everything that exists for knowledge, and hence the whole of this world, is only object in relation to the subject, perception of the perceiver, in a word, franz Brentano challenged this, so also did Frege. Brentano argued that we can join the concept represented by a phrase an A to the concept represented by an adjective B to give the concept represented by the noun phrase a B-A. For example, we can join a man to wise to give a wise man, but the noun phrase a wise man is not a sentence, whereas some man is wise is a sentence
6.
Logical entailment
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Logical consequence is a fundamental concept in logic, which describes the relationship between statements that holds true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusions are entailed by 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 consequence of a set of sentences, for a given language, if and only if. The most widely prevailing view on how to best account for logical consequence is to appeal to formality and 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 A are B. All C are A. Therefore, all C are B and this argument is formally valid, because every instance of arguments constructed using this scheme are valid. This is in contrast to an argument like Fred is Mikes brothers son, if you know that Q follows logically from P no information about the possible interpretations of P or Q will affect that knowledge. Our knowledge that Q is a consequence of P cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, however, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a property of logical consequence is considered to be independent of formality. The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of proofs, the study of the syntactic consequence is called proof theory whereas the study of semantic consequence is called model theory. A formula A is a syntactic consequence within some formal system F S of a set Γ of formulas if there is a proof in F S of A from the set Γ. Γ ⊢ F S A Syntactic consequence does not depend on any interpretation of the formal system, or, in other words, the set of the interpretations that make all members of Γ true is a subset of the set of the interpretations that make A true. Modal accounts of logical consequence are variations on the basic idea, Γ ⊢ A is true if and only if it is necessary that if all of the elements of Γ are true. Alternatively, Γ ⊢ A is true if and only if it is impossible for all of the elements of Γ to be true, such accounts are called modal because they appeal to the modal notions of logical necessity and logical possibility. Consider the modal account in terms of the argument given as an example above, the conclusion is a logical consequence of the premises because we cant imagine a possible world where all frogs are green, Kermit is a frog, and Kermit is not green
7.
Logical necessity
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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 and it is a type of analytic statement. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, Logical truths 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 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 are true in this world, as it has historically unfolded, later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations. Empiricists commonly respond to this objection by arguing that logical truths, are analytic, Logical truths, being analytic statements, do not contain any information about any matters of fact. Other than logical truths, there is also a class of analytic statements. The characteristic of such a statement is that it can be turned into a logical truth by substituting synonyms for synonyms salva veritate, 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, 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, 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, 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. 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. Not all logical truths are tautologies of such a kind, 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 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
8.
Necessary truth
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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 and it is a type of analytic statement. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, Logical truths 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 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 are true in this world, as it has historically unfolded, later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations. Empiricists commonly respond to this objection by arguing that logical truths, are analytic, Logical truths, being analytic statements, do not contain any information about any matters of fact. Other than logical truths, there is also a class of analytic statements. The characteristic of such a statement is that it can be turned into a logical truth by substituting synonyms for synonyms salva veritate, 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, 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, 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, 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. 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. Not all logical truths are tautologies of such a kind, 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 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
9.
Logical truth
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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 and it is a type of analytic statement. All of philosophical logic can be thought of as providing accounts of the nature of logical truth, Logical truths 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 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 are true in this world, as it has historically unfolded, later, with the rise of formal logic a logical truth was considered to be a statement which is true under all possible interpretations. Empiricists commonly respond to this objection by arguing that logical truths, are analytic, Logical truths, being analytic statements, do not contain any information about any matters of fact. Other than logical truths, there is also a class of analytic statements. The characteristic of such a statement is that it can be turned into a logical truth by substituting synonyms for synonyms salva veritate, 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, 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, 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, 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. 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. Not all logical truths are tautologies of such a kind, 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 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
10.
Logical consequence
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Logical consequence is a fundamental concept in logic, which describes the relationship between statements that holds true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusions are entailed by 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 consequence of a set of sentences, for a given language, if and only if. The most widely prevailing view on how to best account for logical consequence is to appeal to formality and 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 A are B. All C are A. Therefore, all C are B and this argument is formally valid, because every instance of arguments constructed using this scheme are valid. This is in contrast to an argument like Fred is Mikes brothers son, if you know that Q follows logically from P no information about the possible interpretations of P or Q will affect that knowledge. Our knowledge that Q is a consequence of P cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, however, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a property of logical consequence is considered to be independent of formality. The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of proofs, the study of the syntactic consequence is called proof theory whereas the study of semantic consequence is called model theory. A formula A is a syntactic consequence within some formal system F S of a set Γ of formulas if there is a proof in F S of A from the set Γ. Γ ⊢ F S A Syntactic consequence does not depend on any interpretation of the formal system, or, in other words, the set of the interpretations that make all members of Γ true is a subset of the set of the interpretations that make A true. Modal accounts of logical consequence are variations on the basic idea, Γ ⊢ A is true if and only if it is necessary that if all of the elements of Γ are true. Alternatively, Γ ⊢ A is true if and only if it is impossible for all of the elements of Γ to be true, such accounts are called modal because they appeal to the modal notions of logical necessity and logical possibility. Consider the modal account in terms of the argument given as an example above, the conclusion is a logical consequence of the premises because we cant imagine a possible world where all frogs are green, Kermit is a frog, and Kermit is not green
11.
Entailment
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Logical consequence is a fundamental concept in logic, which describes the relationship between statements that holds true when one statement logically follows from one or more statements. A valid logical argument is one in which the conclusions are entailed by 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 consequence of a set of sentences, for a given language, if and only if. The most widely prevailing view on how to best account for logical consequence is to appeal to formality and 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 A are B. All C are A. Therefore, all C are B and this argument is formally valid, because every instance of arguments constructed using this scheme are valid. This is in contrast to an argument like Fred is Mikes brothers son, if you know that Q follows logically from P no information about the possible interpretations of P or Q will affect that knowledge. Our knowledge that Q is a consequence of P cannot be influenced by empirical knowledge. Deductively valid arguments can be known to be so without recourse to experience, however, formality alone does not guarantee that logical consequence is not influenced by empirical knowledge. So the a property of logical consequence is considered to be independent of formality. The two prevailing techniques for providing accounts of logical consequence involve expressing the concept in terms of proofs, the study of the syntactic consequence is called proof theory whereas the study of semantic consequence is called model theory. A formula A is a syntactic consequence within some formal system F S of a set Γ of formulas if there is a proof in F S of A from the set Γ. Γ ⊢ F S A Syntactic consequence does not depend on any interpretation of the formal system, or, in other words, the set of the interpretations that make all members of Γ true is a subset of the set of the interpretations that make A true. Modal accounts of logical consequence are variations on the basic idea, Γ ⊢ A is true if and only if it is necessary that if all of the elements of Γ are true. Alternatively, Γ ⊢ A is true if and only if it is impossible for all of the elements of Γ to be true, such accounts are called modal because they appeal to the modal notions of logical necessity and logical possibility. Consider the modal account in terms of the argument given as an example above, the conclusion is a logical consequence of the premises because we cant imagine a possible world where all frogs are green, Kermit is a frog, and Kermit is not green
12.
Paradox
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A paradox is a statement that, despite apparently sound reasoning from true premises, leads to a self-contradictory or a logically unacceptable conclusion. A paradox involves contradictory yet interrelated elements that exist simultaneously and persist over time, some logical paradoxes are known to be invalid arguments but are still valuable in promoting critical thinking. Some paradoxes have revealed errors in definitions assumed to be rigorous, others, such as Currys paradox, are not yet resolved. Examples outside logic include the Ship of Theseus from philosophy, paradoxes can also take the form of images or other media. Escher featured perspective-based paradoxes in many of his drawings, with walls that are regarded as floors from other points of view, and staircases that appear to climb endlessly. In common usage, the word often refers to statements that may be both true and false i. e. ironic or unexpected, such as the paradox that standing is more tiring than walking. Common themes in paradoxes include self-reference, infinite regress, circular definitions, patrick Hughes outlines three laws of the paradox, Self-reference An example is This statement is false, a form of the liar paradox. The statement is referring to itself, another example of self-reference is the question of whether the barber shaves himself in the barber paradox. One more example would be Is the answer to this question No, contradiction This statement is false, the statement cannot be false and true at the same time. Another example of contradiction is if a man talking to a genie wishes that wishes couldnt come true, vicious circularity, or infinite regress This statement is false, if the statement is true, then the statement is false, thereby making the statement true. Another example of vicious circularity is the group of statements. Other paradoxes involve false statements or half-truths and the biased assumptions. This form is common in howlers, for example, consider a situation in which a father and his son are driving down the road. The car crashes into a tree and the father is killed, the boy is rushed to the nearest hospital where he is prepared for emergency surgery. On entering the suite, the surgeon says, I cant operate on this boy. The apparent paradox is caused by a hasty generalization, for if the surgeon is the boys father, the paradox is resolved if it is revealed that the surgeon is a woman — the boys mother. Paradoxes which are not based on a hidden error generally occur at the fringes of context or language, paradoxes that arise from apparently intelligible uses of language are often of interest to logicians and philosophers. Russells paradox, which shows that the notion of the set of all sets that do not contain themselves leads to a contradiction, was instrumental in the development of modern logic
13.
Sense and reference
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The distinction between sense and reference was an innovation of the German philosopher and mathematician Gottlob Frege in 1892, reflecting the two ways he believed a singular term may have meaning. The reference ) of a name is the object it means or indicates. The reference of a sentence is its value, its sense is the thought that it expresses. Frege justified the distinction in a number of ways, sense is something possessed by a name, whether or not it has a reference. For example, the name Odysseus is intelligible, and therefore has a sense, the sense of different names is different, even when their reference is the same. Frege argued that if an identity statement such as Hesperus is the planet as Phosphorus is to be informative. But clearly, if the statement is true, they must have the same reference, the sense is a mode of presentation, which serves to illuminate only a single aspect of the referent. Frege developed his theory of meaning in early works like the Begriffsschrift of 1879. Thus Caesar conquered Gaul divides into the complete term Caesar, whose reference is Caesar himself, only when the empty place is filled by a proper name does the reference of the completed sentence – its truth value – appear. This early theory of meaning explains how the significance or reference of a sentence depends on the significance or reference of its parts. ”This “suggests that he makes a distinction between sense and reference, the reference of the whole is determined by the reference of the parts. Second, sentences which contain proper names that have no reference cannot have a value at all. Yet the sentence Odysseus was set ashore at Ithaca while sound asleep obviously has a sense, the thought remains the same whether or not Odysseus has a reference. Furthermore, a thought cannot contain the objects which it is about, for example, Mont Blanc, with its snowfields, cannot be a component of the thought that Mont Blanc is more than 4,000 metres high. Nor can a thought about Etna contain lumps of solidified lava, Freges notion of sense is somewhat obscure, and neo-Fregeans have come up with different candidates for its role. Accounts based on the work of Carnap and Church treat sense as an intension, for example, the intension of ‘number of planets’ is a function that maps any possible world to the number of planets in that world. John McDowell supplies cognitive and reference-determining roles, devitt treats senses as causal-historical chains connecting names to referents. In his theory of descriptions, Bertrand Russell held the view that most proper names in ordinary language are in fact disguised definite descriptions, for example, Aristotle can be understood as The pupil of Plato and teacher of Alexander, or by some other uniquely applying description. This is known as the descriptivist theory of names, because Frege used definite descriptions in many of his examples, he is often taken to have endorsed the descriptivist theory
14.
Theory of reference
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The distinction between sense and reference was an innovation of the German philosopher and mathematician Gottlob Frege in 1892, reflecting the two ways he believed a singular term may have meaning. The reference ) of a name is the object it means or indicates. The reference of a sentence is its value, its sense is the thought that it expresses. Frege justified the distinction in a number of ways, sense is something possessed by a name, whether or not it has a reference. For example, the name Odysseus is intelligible, and therefore has a sense, the sense of different names is different, even when their reference is the same. Frege argued that if an identity statement such as Hesperus is the planet as Phosphorus is to be informative. But clearly, if the statement is true, they must have the same reference, the sense is a mode of presentation, which serves to illuminate only a single aspect of the referent. Frege developed his theory of meaning in early works like the Begriffsschrift of 1879. Thus Caesar conquered Gaul divides into the complete term Caesar, whose reference is Caesar himself, only when the empty place is filled by a proper name does the reference of the completed sentence – its truth value – appear. This early theory of meaning explains how the significance or reference of a sentence depends on the significance or reference of its parts. ”This “suggests that he makes a distinction between sense and reference, the reference of the whole is determined by the reference of the parts. Second, sentences which contain proper names that have no reference cannot have a value at all. Yet the sentence Odysseus was set ashore at Ithaca while sound asleep obviously has a sense, the thought remains the same whether or not Odysseus has a reference. Furthermore, a thought cannot contain the objects which it is about, for example, Mont Blanc, with its snowfields, cannot be a component of the thought that Mont Blanc is more than 4,000 metres high. Nor can a thought about Etna contain lumps of solidified lava, Freges notion of sense is somewhat obscure, and neo-Fregeans have come up with different candidates for its role. Accounts based on the work of Carnap and Church treat sense as an intension, for example, the intension of ‘number of planets’ is a function that maps any possible world to the number of planets in that world. John McDowell supplies cognitive and reference-determining roles, devitt treats senses as causal-historical chains connecting names to referents. In his theory of descriptions, Bertrand Russell held the view that most proper names in ordinary language are in fact disguised definite descriptions, for example, Aristotle can be understood as The pupil of Plato and teacher of Alexander, or by some other uniquely applying description. This is known as the descriptivist theory of names, because Frege used definite descriptions in many of his examples, he is often taken to have endorsed the descriptivist theory
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Saul Kripke
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Saul Aaron Kripke is an American philosopher and logician. He is a Distinguished Professor of Philosophy at the Graduate Center of the City University of New York and emeritus professor at Princeton University. Since the 1960s Kripke has been a figure in a number of fields related to mathematical logic, philosophy of language, philosophy of mathematics, metaphysics, epistemology. Much of his work remains unpublished or exists only as tape recordings, Kripke was the recipient of the 2001 Schock Prize in Logic and Philosophy. Kripke has made influential and original contributions to logic, especially modal logic and he has also contributed an original reading of Wittgenstein, referred to as Kripkenstein. His most famous work is Naming and Necessity, Saul Kripke is the oldest of three children born to Dorothy K. Kripke and Rabbi Myer S. Kripke. His father was the leader of Beth El Synagogue, the only Conservative congregation in Omaha, Nebraska, Saul and his two sisters, Madeline and Netta, attended Dundee Grade School and Omaha Central High School. He wrote his first completeness theorem in logic at the age of 17. After graduating from school in 1958, Kripke attended Harvard University. During his sophomore year at Harvard, Kripke taught a graduate-level logic course at nearby MIT, upon graduation he received a Fulbright Fellowship, and in 1963 was appointed to the Society of Fellows. After teaching briefly at Harvard, he moved to Rockefeller University in New York City in 1967, in 1988 he received the universitys Behrman Award for distinguished achievement in the humanities. In 2002 Kripke began teaching at the CUNY Graduate Center in midtown Manhattan and he was married to philosopher Margaret Gilbert. He has received degrees from the University of Nebraska, Omaha, Johns Hopkins University, University of Haifa, Israel. He is a member of the American Philosophical Society, an elected Fellow of the American Academy of Arts and Sciences and he won the Schock Prize in Logic and Philosophy in 2001. He is the cousin once removed of television writer, director. The Saul Kripke Center at the Graduate Center of the City University of New York is dedicated to preserving and promoting Kripkes work, the Saul Kripke Center is directed by Gary Ostertag. Kripkes contributions to philosophy include, Kripke semantics for modal and related logics and his 1970 Princeton lectures Naming and Necessity, that significantly restructured philosophy of language. The most familiar logics in the family are constructed from a weak logic called K
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Gottlob Frege
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Friedrich Ludwig Gottlob Frege was a German philosopher, logician, and mathematician. Considered a major figure in mathematics, he is responsible for the development of modern logic and he is also understood by many to be the father of analytic philosophy, where he concentrated on the philosophy of language and mathematics. Though largely ignored during his lifetime, Giuseppe Peano and Bertrand Russell introduced his work to generations of logicians. Frege was born in 1848 in Wismar, Mecklenburg-Schwerin and his father Carl Alexander Frege was the co-founder and headmaster of a girls high school until his death. In childhood, Frege encountered philosophies that would guide his future scientific career, Frege studied at a gymnasium in Wismar and graduated in 1869. His teacher Gustav Adolf Leo Sachse, who was a poet, played the most important role in determining Freges future scientific career, Frege matriculated at the University of Jena in the spring of 1869 as a citizen of the North German Confederation. In the four semesters of his studies he attended approximately twenty courses of lectures and his most important teacher was Ernst Karl Abbe. Abbe was more than a teacher to Frege, he was a trusted friend, after Freges graduation, they came into closer correspondence. His other notable university teachers were Christian Philipp Karl Snell, Hermann Karl Julius Traugott Schaeffer, Frege married Margarete Katharina Sophia Anna Lieseberg on 14 March 1887. Though his education and early work focused primarily on geometry. His Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens, Halle a/S, the Begriffsschrift broke new ground, including a rigorous treatment of the ideas of functions and variables. Previous logic had dealt with the constants and, or. Freges conceptual notation however can represent such inferences, one of Freges stated purposes was to isolate genuinely logical principles of inference, so that in the proper representation of mathematical proof, one would at no point appeal to intuition. If there was an element, it was to be isolated and represented separately as an axiom, from there on. Already in the 1879 Begriffsschrift important preliminary theorems, for example a generalized form of law of trichotomy, were derived within what Frege understood to be pure logic and this idea was formulated in non-symbolic terms in his The Foundations of Arithmetic. Later, in his Basic Laws of Arithmetic, Frege attempted to derive, by use of his symbolism, most of these axioms were carried over from his Begriffsschrift, though not without some significant changes. The one truly new principle was one he called the Basic Law V, the crucial case of the law may be formulated in modern notation as follows. Let denote the extension of the predicate Fx, i. e. the set of all Fs, then Basic Law V says that the predicates Fx and Gx have the same extension iff ∀x
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Meaning (philosophy of language)
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The nature of meaning, its definition, elements, and types, was discussed by philosophers Aristotle, Augustine, and Aquinas. According to them meaning is a relationship between two sorts of things, signs and the kinds of things they mean, One term in the relationship of meaning necessarily causes something else to come to the mind. In other words, a sign is defined as an entity that indicates another entity to some agent for some purpose, as Augustine states, a sign is “something that shows itself to the senses and something other than itself to the mind”. The types of meanings vary according to the types of the thing that is being represented, All subsequent inquiries emphasize some particular perspectives within the general AAA framework. The evaluation of meaning according to one of the five major substantive theories of meaning. Each theory of meaning as evaluated by these theories of truth are each further researched by the individual scholars supporting each one of the respective theories of truth. Both hybrid theories of meaning and alternative theories of meaning and truth have also been researched and this type of theory stresses a relationship between thoughts or statements on one hand, and things or objects on the other. It is a traditional model tracing its origins to ancient Greek philosophers such as Socrates, Plato and this class of theories holds that the truth or the falsity of a representation is determined in principle entirely by how it relates to things, by whether it accurately describes those things. Many modern theorists have stated that this ideal cannot be achieved without analysing additional factors, for example, language plays a role in that all languages have words to represent concepts that are virtually undefined in other languages. Thus, some add an additional parameter to the construction of an accurate truth predicate. Among the philosophers who grappled with this problem is Alfred Tarski, for coherence theories in general, the assessment of meaning and truth requires a proper fit of elements within a whole system. So, for example, the completeness and comprehensiveness of the set of concepts is a critical factor in judging the validity. Some variants of coherence theory are claimed to describe the essential and intrinsic properties of systems in logic. However, formal reasoners are content to contemplate axiomatically independent and sometimes mutually contradictory systems side by side, for example, coherence theories distinguish the thought of rationalist philosophers, particularly of Spinoza, Leibniz, and G. W. F. Hegel, along with the British philosopher F. H. Bradley, other alternatives may be found among several proponents of logical positivism, notably Otto Neurath and Carl Hempel. Constructivism views all of our knowledge as constructed, because it does not reflect any external transcendent realities, rather, perceptions of truth are viewed as contingent on convention, human perception, and social experience. It is believed by constructivists that representations of physical and biological reality, including race, sexuality, giambattista Vico was among the first to claim that history and culture along with their meaning were man-made. Vicos epistemological orientation gathers the most diverse rays and unfolds in one axiom – verum ipsum factum – truth itself is constructed, hegel and Marx were among the other early proponents of the premise that truth is, or can be, socially constructed
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Denotation
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Denotation is a translation of a sign to its meaning, precisely to its literal meaning, more or less like dictionaries try to define it. Denotation is sometimes contrasted to connotation, which translates a sign to its associated meanings, in semiotics, the surface or literal meaning of a signifier. In logic, formal semantics and parts of linguistics, the extension of a term, in computer science, denotational semantics is contrasted with operational semantics. In media studies terminology, denotation is an example of the first level of analysis, denotation often refers to something literal, and avoids being a metaphor. Here it is coupled with connotation which is the second level of analysis. Semiotics for Beginners VirtuaLit Elements of Poetry
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Connotation
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A connotation is frequently described as either positive or negative, with regards to its pleasing or displeasing emotional connection. Connotation branches into a mixture of different meanings and these could include the contrast of a word or phrase with its primary, literal meaning, with what that word or phrase specifically denotes. The connotation essentially relates to how anything may be associated with a word or phrase, for example and it is often useful to avoid words with strong connotations when striving to achieve a neutral point of view. A desire for more positive connotations, or fewer negative ones, is one of the reasons for using euphemisms. In logic and semantics, connotation is roughly synonymous with intension, connotation is often contrasted with denotation, which is more or less synonymous with extension. Alternatively, the connotation of the word may be thought of as the set of all its possible referents, a words denotation is the collection of things it refers to, its connotation is what it implies about the things it is used to refer to. The denotation of dog is four-legged canine carnivore, so saying, You are a dog would imply that you were ugly or aggressive rather than stating that you were canine. Denotation Double entendre Extension Extensional definition Intension Intensional definition Metacommunicative competence Pun Subtext