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There is no general agreement on how the two are to be distinguished. One prominent approach associates their difference with the study of arguments expressed in formal or informal languages. Another characterizes informal logic as the study of ampliative inferences, in contrast to the deductive inferences studied by formal logic. But it is also common to link their difference to the distinction between formal and informal fallacies. Logic is based on various fundamental concepts.
It studies arguments, which are made up of a set of premises together with a conclusion. Premises and conclusions are usually understood either as sentences or as propositions and are characterized by their internal structure. Complex propositions are made up of other propositions linked to each other by propositional connectives. Simple propositions have subpropositional parts, like singular terms and predicates. In either case, the truth of a proposition usually depends on the denotations of its constituents.
Logically true propositions constitute a special case since their truth depends only on the logical vocabulary used in them. The arguments or inferences made up of these propositions can be either correct or incorrect. An argument is correct if its premises support its conclusion. The strongest form of support is found in deductive arguments: it is impossible for their premises to be true and their conclusion to be false. This is the case if they follow a rule of inference , which ensures the truth of the conclusion if the premises are true.
A consequence of this is that deductive arguments cannot arrive at any substantive new information not already found in their premises. They contrast in this respect with ampliative arguments, which may provide genuinely new information.
This comes with an important drawback: it is possible for all their premises to be true while their conclusion is still false. Many arguments found in everyday discourse and the sciences are ampliative arguments. They are sometimes divided into inductive and abductive arguments. Inductive arguments usually take the form of statistical generalizations while abductive arguments are inferences to the best explanation.
Arguments that fall short of the standards of correct reasoning are called fallacies. For formal fallacies, the source of the error is found in the form of the argument while informal fallacies usually contain errors on the level of the content or the context. Besides the definitory rules of logic, which determine whether an argument is correct or not, there are also strategic rules, which describe how a chain of correct arguments can be used to arrive at one's intended conclusion.
In formal logic, formal systems are often used to give a precise definition of correct reasoning using a formal language. Systems of logic are theoretical frameworks for assessing the correctness of reasoning and arguments. Aristotelian logic focuses on reasoning in the form of syllogisms. Its traditional dominance was replaced by classical logic in the modern era. Classical logic is "classical" in the sense that it is based on various fundamental logical intuitions shared by most logicians.
It consists of propositional logic and first-order logic. Propositional logic ignores the internal structure of simple propositions and only considers the logical relations on the level of propositions. First-order logic, on the other hand, articulates this internal structure using various linguistic devices, such as predicates and quantifiers.
Extended logics accept the basic intuitions behind classical logic and extend it to other fields, such as metaphysics , ethics , and epistemology. This happens usually by introducing new logical symbols, such as modal operators.
Deviant logics, on the other hand, reject certain classical intuitions and provide alternative accounts of the fundamental laws of logic. While most systems of logic belong to formal logic, some systems of informal logic have also been proposed.
One prominent approach understands reasoning as a dialogical game of persuasion while another focuses on the epistemic role of arguments. Logic is studied in and applied to various fields, such as philosophy, mathematics , computer science , and linguistics.
Logic has been studied since Antiquity , early approaches including Aristotelian logic, Stoic logic , Anviksiki , and the mohists. Modern formal logic has its roots in the work of late 19th-century mathematicians such as Gottlob Frege.
The word "logic" originates from the Greek word "logos", which has a variety of translations, such as reason , discourse , or language. Logic is interested in whether arguments are good or inferences are valid, i. These general characterizations apply to logic in the widest sense since they are true both for formal and informal logic.
In this narrower sense, logic is a formal science that studies how conclusions follow from premises in a topic-neutral way. This means that it is impossible for the premises to be true and the conclusion to be false. This means that it is true in all possible worlds and under all interpretations of its non-logical terms.
The term "logic" can also be used in a slightly different sense as a countable noun. In this sense, a logic is a logical formal system. Different logics differ from each other concerning the formal languages used to express them and, most importantly, concerning the rules of inference they accept as valid.
There is an ongoing debate about which of these systems should be considered logics in the strict sense instead of non-logical formal systems.
According to these criteria, it has been argued, for example, that higher-order logics and fuzzy logic should not be considered logics when understood in a strict sense. When understood in the widest sense, logic encompasses both formal and informal logic.
These difficulties often coincide with the wide disagreements about how informal logic is to be defined. The most literal approach sees the terms "formal" and "informal" as applying to the language used to express arguments.
Formal languages are characterized by their precision and simplicity. Another approach draws the distinction according to the different types of inferences analyzed. This means that if all the premises are true, it is impossible for the conclusion to be false. They achieve this at the cost of certainty: even if all premises are true, the conclusion of an ampliative argument may still be false. One more approach tries to link the difference between formal and informal logic to the distinction between formal and informal fallacies.
In the case of formal fallacies, the error is found on the level of the argument's form, whereas for informal fallacies, the content and context of the argument are responsible. Informal logic, on the other hand, also takes the content and context of an argument into consideration.
But in another context, against an opponent that actually defends the strawman position, the argument is correct. Other accounts draw the distinction based on investigating general forms of arguments in contrast to particular instances, on the study of logical constants instead of substantive concepts , on the discussion of logical topics with or without formal devices, or on the role of epistemology for the assessment of arguments.
Premises and conclusions are the basic parts of inferences or arguments and therefore play a central role in logic. In the case of a valid inference or a correct argument, the conclusion follows from the premises or the premises support the conclusion.
It is generally accepted that premises and conclusions have to be truth-bearers. Thus contemporary philosophy generally sees them either as propositions or as sentences. Propositional theories of premises and conclusions are often criticized because of the difficulties involved in specifying the identity criteria of abstract objects or because of naturalist considerations. But this approach comes with new problems of its own: sentences are often context-dependent and ambiguous , meaning that whether an argument is valid would not only depend on its parts but also on its context and on how it is interpreted.
In earlier work, premises and conclusions were understood in psychological terms as thoughts or judgments, an approach known as " psychologism ". This position was heavily criticized around the turn of the 20th century. A central aspect of premises and conclusions for logic, independent of how their nature is conceived, concerns their internal structure.
As propositions or sentences, they can be either simple or complex. Simple propositions, on the other hand, do not have propositional parts. But they can also be conceived as having an internal structure: they are made up of subpropositional parts, like singular terms and predicates. Whether a proposition is true depends, at least in part, on its constituents. These subpropositional parts have meanings of their own, like referring to objects or classes of objects.
This topic is studied by theories of reference. In some cases, a simple or a complex proposition is true independently of the substantive meanings of its parts.
In such cases, the truth is called a logical truth : a proposition is logically true if its truth depends only on the logical vocabulary used in it. In some modal logics , this notion can be understood equivalently as truth at all possible worlds.
Logic is commonly defined in terms of arguments or inferences as the study of their correctness. Sometimes a distinction is made between simple and complex arguments. These simple arguments constitute a chain because the conclusions of the earlier arguments are used as premises in the later arguments.
For a complex argument to be successful, each link of the chain has to be successful. A central aspect of arguments and inferences is that they are correct or incorrect. If they are correct then their premises support their conclusion. In the incorrect case, this support is missing. It can take different forms corresponding to the different types of reasoning.
But even arguments that are not deductively valid may still constitute good arguments because their premises offer non-deductive support to their conclusions. For such cases, the term ampliative or inductive reasoning is used. A deductively valid argument is one whose premises guarantee the truth of its conclusion.
Alfred Tarski holds that deductive arguments have three essential features: 1 they are formal, i. Because of the first feature, the focus on formality, deductive inference is usually identified with rules of inference. Arguments that do not follow any rule of inference are deductively invalid. It has the form "if A, then B; A; therefore B". The third feature can be expressed by stating that deductively valid inferences are truth-preserving: it is impossible for the premises to be true and the conclusion to be false.
A different characterization distinguishes between surface and depth information. Ampliative inferences, on the other hand, are informative even on the depth level. They are more interesting in this sense since the thinker may acquire substantive information from them and thereby learn something genuinely new. This characteristic is closely related to non-monotonicity and defeasibility : it may be necessary to retract an earlier conclusion upon receiving new information or in the light of new inferences drawn.
Ampliative arguments are not automatically incorrect. Instead, they just follow different standards of correctness. An important aspect of most ampliative arguments is that the support they provide for their conclusion comes in degrees.
This contrasts with deductive arguments, which are either valid or invalid with nothing in-between. The terminology used to categorize ampliative arguments is inconsistent. Some authors use the term "induction" to cover all forms of non-deductive arguments. The conclusion then is a general law that this pattern always obtains. Abductive inference may or may not take statistical observations into consideration.
In either case, the premises offer support for the conclusion because the conclusion is the best explanation of why the premises obtain. This conclusion is justified because it is the best explanation of the current state of the kitchen.
For example, the conclusion that a burglar broke into the house last night, got hungry on the job, and had a midnight snack, would also explain the state of the kitchen. But this conclusion is not justified because it is not the best or most likely explanation.
Not all arguments live up to the standards of correct reasoning. When they do not, they are usually referred to as fallacies. Their central aspect is not that their conclusion is false but that there is some flaw with the reasoning leading to this conclusion. Some theorists give a more restrictive definition of fallacies by additionally requiring that they appear to be correct. This explains why people tend to commit fallacies: because they have an alluring element that seduces people into committing and accepting them.
Fallacies are usually divided into formal and informal fallacies. For example, denying the antecedent is one type of formal fallacy, as in "if Othello is a bachelor, then he is male; Othello is not a bachelor; therefore Othello is not male".
The source of their error is usually found in the content or the context of the argument. For fallacies of ambiguity, the ambiguity and vagueness of natural language are responsible for their flaw, as in "feathers are light; what is light cannot be dark; therefore feathers cannot be dark".
The main focus of most logicians is to investigate the criteria according to which an argument is correct or incorrect. A fallacy is committed if these criteria are violated. In the case of formal logic, they are known as rules of inference. Definitory rules contrast with strategic rules.
In chess , for example, the definitory rules dictate that bishops may only move diagonally while the strategic rules describe how the allowed moves may be used to win a game, for example, by controlling the center and by defending one's king. They belong to the field of psychology and generalize how people actually draw inferences. A formal system of logic consists of a language , a proof system , and a semantics. The term "a logic" is often used a countable noun to refer to a particular formal system of logic.
Different logics can differ from each other in their language, proof system, or their semantics. A language is a set of well formed formulas. Languages are typically defined by providing an alphabet of basic expressions and recursive syntactic rules which build them into formulas.
A proof system is a collection of formal rules which define when a conclusion follows from given premises. Rules in a proof systems are always defined in terms of formulas' syntactic form, never in terms of their meanings. Such rules can be applied sequentially, giving a mechanical procedure for generating conclusions from premises. There are a number of different types of proof systems including natural deduction and sequent calculi.
A semantics is a system for mapping expressions of a formal language to their denotations. In many systems of logic, denotations are truth values. Entailment is a semantic relation which holds between formulas when the first cannot be true without the second being true as well. A system of logic is sound when its proof system cannot derive a conclusion from a set of premises unless it is semantically entailed by them.
In other words, its proof system cannot lead to false conclusions, as defined by the semantics. A system is complete when its proof system can derive every conclusion that is semantically entailed by its premises.
In other words, its proof system can lead to any true conclusion, as defined by the semantics. Thus, soundness and completeness together describe a system whose notions of validity and entailment line up perfectly. The study of properties of formal systems is called metalogic. Other important metalogical properties include consistency , decidability , and expressive power.
For over two thousand years, Aristotelian logic was treated as the cannon of logic. It encompasses propositional logic and first-order logic. Because of this focus on mathematics, it does not include logical vocabulary relevant to many other topics of philosophical importance, like the distinction between necessity and possibility, the problem of ethical obligation and permission, or the relations between past, present, and future.
They build on the fundamental intuitions of classical logic and expand it by introducing new logical vocabulary. This way, the exact logical approach is applied to fields like ethics or epistemology that lie beyond the scope of mathematics.
Deviant logics, on the other hand, reject some of the fundamental intuitions of classical logic. Deviant logical systems differ from each other either because they reject different classical intuitions or because they propose different alternatives to the same issue. Informal logic is usually done in a less systematic way. It often focuses on more specific issues, like investigating a particular type of fallacy or studying a certain aspect of argumentation.
When understood in the widest sense, Aristotelian logic encompasses a great variety of topics, including metaphysical theses about ontological categories and problems of scientific explanation. A syllogism is a certain form of argument involving three propositions: two premises and a conclusion. Each proposition has three essential parts: a subject , a predicate , and a copula connecting the subject to the predicate.
In this sense, Aristotelian logic does not contain complex propositions made up of various simple propositions. Aristotelian logic differs from predicate logic in that the subject is either universal , particular , indefinite , or singular.
A similar proposition could be formed by replacing it with the particular term "some humans", the indefinite term "a human", or the singular term "Socrates". Using different combinations of subjects and predicates, a great variety of propositions and syllogisms can be formed.
Syllogisms are characterized by the fact that the premises are linked to each other and to the conclusion by sharing one predicate in each case.
The syllogism "all cats are mortal; Socrates is mortal; therefore Socrates is a cat", on the other hand, is invalid. Propositional logic comprises formal systems in which formulae are built from atomic propositions using logical connectives. Unlike predicate logic where terms and predicates are the smallest units, propositional logic takes full propositions with truth values as its most basic component.
First-order logic provides an account of quantifiers general enough to express a wide set of arguments occurring in natural language. The development of first-order logic is usually attributed to Gottlob Frege , who is also credited as one of the founders of analytic philosophy , but the formulation of first-order logic most often used today is found in Principles of Mathematical Logic by David Hilbert and Wilhelm Ackermann in The analytical generality of first-order logic allowed the formalization of mathematics, drove the investigation of set theory , and allowed the development of Alfred Tarski 's approach to model theory.
It provides the foundation of modern mathematical logic. Many extended logics take the form of modal logic by introducing modal operators. Modal logic were originally developed to represent statements about necessity and possibility. Modal logics can be used to represent different phenomena depending on what flavor of necessity and possibility is under consideration.
Within philosophy, modal logics are widely used in formal epistemology , formal ethics , and metaphysics. Within linguistic semantics , systems based on modal logic are used to analyze linguistic modality in natural languages. Higher-order logics extend classical logic not by using modal operators but by introducing new forms of quantification. In classical first-order logic, quantifiers are only applied to individuals. In higher-order logics, quantification is also allowed over predicates.
This increases its expressive power. A great variety of deviant logics have been proposed. One major paradigm is intuitionistic logic , which rejects the law of the excluded middle. Intuitionism was developed by the Dutch mathematicians L. Brouwer and Arend Heyting to underpin their constructive approach to mathematics , in which the existence of a mathematical object can only be proven by constructing it.
Intuitionistic logic is of great interest to computer scientists, as it is a constructive logic and sees many applications, such as extracting verified programs from proofs and influencing the design of programming languages through the formulae-as-types correspondence. Multi-valued logics depart from classicality by rejecting the principle of bivalence which requires all propositions to be either true or false.
Fuzzy logics are multivalued logics that have an infinite number of "degrees of truth", represented by a real number between 0 and 1. The pragmatic or dialogical approach to informal logic sees arguments as speech acts and not merely as a set of premises together with a conclusion. Walton understands a dialogue as a game between two players.
Dialogues are games of persuasion: each player has the goal of convincing the opponent of their own conclusion. A winning move is a successful argument that takes the opponent's commitments as premises and shows how one's own conclusion follows from them.
For this reason, it is normally necessary to formulate a sequence of arguments as intermediary steps, each of which brings the opponent a little closer to one's intended conclusion. Besides these positive arguments leading one closer to victory, there are also negative arguments preventing the opponent's victory by denying their conclusion.
Fallacies , on the other hand, are violations of the standards of proper argumentative rules. The epistemic approach to informal logic, on the other hand, focuses on the epistemic role of arguments. They achieve this by linking justified beliefs to beliefs that are not yet justified. Degrees of belief are understood as subjective probabilities in the believed proposition, i. Bad or irrational reasoning, on the other hand, violates these laws. Logic is studied in various fields.
In many cases, this is done by applying its formal method to specific topics outside its scope, like to ethics or computer science. This can happen in diverse ways, like by investigating the philosophical presuppositions of fundamental logical concepts, by interpreting and analyzing logic through mathematical structures, or by studying and comparing abstract properties of formal logical systems.
Philosophy of logic is the philosophical discipline studying the scope and nature of logic. It studies the application of logical methods to philosophical problems in fields like metaphysics , ethics, and epistemology.
Mathematical logic is the study of logic within mathematics. Major subareas include model theory , proof theory , set theory , and computability theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic.
However, it can also include attempts to use logic to analyze mathematical reasoning or to establish logic-based foundations of mathematics. Mathematical theories were supposed to be logical tautologies , and the programme was to show this by means of a reduction of mathematics to logic.
Set theory originated in the study of the infinite by Georg Cantor , and it has been the source of many of the most challenging and important issues in mathematical logic, from Cantor's theorem , through the status of the Axiom of Choice and the question of the independence of the continuum hypothesis , to the modern debate on large cardinal axioms. Recursion theory captures the idea of computation in logical and arithmetic terms; its most classical achievements are the undecidability of the Entscheidungsproblem by Alan Turing , and his presentation of the Church—Turing thesis.
In computer science, logic is studied as part of the theory of computation. Key areas of logic that are relevant to computing include computability theory , modal logic , and category theory. Early computer machinery was based on ideas from logic such as the lambda calculus.
For instance, modern artificial intelligence builds on logicians' work in argumentation theory , while automated theorem proving can assist logicians in finding and checking proofs. In logic programming languages such as Prolog , a program computes the consequences of logical axioms and rules to answer a query. Formal semantics is a subfield of both linguistics and philosophy which uses logic to analyze meaning in natural language. It is an empirical field which seeks to characterize the denotations of linguistic expressions and explain how those denotations are composed from the meanings of their parts.
The field was developed by Richard Montague and Barbara Partee in the s, and remains an active area of research. Central questions include scope , binding , and linguistic modality. What is the epistemological status of the laws of logic?
What sort of argument is appropriate for criticizing purported principles of logic? In an influential paper entitled " Is Logic Empirical? Quine , argued that in general the facts of propositional logic have a similar epistemological status as facts about the physical universe, for example as the laws of mechanics or of general relativity , and in particular that what physicists have learned about quantum mechanics provides a compelling case for abandoning certain familiar principles of classical logic: if we want to be realists about the physical phenomena described by quantum theory, then we should abandon the principle of distributivity , substituting for classical logic the quantum logic proposed by Garrett Birkhoff and John von Neumann.
Another paper of the same name by Michael Dummett argues that Putnam's desire for realism mandates the law of distributivity. In this way, the question, "Is Logic Empirical? Georg Wilhelm Friedrich Hegel was deeply critical of any simplified notion of the law of non-contradiction. It was based on Gottfried Wilhelm Leibniz 's idea that this law of logic also requires a sufficient ground to specify from what point of view or time one says that something cannot contradict itself.
A building, for example, both moves and does not move; the ground for the first is our solar system and for the second the earth. In Hegelian dialectic, the law of non-contradiction, of identity, itself relies upon difference and so is not independently assertable. Closely related to questions arising from the paradoxes of implication comes the suggestion that logic ought to tolerate inconsistency.
Relevance logic and paraconsistent logic are the most important approaches here, though the concerns are different: a key consequence of classical logic and some of its rivals, such as intuitionistic logic , is that they respect the principle of explosion , which means that the logic collapses if it is capable of deriving a contradiction.
Graham Priest , the main proponent of dialetheism , has argued for paraconsistency on the grounds that there are in fact, true contradictions. Logic arose from a concern with correctness of argumentation. Modern logicians usually wish to ensure that logic studies just those arguments that arise from appropriately general forms of inference. For example, Thomas Hofweber writes in the Stanford Encyclopedia of Philosophy that logic "does not, however, cover good reasoning as a whole. That is the job of the theory of rationality.
Rather it deals with inferences whose validity can be traced back to the formal features of the representations that are involved in that inference, be they linguistic, mental, or other representations. The idea that logic treats special forms of argument, deductive argument, rather than argument in general, has a history in logic that dates back at least to logicism in mathematics 19th and 20th centuries and the advent of the influence of mathematical logic on philosophy.
A consequence of taking logic to treat special kinds of argument is that it leads to identification of special kinds of truth, the logical truths with logic equivalently being the study of logical truth , and excludes many of the original objects of study of logic that are treated as informal logic.
Robert Brandom has argued against the idea that logic is the study of a special kind of logical truth, arguing that instead one can talk of the logic of material inference in the terminology of Wilfred Sellars , with logic making explicit the commitments that were originally implicit in informal inference.
The philosophical vein of various kinds of skepticism contains many kinds of doubt and rejection of the various bases on which logic rests, such as the idea of logical form, correct inference, or meaning, sometimes leading to the conclusion that there are no logical truths.
This is in contrast with the usual views in philosophical skepticism , where logic directs skeptical enquiry to doubt received wisdoms, as in the work of Sextus Empiricus. Friedrich Nietzsche provides a strong example of the rejection of the usual basis of logic: his radical rejection of idealization led him to reject truth as a " Innumerable beings who made inferences in a way different from ours perished".
This position held by Nietzsche however, has come under extreme scrutiny for several reasons. Logic was developed independently in several cultures during antiquity.
One major early contributor was Aristotle , who developed term logic in his Organon and Prior Analytics. Inferences are expressed by means of syllogisms that consist of two propositions sharing a common term as premise, and a conclusion that is a proposition involving the two unrelated terms from the premises. Aristotle's monumental insight was the notion that arguments can be characterized in terms of their form.
Aristotelian logic was highly regarded in classical and medieval times, both in Europe and the Middle East. It remained in wide use in the West until the early 19th century. Ibn Sina Avicenna — CE was the founder of Avicennian logic , which replaced Aristotelian logic as the dominant system of logic in the Islamic world , [] and also had an important influence on Western medieval writers such as Albertus Magnus [] and William of Ockham. In Europe during the later medieval period, major efforts were made to show that Aristotle's ideas were compatible with Christian faith.
During the High Middle Ages , logic became a main focus of philosophers, who would engage in critical logical analyses of philosophical arguments, often using variations of the methodology of scholasticism.
Initially, medieval Christian scholars drew on the classics that had been preserved in Latin through commentaries by such figures such as Boethius , later the work of Islamic philosophers such as Ibn Sina and Ibn Rushd Averroes — CE were drawn on, which expanded the range of ancient works available to medieval Christian scholars since more Greek work was available to Muslim scholars that had been preserved in Latin commentaries.
In , William of Ockham 's influential Summa Logicae was released. Logic Pro, being a Mac-based software follows suit with quality aesthetics and is intuitively easy to use. A good alternative to Logic should share that aspect. A strong community to learn how to utilize certain aspects is almost as important as the software itself. Choosing a DAW comparable to Logic should have a strong community to tap into or a library of online tutorials. We took a look at alternatives that are just as affordable from a price point, including the value you can get out of it.
Here are the top Logic Pro X alternatives with details on pricing, platform support, features, and more. We looked for support on Windows machines as a necessary component as Logic is only available on Mac. Soundtrap is a completely online DAW acquired by Spotify in Above all, the learning curve to accomplish recording a song is made easy with Soundtrap.
You do not need to worry about compatibility with your computer, as it runs in the browser independently, while still operating as professionally built DAW. One of the more exciting features of Soundtrap is its online collaboration tools.
You can easily sync up and work with your producer of band members, as well as find new talent easily by searching. Reason is a DAW that many others have used as a basis for new ideas and creativity. With their unique rack and patching system, you are able to work with synths and other outboard style gear virtually like it was the real thing!
Take the way you like to approach effects routing to Pro Tools or Logic with ease! Hey producers! Looking for fresh new sounds? There are countless ways to learn how Pro Tools works and is a great option for a home studio or professional studio.
You can learn about these here! PreSonus is another brilliantly designed platform that reduces the learning curve when new to music production or starting with a new DAW. Additionally, the community you can tap into for inspiration, collaboration, and general support is now extensive.
User reviews have reported transition from Logic to Studio One as an alternative easier than you might expect. Another great feature with Studio One is the 40GBs of samples , loops, and one-shots that it comes with by default. Many other DAWs require annual upgrade purchases. They have a very active community on their forum, and FL studio has some of the most tutorials and courses online.
Sound Forge is focused on the mastering aspects of the production process. With Sound Forge, you should be able to achieve commercial standard volume in your mastering process, while also reducing noise. Outside of being a free DAW desktop platform, their community is essentially a social network for musicians, producers, and music fans that help you collaborate and make money off of your music. In addition to Cakewalk, Bandlab has an online mix editor to start making beats, recording audio, and much more.
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