First-order logic Essay Example for Free

First- redact system of logic Essay1. (Philosophy) the academic discipline relate with do explicit the record and significance of ordinary and scientific beliefs and investigating the intelligibility of concepts by means of rational object concerning their presuppositions, implications, and interrelationships in particular, the rational investigation of the nature and structure of reality (metaphysics), the resources and limits of familiarity (epistemology), the principles and import of moral view (ethics), and the relationship amidst spoken communication and reality (semantics) 2.(Philosophy) the particular doctrines relating to these issues of some(a) specific individual or instruct the philosophic system of Descartes 3. (Philosophy) the critical guide of the basic principles and concepts of a discipline the philosophy of law 4. (Literary Literary fine Terms) Archaic or literary the investigation of natural phenomena, esp alchemy, astrology, and astronomy 5. distributively system of belief, judge, or tenets 6. a personal outlook or viewpoint 7. serenity of temper philosophy (f l? s ? fi) n. , pl. -phies. 1. the rational investigation of the loyaltys and principles of being, knowledge, or conduct. 2.a system of philosophical doctrine the philosophy of Spinoza. 3. the critical hit the books of the basic principles and concepts of a particular branch of knowledge the philosophy of science. 4. a system of principles for guidance in practical affairs a philosophy of life. 5. a calm or philosophical attitude. Philosophy is the depicted object of general and fundamental problems, such(prenominal) as those connected with reality, existence, knowledge, values, reason,mind, and dustup. Philosophy is distinguished from separate ways of addressing such problems by its critical, gener eithery systematic approach and its reliance on rational argument.3 In more nonchalant speech, by extension, philosophy can refer to the nearly basic belief s, concepts, and attitudes of an individual or group. The word philosophy comes from the Ancient Greek (philosophia), which literally means love of wisdom. 567 The introduction of the margins philosopher and philosophy has been ascribed to the Greek head Pythagoras. 8 https//en. wikipedia. org/wiki/Philosophy http//www. thefreedictionary. com/philosophy Branches of Philosophy Main branches of philosophy Traditionally, there be five main branches of philosophy.They be Metaphysics, which deals with the fundamental questions of reality. Epistemology, which deals with our concept of knowledge, how we learn and what we can know. Logic, which studies the rules of valid reasoning and argumentation Ethics, or moral philosophy, which is concerned with human values and how individuals should act. aesthetics or esthetics, which deals with the nonion of apricot and the philosophy of art. http//www. philosophy-index. com/philosophy/branches/ Aesthetics Aesthetics is the argona of p hilosophy which covers the concepts of beauty and art.Beauty is in the eye of the beholder There are dickens basic standings on the nature of beauty objective and subjective judgement. Subjective judgement of beauty suggests that beauty is not the same to incessantlyyone that which aesthetically pleases the observer is beautiful (to the observer). Alternatively, those partial to the objective description of beauty take heed to measaure it. They suggest that accepted properties of an object create an inherent beauty such as symmetry and balance. some(prenominal) Plato and Aristotlesupported the objective judgement.Some, such as Immanuel Kant, took a middle path, holding that beauty is of a subjective nature, studyly there are qualities of beauty which welcome universal validity. Classical and recent Aesthetics The classical concepts privy aesthetics saw beauty in nature, and that art should mimic those qualities found in nature. AristotlesPoetics describes this base, w hich he develops from Platos teachings. Modern aesthetic ideas, including those of Kant, stress the creative and tokenic side of art that nature does not always have to guide art for it to be beautiful. EpistemologyEpistemology is the area of philosophy that is concerned with knowledge. The main concerns of epistemology are the definition of knowledge, the sources of knowledge (innate ideas, experience, etc. ), the process of acquiring knowledge and the limits of knowledge. Epistemology imagines that knowledge can be obtained throughexperience and/or reason. Defining friendship A primary concern of epistemology is the very definition of knowledge itself. The traditional definition, since Plato, is that knowledge is justified professedly belief, notwithstanding recent evaluations of the concept have shown so-called counterexamples to this definition.In order to fully explore the nature of knowledge and how we come to know things, the various conceptions of what knowledge is mu st first be understood. Definition of knowledge Sources of Knowledge The sources of knowledge must also be considered. Perception, reason, reminiscence, testimony, introspection and innate ideas are all supposed sources of knowledge. Scepticism There also take cares to be reason to doubt each of these sources of knowledge. Could it be that all knowledge is fallible? If that is the case, do we really know anything? This is the central question to the problem of scepticism.Ethics Ethics or moral philosophy is the branch of philosophy concerned with human conduct and its moral value. There are broadly three branches of ethics Meta-ethics, which is concerned with questions approximately what whether or not morality exists, and what it consists of if it does Normative ethics, which is concerned with how moral values should be developed and Applied ethics, which deals with how moral values can be use to specific cases. Logic Logic is the systematic process of valid reasoning throug h inference filiation demonstrations from in figure of speechation that is known to be true.It is the area of philosophy that is concerned with the laws of valid reasoning. symbolical Logic Symbolic logic is the method of representing arranged expressions through the use of symbols and variables, rather than in ordinary language. This has the benefit of removing the equivocalness that normally accompanies ordinary languages, such as English, and allows easier operation. There are many systems of symbolic logic, such as classical propositional logic, first-order logic and modal logic. Each whitethorn have seperate symbols, or exclude the use of certain symbols.Logical Symbols The following table presents several discursive symbols, their name and meaning, and any relevant notes. The name of the symbol (under meaning links to a page explaining the symbol or experimental condition and its use). Note that diametric symbols have been use by different logicians and systems of logic . For the saki of clarity, this site consistently uses the symbols in the left column, while the Notes column may indicate other commonly-used symbols. Symbol Meaning Notes Operators (Connectives) negation (NOT) The tilde ( ? ) is also often used. ?conjunction (AND) The ampersand ( ) or window pane ( ) are also often used. ? disconnectedness (OR) This is the inclusive disjunction, equivalent to and/or in English. ? exclusive disjunction (XOR) ? means that only one of the connected propositions is true, equivalent to eitheror. Sometimes ? is used. transfernative denial(NAND) Means not both. Sometimes written as v joint denial (NOR) Means neither/nor. conditional(if/ then) many a(prenominal) logicians use the symbol? behavior of. This is also known as material implication. - biconditional (iff) Means if and only if ?is sometimes used, but this site reserves that symbol for equivalence. Quantifiers ? universal quantifier Means for all, so ? xPx means that Px is true for every x. ? existential quantifier Means there exists, so ? xPxmeans that Px is true for at least one x. Relations ? implication ? ? ? means that ? follows from? ? equivalence Also ?. Equivalence is two-way implication, so ? ? ? means? pic ? and ? pic ?. ? provability Shows provable inference. ? pic ? means that from ? we can prove that ?. ? therefore apply to signify the conclusion of an argument.Usually taken to mean implication, but often used to present arguments in which the premises do not deductively imply the conclusion. ? forces A relationship between accomplishable worlds and sentences in modal logic. Truth-Values ? tautology may be used to replace any tautologous (always true) formula. ? contradiction May be used to replace any contradictory (always false) formula. Sometimes F is used. Parentheses ( ) parentheses apply to group expressions to show precedence of operations. Square brackets are sometimes used to clarify groupings. Set T heory ?.membership Denotes membership in a set. Ifa ? ?, then a is a member (or an element) of set ?. ? union Used to join sets. If S and T are sets of formula, S ? T is a set containing all members of both. ? intersection The overlap between sets. If S and T are sets of formula, S ? Tis a set containing those elemenets that are members of both. ? subset A subset is a set containing some or all elements of another set. ? proper subset A proper subset contains some, but not all, elements of another set. = set equality twain sets are equal if they contain exactly the same elements. ? absolute balance ? (S) is the set of all things that are not in the set S. Sometimes written as C(S), S or SC. - relative complement T S is the set of all elements in T that are not also in S. Sometimes written as T S. ? empty set The set containing no elements. Modalities ? necessarily Used only in modal logic systems. Sometimes expressed as where the symbol is unavailable. ? possibly Used only in modal logic systems. Sometimes expressed as where the symbol is unavailable. Propositions, Variables and Non-Logical Symbols.The use of variables in logic varies depending on the system and the author of the logic being presented. However, some common uses have emerged. For the sake of clarity, this site allow use the system defined below. Symbol Meaning Notes A, B, C Z propositions capital letter roman type letters signify individual propositions.For example, P may symbolize the proposition Pat is senseless. P and Q are traditionally used in most examples. ? , ? , ? ? formulae Lowercase Greek letters signify formulae, which may be themselves a proposition (P), a formula (P ?Q) or several connected formulae (? ? ? ). x, y, z variables Lowercase Roman letters towards the end of the alphabet are used to signify variables. In limpid systems, these are usually coupled with a quantifier, ? or ? , in order to signify some or all of some unspecified subject or object. By convention, these begin with x, but any other letter may be used if needed, so long as they are defined as a variable by a quantifier. a, b, c, z constants Lowercase Roman letters, when not assigned by a quantifier, signifiy a constant, usually a proper noun.For instance, the letter j may be used to signify Jerry. Constants are given a meaning before they are used in consistent expressions. Ax, Bx Zx predicate symbols Uppercase Roman letters appear again to indicate predicate relationships between variables and/or constants, coupled with one or more variable places which may be filled by variables or constants. For instance, we may definite the relation x is green as Gx, and x likes y as Lxy. To narrow down them from propositions, they are often presented in italics, so while P may be a proposition, Px is a predicate relation for x.Predicate symbols are non-logical they describe relations but have neither operational function nor truth val ue in themselves. ? , ? , ? sets of formulae Uppercase Greek letters are used, by convention, to refer to sets of formulae. ? is usually used to represent the first site, since it is the first that does not look like Roman letters. (For instance, the capital letter Alpha (? ) looks identical to the Roman letter A) ? , ? , ? possible worlds In modal logic, uppercase greek letters are also used to represent possible worlds.Alternatively, an uppercase W with a subscript numeral is sometimes used, representing worlds as W0, W1, and so on. sets Curly brackets are generally used when particularization the contents of a set, such as a set of formulae, or a set of possible worlds in modal logic. For instance, ? = ? , ? , ? , ? Systems of Logic A system of logic, also known as a logical calculus, or evidently a logic, is a method by which to express and evaluate information in a logical manner. Formal Language and Rules of Inference Logical systems consist of a formal language of symbolic logic.This language defines A set of symbols to refer to formulae, including propositions and operators. Grammar, that is rules of well-formation, on how formulae must be expressed. The formal language of a system consists of, on one hand, the syntax of the language, and on the other, a method for expressing semantics within the system. The semantics of a system may be as simple as assigning truth-value to propositions and formulae, or more complicated, victimization predicate symbols to define non-logical relationships between formulae.Systems also consist of rules of inference, which determine how expressions in the language may be used to draw new, previously unstated conclusions. Common Systems of Logic Classical Logics, the most common form of logical expression, including o Aristotelian logic o Propositional logic o First-order logic o Second-order logic o Higher-order logics Contextual Logics, which deal with non-truth-functionaloperators, and includ e o Modal Logic, which deals with modal operators neccessarily and possibly. o Epistemic Logic, which reasons intimately knowledge o Doxastic Logic, which reasons about belief.o Deontic Logic, which reasons about ethical obligation and permissibility o Temporal Logic, which reasons about propositions over time Free Logic, which rejects the assumption that the domain is non-empty, that something exists Fuzzy Logic, which rejects the law of the excluded middle Intuitionistic Logic, which redefines truth values based on proof Paraconsistent Logic, which allows contradictions without entailment of any other formulae Relevance Logic, which requires a stronger link of relevancy between premises and conclusion Metaphysics Metaphysics is the area of philosophy which deals with the ultimate nature of reality.Metaphysics can emcompass large areas of philosophy, and most other philosophical schools turn back to it for basic definition. In that respect, the term metaphysics is a broad one , encompassing the philosophical ideas of cosmology and ontology. Metaphysics or First Philosophy The term metaphysics comes from Greek, meaning after the Physics. Although the term metaphysics generally makes sense in the way that it partially refers to things outisde of and beyond the natural sciences, this is not the seed of the term (as opposted to, say, meta-ethics, which refers to the nature of ethics itself).Instead, the term was used by later editors of Aristotle. Aristotle had written several books on effect and physics, and followed those volumes with work on ontology, and other broad subjects. These editors referred to them as the books that came after the books on physics or metaphysics. Aristotle himself refers to metaphysics as first philosophy. This term was also used by some later philosophers, such as Descartes, whose primary work on the subject of metaphysics is calledMeditations on First Philosophy. Branches of Metaphysics The main branches of metaphysics are Ontology Cosmology Ontology is a branch of metaphysics which studies being. Ontology is concerned with the ultimate nature of being, and of all reality in general. The process of postulateing ontology generally consists of describing being as well as determining how reality may be organized and categorized, and how different types of beings relate to one another. The term an ontology refers to the things counted as being in a metaphysical system. Generally, an ontology is a contention of things that exist the furniture of the universe as it is sometimes put.Differences in ontology among philosophers generally deal with whether or not there are non-physical entities, and whether those things can be counted as being, existing, both or neither. Examples of candidates for ontological status as non-physical being include the mind, mathematical objects and universals. Ontologists Philosophers who do work on ontology are referred to asontologists. The following are some of the prominen t ontologists discussed on this site Aristotle Saint Anselm Georg Wilhelm Friedrich Hegel Martin Heidegger. Immanuel Kant Plato W. V. O. Quine Jean-Paul Sartre Baruch Spinoza Cosmology Cosmology is the area of metaphysics and science that studies the origin, phylogeny and nature of the universe. Cosmology is concerned with the contents and astrophysical phenomena of space and time, as well as their origin and progression. Although cosmology is most often concerned with physics and astronomy in the scientific world, it directly relates to a number of philosophical and theological views. The scientific theories related to .While ontology studies the nature of being and reality itself, cosmology is the see of those things that are in reality, and how they, and perhaps reality, came to be. Divisions of Philosophy Abstract Philosophy, philosophical inquiry, and the main branches of philosophy are characterized. 1. What is Philosophy? 1. The derivation of the word philosophy fro m the Greek is suggested by the following words and word-fragments. ? philolove of, coincidence for, liking of ? philanderto engage in love affairs frivolously ? philanthropylove of cosmos in general ? philatelypostage stamps hobby? phile(as in anglophile) one having a love for ? philologyhaving a liking for words ? sophoswisdom ? sophistlit. one who loves knowledge ? sophomorewise and moros laughablei. e. one who thinks he knows many things ? sophisticatedone who is knowledgeable 2. A suggested definition for our beginning study is as follows. Philosophy is the systematic inquiry into the principles and presuppositions of any field of study. ? From a psychological point of view, philosophy is an attitude, an approach, or a calling to answer or to ask, or even to comment upon certain eccentric problems (i. e. , specifically the kinds of problems usually relegated to the main branches discussed below in Section II).? There is, perhaps, no one atomic number 53 sense of the word phi losophy. Eventually many writers abandon the attempt to define philosophy and, instead, turn to the kinds of things philosophers do. ? What is complex in the study of philosophy involves is described by the London Times in an article dealing with the 20th World Congress of Philosophy The great virtue of philosophy is that it teaches not what to think, but how to think.It is the study of meaning, of the principles underlying conduct, thought and knowledge. The skills it hones are the ability to analyse, to question orthodoxies and to express things clearly. However arcane some philosophical texts may be the ability to formulate questions and follow arguments is the essence of education. 1. The Main Branches of Philosophy are shared as to the nature of the questions asked in each area. The integrity of these divisions cannot be rigidly maintained, for one area overlaps into the others. 1.Axiology the study of value the investigation of its nature, criteria, and metaphysical stat us. More often than not, the term value theory is used instead of axiology in contemporary discussions even though the term theory of value is used with respect to the value or price of goods and services in economics. ? Some significant questions in axiology include the following 1. genius of value is value a fulfillment of desire, a pleasure, a preference, a behavioral disposition, or simply a human interest of some kind? 2.Criteria of value de gustibus non (est) disputandum (i. e. , (theres no accounting for tastes) or do objective standards apply? 3. Status of value how are values related to (scientific) facts? What ultimate worth, if any, do human values have? ? Axiology is usually divided into two main parts. 1. Ethics the study of values in human behavior or the study of moral problems e. g. , (1) the rightness and wrongness of actions, (2) the kinds of things which are good or desirable, and (3) whether actions are blameworthy or praiseworthy.1. Consider this example analyz ed by J.O. Urmson in his well-known essay, Saints and Heroes We may imagine a squad of soldiers to be practicing the throwing of live hand grenades a grenade slips from the hand of one of them and rolls on the ground near the squad one of them sacrifices his life by throwing himself on the grenade and protecting his comrades with his own body. It is preferably unreasonable to suppose that such a man must be impelled by the sort of emotion that he might be impelled by if his best friend were in the squad. 2. Did the soldier who threw himself on the grenade do the right thing?If he did not cover the grenade, several soldiers might be injured or be killed. His action probably saved lives certainly an action which saves lives is a chastely correct action. One might even be inclined to conclude that saving lives is a duty. moreover if this were so, wouldnt each of the soldiers have the moral obligation or duty to save his comrades? Would we thereby expect each of the soldiers to vie for the opportunity to cover the grenade? 1. ?sthetics the study of value in the arts or the inquiry into feelings, judgments, or standards of beauty and related concepts.Philosophy of art is concerned with judgments of sense, taste, and emotion. 1. E. g. , Is art an intellectual or representational exercise? What would the realistic representations in pop art represent? Does art represent sensible objects or high-flown objects? 2. Is artistic value objective? Is it merely coincidental that many forms in architecture and painting seem to illustrate mathematical principles? Are there standards of taste? 3. Is there a clear distinction between art and reality? 1. Epistemology the study of knowledge.In particular, epistemology is the study of the nature, scope, and limits of human knowledge. ? Epistemology investigates the origin, structure, methods, and integrity of knowledge. ? Consider the degree of truth of the statement, The earth is round. Does its truth depend upon the contex t in which the statement is uttered? For example, this statement can be successively more accurately translated as 1. The earth is spherical 2. The earth is an oblate spheroid (i. e. , shape at the poles). 3. But what about the Himalayas and the Marianas Trench?Even if we surveyed exactly the shape of the earth, our process of surveying would alter the surface by the footprints left and the impressions of the survey stakes and instruments. Hence, the exact shape of the earth cannot be known. every(prenominal) rain shower changes the shape. 4. (Note here as well the implications for skepticism and relativism simply because we cannot exactly describe the exact shape of the earth, the conclusion does not logically follow that the earth does not have a shape. ) ? merelymore, consider two well-known problems in epistemology 1.Russells Five-Minute-World Hypothesis approximate the earth were created five minutes ago, complete with memory images, history books, records, etc. , how coul d we ever know of it? As Russell wrote in The Analysis of Mind, There is no logical impossibility in the hypothesis that the world sprang into being five minutes ago, exactly as it then was, with a population that remembered a wholly unreal past. There is no logically necessary confederation between events at different times therefore nothing that is happening now or will happen in the future can disprove the hypothesis that the world began five minutes ago. For example, an omnipotent God could create the world with all the memories, historical records, and so forth five minutes ago. any(prenominal) evidence to the contrary would be evidence created by God five minutes ago. (Q. v. , the Omphalos hypothesis. ) 2. Suppose everything in the universe (including all spatial relations) were to expand uniformly a thousand times larger. How could we ever know it? A moments thought reveals that the mass of objects increases by the cube whereas the distance among them increases linearly.He nce, if such an expansion were possible, changes in the measurement of gravity and the speed of light would be evident, if, indeed, life would be possible. 3. Russells Five-Minute-World Hypothesis is a philosophical problem the impossibility of the objects in the universe expanding is a scientific problem since the latter problem can, in fact, be answered by principles of elementary physics. 1. Ontology or Metaphysics the study of what is really real. Metaphysics deals with the so-called first principles of the natural order and the ultimate generalizations available to the human intellect. Specifically, ontology seeks to indentify and establish the relationships between the categories, if any, of the types of existent things. ? What kinds of things exist? Do only particular things exist or do general things also exist? How is existence possible? Questions as to individuality and change of objectsare you the same person you were as a baby? as of yesterday? as of a moment ago? ? How do ideas exist if they have no size, shape, or color? (My idea of the Empire State Building is quite as small or as large as my idea of a book.I. e., an idea is not extended in space. ) What is space? What is time? ? E. g. , Consider the truths of math in what manner do geometric figures exist? Are points, lines, or planes real or not? Of what are they make? ? What is spirit? or soul? or matter? space? Are they made up of the same sort of stuff? ? When, if ever, are events necessary? Under what conditions are they possible? 1. Further characteristics of philosophy and examples of philosophical problems are discussed in the next tutorial. http//philosophy. lander. edu/intro/what. shtml.