First-order logic Essay Example for Free

First-order logic Essay 1. (Philosophy) the academic discipline concerned with making explicit the nature and significance of ordinary and scientific beliefs and investigating the intelligibility of concepts by means of rational argument concerning their presuppositions, implications, and interrelationships; in particular, the rational investigation of the nature and structure of reality (metaphysics), the resources and limits of knowledge (epistemology), the principles and import of moral judgment (ethics), and the relationship between language and reality (semantics) 2. (Philosophy) the particular doctrines relating to these issues of some specific individual or school the philosophy of Descartes 3. (Philosophy) the critical study of the basic principles and concepts of a discipline the philosophy of law 4. (Literary Literary Critical Terms) Archaic or literary the investigation of natural phenomena, esp alchemy, astrology, and astronomy 5. any system of belief, values, or tenets 6. a personal outlook or viewpoint 7. serenity of temper phi†¢los†¢o†¢phy (f l? s ? fi) n. , pl. -phies. 1. the rational investigation of the truths and principles of being, knowledge, or conduct. 2. a system of philosophical doctrine: the philosophy of Spinoza. 3. the critical study 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 study of general and fundamental problems, such as those connected with reality, existence, knowledge, values, reason,mind, and language. Philosophy is distinguished from other ways of addressing such problems by its critical, generally systematic approach and its reliance on rational argument. [3] In more casual speech, by extension, philosophy can refer to the most basic beliefs, concepts, and attitudes of an individual or group. The word philosophy comes from the Ancient Greek (philosophia), which literally means love of wisdom. [5][6][7] The introduction of the terms philosopher and philosophy has been ascribed to the Greek thinker Pythagoras. [8] https://en. wikipedia. org/wiki/Philosophy http://www. thefreedictionary. com/philosophy Branches of Philosophy Main branches of philosophy Traditionally, there are five main branches of philosophy. They are: †¢ 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 notion of beauty and the philosophy of art. http://www. philosophy-index. com/philosophy/branches/ Aesthetics Aesthetics is the area of philosophy which covers the concepts of beauty and art. â€Å"Beauty is in the eye of the beholder† There are two basic standings on the nature of beauty: objective and subjective judgement. Subjective judgement of beauty suggests that beauty is not the same to everyone — that which aesthetically pleases the observer is beautiful (to the observer). Alternatively, those partial to the objective description of beauty try to measaure it. They suggest that certain properties of an object create an inherent beauty — such as symmetry and balance. Both Plato and Aristotlesupported the objective judgement. Some, such as Immanuel Kant, took a middle path, holding that beauty is of a subjective nature, but there are qualities of beauty which have universal validity. Classical and Modern Aesthetics The classical concepts behind aesthetics saw beauty in nature, and that art should mimic those qualities found in nature. AristotlesPoetics describes this idea, which he develops from Platos teachings. Modern aesthetic ideas, including those of Kant, stress the creative and symbolic side of art — that nature does not always have to guide art for it to be beautiful. Epistemology Epistemology 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 considers that knowledge can be obtained throughexperience and/or reason. Defining Knowledge A primary concern of epistemology is the very definition of knowledge itself. The traditional definition, since Plato, is that knowledge is justified true belief, but recent evaluations of the concept have shown supposed 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 must first be understood. Definition of knowledge Sources of Knowledge The sources of knowledge must also be considered. Perception, reason, memory, testimony, introspection and innate ideas are all supposed sources of knowledge. Scepticism There also seems 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 generally three branches of ethics: †¢ Meta-ethics, which is concerned with questions about 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 applied to specific cases. Logic Logic is the systematic process of valid reasoning through inference — deriving conclusions from information that is known to be true. It is the area of philosophy that is concerned with the laws of valid reasoning. Symbolic Logic Symbolic logic is the method of representing logical expressions through the use of symbols and variables, rather than in ordinary language. This has the benefit of removing the ambiguity 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 may have seperate symbols, or exclude the use of certain symbols. Logical Symbols The following table presents several logical symbols, their name and meaning, and any relevant notes. The name of the symbol (under â€Å"meaning† links to a page explaining the symbol or term and its use). Note that different symbols have been used by different logicians and systems of logic. For the sake of clarity, this site consistently uses the symbols in the left column, while the â€Å"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 dot (  · ) are also often used. | |? |disjunction (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 either†¦or. Sometimes ? is | | | |used. | || |alternative denial(NAND) |Means â€Å"not both†. Sometimes written as ^ | |v |joint denial (NOR) |Means â€Å"neither/nor†. | | |conditional(if/then) |Many logicians use the symbol? instead. This is also known as material implication. | |- |biconditional (iff) |Means â€Å"if and only if† ? is sometimes used, but this site reserves that symbol for equivalence. | |Quantifiers | |? |universal quantifier |Means â€Å"for all†, so ? xPx means that Px is true for every x. | |? |existential quantifier |Means â€Å"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 |Used 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 possible 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 â€Å"F† is used. | |Parentheses | |( ) |parentheses |Used to group expressions to show precedence of operations. Square brackets [ ] are sometimes used to | | | |clarify groupings. | |Set Theory | |?. |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 |Two sets are equal if they contain exactly the same elements. | |? |absolute complement |? (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 will use the system defined below. |Symbol |Meaning |Notes | |A, B, C †¦ Z |propositions |Uppercase Roman letters signify individual propositions. For example, P may symbolize the proposition â€Å"Pat is | | | |ridiculous†. 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 logical 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 â€Å"j† may be used to signify â€Å"Jerry†. Constants are given a meaning before they are used in | | | |logical 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 â€Å"x is green† as Gx, and â€Å"x likes y† as Lxy. To differentiate 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 value 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 uppercase Alpha (? ) | | | |looks identical to the Roman letter â€Å"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 detailing 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 simply 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, using 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 include: o Modal Logic, which deals with modal operators neccessarily and possibly. o Epistemic Logic, which reasons about 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 relevance 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 â€Å"metaphysics† comes from Greek, meaning â€Å"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 origin 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 matter and physics, and followed those volumes with work on ontology, and other broad subjects. These editors referred to them as â€Å"the books that came after the books on physics† or â€Å"metaphysics†. Aristotle himself refers to metaphysics as â€Å"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 studying 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 â€Å"an ontology† refers to the things counted as being in a metaphysical system. Generally, an ontology is a list of things that exist — the â€Å"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 prominent 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, evolution 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 study 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 from the Greek is suggested by the following words and word-fragments. ? philo—love of, affinity for, liking of ? philander—to engage in love affairs frivolously ? philanthropy—love of mankind in general ? philately—postage stamps hobby ? phile—(as in anglophile) one having a love for ? philology—having a liking for words ? sophos—wisdom ? sophist—lit. one who loves knowledge ? sophomore—wise and moros—foolish;i. e. one who thinks he knows many things ? sophisticated—one 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  peculiar problems (i. e. , specifically the kinds of problems usually relegated to the main branches discussed below in Section II). ? There is, perhaps, no one single sense of the word philosophy. Eventually many writers abandon the attempt to define philosophy and, instead, turn to the kinds of things philosophers do. ? What is involved 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 divided 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 status. More often than not, the term value theory is used instead of axiology in contemporary discussions even though the term â€Å"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. Nature 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. , (â€Å"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 analyzed 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 quite 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 morally correct action. One might even be inclined to conclude that saving lives is a duty. But 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 activity? What would the realistic representations in pop art represent? Does art represent sensible objects or ideal 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 context 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. , flattened 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 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. ) ? Furthermore, consider two well-known problems in epistemology: 1. Russells Five-Minute-World Hypothesis: Suppose the earth were created five minutes ago, complete with memory images, history books, records, etc. , how could 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 connection 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 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. Hence, 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 identity and change of objects—are 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 mathematics: in what manner do geometric figures exist? Are points, lines, or planes real or not? Of what are they made? ? 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.