Difference between revisions of "Logic"

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[[Logic]] ([[Greek]] λογίζω I reckon, I count, from λόγος a word) is a branch of [[philosophy]] that deals with and attempts to guide the faculty of human reason.
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[[File:Aristotle.jpg|thumb|250px|[[Aristotle]] is often called the father of logic in the [[West]]. ]]
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'''Logic''' ([[Greek language|Greek]] ''λογίζω'' – I think, I reason; from ''λόγος'' – reason) refers the patterns in reasoning behind [[argument]]s. In [[philosophy]], logic is a sub-branch of [[epistemology]] that deals with and attempts to guide the faculty of human [[reason]]. It is often studied alongside [[mathematics]].
  
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[[Aristotle]] was the first to formalize the practice of reasoning.  In particular, Aristotle developed a formalization of the [[syllogism]] and the [[Square of Opposition]].  While [[Modus ponens]] and its complement [[Modus tollens]] were known to the Medievals, it is not clear when these central laws of deduction were first formalized.{{fact}}
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''[[Encyclopedia Britannica]]'' declares:
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:"Laws of thought, traditionally, the three fundamental laws of logic: (1) the [[Law of Non-Contradiction|law of contradiction]], (2) the [[Law of the excluded middle|law of excluded middle]] (or third), and (3) the [[Law of identity|principle of identity]]. That is, (1) for all [[proposition]]s p, it is impossible for both p and not p to be true, or symbolically ∼(p · ∼p), in which ∼ means “not” and · means “and”; (2) either p or ∼p must be true, there being no third or middle true proposition between them, or symbolically p ∨ ∼p, in which ∨ means “or”; and (3) if a propositional function F is true of an individual variable x, then F is indeed true of x, or symbolically F(x) ⊃ F(x), in which ⊃ means “formally implies.” Another formulation of the principle of identity asserts that a thing is identical with itself, or (∀x) (x = x), in which ∀ means “for every”; or simply that x is x."<ref>[https://www.britannica.com/topic/laws-of-thought Laws of Thought], Encyclopedia Britannica</ref>
  
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The next easily-identifiable major development in logic comes from [[Boole]] in 1847 with the creation of [[Boolean algebra]]. In the 1870's, Peirce introduces a logic of quantifiers, followed by [[Gottlob Frege]]'s 1879 ''Begriffsschrift'', which contains the first complete formalization of the propositional calculus.  Frege's work was developed in service of his [[logicism|logicist]] project, which was shown to be inconsistent by [[Bertrand Russell]] in 1903 with what is famously called [[Russell's paradox]].
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[[Aristotle]] was the first to formalize the practice of [[Logical reasoning|reasoning]]. In particular, Aristotle developed a formalization of the [[Syllogism|categorical syllogism]], the three laws of logic, and the [[Square of opposition|Square of Opposition]]. While modus ponens and its complement modus tollens were known to the medieval philosophers, it is not clear when these central laws of deduction were first formalized.
  
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By the 1950's, with the work of many logicians including [[Hilbert]], [[Emile Post]], [[Alfred Tarski]] and [[Kurt Goedel]], most of the major results in first-order logic had been proved, and in the 1960's [[Saul Kripke]] added a completeness proof for [[modal logic]].
+
The next easily identifiable major development in logic comes from Boole in 1847 with the creation of [[Boolean algebra]] and Boolean logic. In the 1870s, Peirce introduces a logic of quantifiers, followed by [[Gottlob Frege]]'s 1879 ''Begriffsschrift'', which contains the first complete formalization of the propositional calculus. Frege's work was developed in service of his logicist project, which was shown to be inconsistent by [[Bertrand Russell]] in 1903 with what is famously called [[Russell's paradox]]. In 1913 Russell and Whitehead published ''Principia Mathematica'', a landmark work that sought to define mathematics with first-order logic.
  
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Most logical systems are [[bivalent]]; that is, they admit only two truth-values.  However, there is a fair amount of work done on non-bivalent systems of logic, especially [[intuitionist logic]] and [[relevance logic]].  In intuitionist logic, "true" and "false" are replaced with "proven true", "proven contradictory", and "not proven".  In relevance logic, "neither true nor false" and "both true and false" are added to the standard two truth values.  There is some debate about the value of these systems in philosophical circles, and [[Timothy Williamson]] claims to have a proof that any non-bivalent logic can be converted into a bivalent logic.
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By the 1950s, with the work of many logicians including [[Hilbert]], Emile Post, Alfred Tarski and [[Kurt Gödel]], most of the major results in first-order logic had been proved, and in the 1960s Saul Kripke added a completeness proof for modal logic.
  
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Formal logic requires, since Frege, a distinction between the [[object language]] and the [[metalanguage]].  The metalanguage is ordinarily a natural language like English or German, while the object language is a symbolic language with a limited alphabet and syntax.
+
Most logical systems are bivalent; that is, they admit only two truth-values, true or false. However, there is a fair amount of work done on trivalent systems of logic, specifically paraconsistent logic and relevance logic (a subset of paraconsistent logic). These logics were developed in response to the paradoxes that classical logic can create, namely, that anything can follow from a falsehood and the [[principle of explosion]].
  
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== Informal Logic ==
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Formal logic requires, since Frege, a distinction between the object language and the metalanguage. The metalanguage is ordinarily a natural language like English or German, while the object language is a symbolic language with an alphabet limited to variable meanings and a symbolic system for describing the relationships between the variables.
  
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The art of [[rhetoric]] is sometimes called "informal logic", and the classic [[logical fallacies|fallacies]] are often described as "logical fallacies", though, strictly speaking, most of them are simply emotionally effective ways to build an invalid argument.  Generally speaking, "informal logic" consists of "common sense" and a collection of other rather loose rules that people employ while making most decisions and even in debate. It is unstructured, and depends largely on one's view of "the reasonable". The thresholds of what is "reasonable" and what is not, are inexact and subject to change with the receipt of sufficient contrary evidence--and again, what constitutes "sufficiency" in this context might vary from person to person.
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== Logical reasoning ==
  
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== Deduction and Induction ==
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''See also:'' [[Logical reasoning]]
  
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Deductive logic is characterized by certainty: in a valid argument, when the premises are true, the conclusion '''must''' be true. The certainty, however, comes at a price. Take a classic argument of deductive logic:
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[[Logical reasoning]] is a form of [[rational thinking]] that focuses on drawing conclusions from information using [[structural thinking]] and a rigorous approach. It involves analyzing premises and assumptions to see if they sufficently support a conclusion and ensuring the conclusion is [[Reasonable person|reasonable based on the given evidence]]. It's a vital skill for [[Evidence-based thinking]], [[critical thinking]], [[decision making]], [[problem solving]], [[analytical thinking]] and [[systems thinking]] in various contexts, including [[law|legal]] reasoning.<ref>
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*[https://www.google.com/search?q=concepts+related+to+analytical+thinking&oq=concepts+related+to+analytical+thinking&gs_lcrp=EgZjaHJvbWUyBggAEEUYOTIICAEQABgWGB4yDQgCEAAYhgMYgAQYigUyDQgDEAAYhgMYgAQYigUyCggEEAAYgAQYogQyCggFEAAYgAQYogQyCggGEAAYgAQYogQyCggHEAAYogQYiQUyBggIEC4YQNIBCTExMzEwajBqMagCALACAA&sourceid=chrome&ie=UTF-8 Concepts related to analytical thinking]
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*[https://www.lsac.org/lsat/taking-lsat/test-format/logical-reasoning#:~:text=Each%20Logical%20Reasoning%20question%20requires,be%20central%20to%20legal%20reasoning. Logical reasoning], Law School Admission Council
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*[https://www.hipeople.io/glossary/logical-reasoning Logical reasoning]
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*[https://www.mentorink.com/blog/logical-reasoning/#:~:text=Logical%20reasoning%20can%20best%20be,reaching%20role%20in%20workplace%20performance. Logical reasoning], Mentor Link
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*[https://www.coursera.org/articles/analytical-thinking What Is Analytical Thinking and How Can You Improve It?], Coursera
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*[https://www.vaia.com/en-us/explanations/psychology/cognitive-psychology/systematic-thinking/#:~:text=Systematic%20thinking%20involves%20a%20structured%2C%20logical%2C%20and,on%20established%20procedures%20and%20clear%2C%20step%2Dby%2Dstep%20processes. Systematic thinking]
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</ref>
  
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'''Premises:'''
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== Informal Logic ==
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All men are mortal,
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Socrates is a man
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'''Conclusion:'''
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Socrates is mortal
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The conclusion is "contained" in the premises.  In a sense the conclusion is "known" before it is elucidated.  The conclusion, "Socrates is mortal" is also less informative than the premises which imply not just that Socrates is mortal but that a lot of other beings (anyone for whom the term "is a man" applies) are also mortal.  As a result deductive logic is not thought to add to knowledge, merely to clarify it.
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''See also:'' [[Rhetoric]]
  
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Inductive logic is ampliative, but is famously less certain.  In a good inductive argument, even when the premises are true, it is still possible for the conclusion to be false. A classic example of an inductive argument is:
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The art of [[rhetoric]] is sometimes called "informal logic", and the classic [[logical fallacies|fallacies]] are often described as "logical fallacies", though, strictly speaking, most of them are simply emotionally effective ways to build an invalid argument.  Generally speaking, "informal logic" consists of "common sense" and a collection of other rather loose rules that people employ while making most decisions and even in debate. It is unstructured, and depends largely on one's view of "the reasonable". The thresholds of what is "reasonable" and what is not are inexact and subject to change with the receipt of sufficient contrary evidence &ndash; and again, what constitutes "sufficiency" in this context might vary from person to person.
  
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'''Premise:'''
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== Deduction and Induction ==
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All the many swans obversed have been white
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Deductive logic is characterized by certainty: in a valid argument, when the premises are true, the conclusion '''must''' be true. Take a classic argument of deductive logic:
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'''Conclusion:'''
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All swans are white
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Before the discovery of Australia the premise was true someone following inductive logic would have reached the conclusion.  Given the existence of black swans in Australia, however, that conclusion is false. [[David Hume]]'s 18th-century critique of induction remains a very pressing problem for disciplines like [[science]] which are commonly held to rely on inductive reasoning.
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:'''P1:''' All men are mortal
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:'''P2:''' Socrates is a man.
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:'''C:''' Socrates is mortal.  
  
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Another example of faulty logic would be when it comes to atheism. Atheists typically use many logical fallacies in an effort to convert people to their religion.  
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The conclusion is "contained" in the premises. In this sense the conclusion can be said to "follow" from the premises. The conclusion, "Socrates is mortal", is also less informative than the premises, which imply not just that Socrates is mortal but that all men are also mortal.  As a result, deductive logic is not thought to add to knowledge, merely to clarify it.  
  
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'''Premise:'''
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Inductive logic is ampliative, but is famously less certain.  In a strong inductive argument, even when the premises are true, it is still possible for the conclusion to be false. An example of an inductive argument is:
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There is no emperical evidence for something.
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'''Conclusion:'''
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The object does not exist.
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This logical fallacy acts as if absence of evidence is evidence of absence. This can be proven to be a faulty supposition because before America was discovered by Europeans, one could say there are only four continents based upon the evidence. This would be treated as a fact despite it being proven false later on. Also, this premise ignores the fact that there are multiple types of knowledge. There is empirical knowledge and then there is knowledge one has through faith. An example would be the knowledge that a mother loves her son. This cannot be proven to be a lie, but people have faith that is it true.
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:'''P1:''' The first student is wearing red
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:'''P2:''' The second student is wearing red
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:'''P(n):''' The ''n''th student is wearing red
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:'''Conclusion:''' ''All'' students are wearing red
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Given the amount of evidence, it is reasonable to inductively conclude that all students are wearing red; however, one's senses could fail or a student could be hiding who is wearing blue. Given this, the conclusion is never certainly true and can only be highly probably true. The event of moving from "''n'' examples are this way" to a universal statement that "''all'' examples are this way" is commonly known as the problem of induction.
  
 
== Uses of logic in other disciplines ==
 
== Uses of logic in other disciplines ==
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Logic is a necessary discipline in [[philosophy]], because it deals with how we study and interact with the world and with other people in it. Logic and [[mathematics]] are also closely connected, and much of mathematics can be reduced to first-order logic, though Goedel's [[Incompleteness theorem]] shows that not all of mathematics can be so reduced.
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Logic is a necessary discipline in [[philosophy]], because it deals with how we study and interact with propositions. Logic and [[mathematics]] are also closely connected, and much of mathematics can be reduced to first-order logic, though [[Godel's Incompleteness Theorems|Gödel's Incompleteness Theorems]] shows that not all of mathematics can be so reduced.
  
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In [[computer science]], logic dictates how a machine will follow a set of instructions, including how to test its "world," evaluate it, and act according to that evaluation. Every computer language includes its own version of the language of symbolic logic, except that instead of establishing propositions, a computer following a program is usually choosing between and among different commands to execute.
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*In [[computer science]], logic dictates how a machine will follow a set of instructions, including how to test its "world," evaluate it, and act according to that evaluation, except that instead of establishing propositions, a computer following a program is usually choosing between and among different commands to execute.
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*Logic is also used extensively in [[theology]], and especially the study of the [[Bible]]
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*Logic, and especially formal logic, informs the discipline of [[critical thinking]] &ndash; which, by no coincidence, takes its name from the Greek word for a judge. Indeed, judges and juries in courts of [[law]] must apply logic to evaluate the arguments and evidence and arrive at their decisions.
  
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Logic is also used extensively in [[theology]], and especially the study of the [[Bible]], which, due to its divine inspiration, is very logical.  
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== Propositional logic ==
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Propositional logic is the study of logical propositions. It is sometimes called ''zeroth''-order logic because it does not deal with subjects or predicates but is specifically the study of the truth values of logical propositions and the logical connectives used to make compound propositions. [[Lewis Carroll]] wrote a book on it for young students.
  
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Logic, and especially formal logic, inform the discipline of [[critical thinking]]--which, by no coincidence, takes its name from the [[Greek]] word for a judge. Indeed, judges and juries in courts of [[law]] must apply logic, both formal and informal, to arrive at their decisions. Formal logic will usually serve to state what obedience to a given body of law requires; informal logic must usually serve a trier of fact charged with deciding whether a given person was in obedience or in violation. The latter principle holds primarily because plaintiff and defendant in a court of law quite often ''do not'' agree on matters of fact.
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== Logical arguments for the existence of God ==
  
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See: [[Logical arguments for the existence of God]]
  
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== See Also ==
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== Recommended reading ==
 
* [[Aristotle]], "Organon"
 
* [[Aristotle]], "Organon"
 
* [[Gottlob Frege]], "Begriffsschrift" and "The Foundations of Arithmetic"
 
* [[Gottlob Frege]], "Begriffsschrift" and "The Foundations of Arithmetic"
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* [[Bertrand Russell]] and [[Albert N Whitehead]], "Principia Mathematica"
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* [[Bertrand Russell]] and Albert N Whitehead, "Principia Mathematica"
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* [[Bas van Fraassen]] and [[J C Beall]], "Possibility and Paradox"
+
* Bas van Fraassen and J C Beall, "Possibility and Paradox"
 
* Geoffrey Hunter, "Metalogic"
 
* Geoffrey Hunter, "Metalogic"
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* Patrick Hurley, "A Concise Introduction to Logic, 12th Edition"
 +
 +
== See also ==
 +
 +
*[[Evidence-based thinking]]
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*[[Graham's hierarchy of disagreement]]
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*[[Atheism and logic]]
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*[[Christianity and logic]]
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*[[Specious reasoning]]
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*[[Abstract thinking]]
  
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==References==
 +
{{reflist|2}}
 
[[Category:Philosophy]]
 
[[Category:Philosophy]]
 +
[[Category:Conservatism]]

Latest revision as of 12:41, December 2, 2025

Aristotle is often called the father of logic in the West.

Logic (Greek λογίζω – I think, I reason; from λόγος – reason) refers the patterns in reasoning behind arguments. In philosophy, logic is a sub-branch of epistemology that deals with and attempts to guide the faculty of human reason. It is often studied alongside mathematics.

Encyclopedia Britannica declares:

"Laws of thought, traditionally, the three fundamental laws of logic: (1) the law of contradiction, (2) the law of excluded middle (or third), and (3) the principle of identity. That is, (1) for all propositions p, it is impossible for both p and not p to be true, or symbolically ∼(p · ∼p), in which ∼ means “not” and · means “and”; (2) either p or ∼p must be true, there being no third or middle true proposition between them, or symbolically p ∨ ∼p, in which ∨ means “or”; and (3) if a propositional function F is true of an individual variable x, then F is indeed true of x, or symbolically F(x) ⊃ F(x), in which ⊃ means “formally implies.” Another formulation of the principle of identity asserts that a thing is identical with itself, or (∀x) (x = x), in which ∀ means “for every”; or simply that x is x."[1]

Aristotle was the first to formalize the practice of reasoning. In particular, Aristotle developed a formalization of the categorical syllogism, the three laws of logic, and the Square of Opposition. While modus ponens and its complement modus tollens were known to the medieval philosophers, it is not clear when these central laws of deduction were first formalized.

The next easily identifiable major development in logic comes from Boole in 1847 with the creation of Boolean algebra and Boolean logic. In the 1870s, Peirce introduces a logic of quantifiers, followed by Gottlob Frege's 1879 Begriffsschrift, which contains the first complete formalization of the propositional calculus. Frege's work was developed in service of his logicist project, which was shown to be inconsistent by Bertrand Russell in 1903 with what is famously called Russell's paradox. In 1913 Russell and Whitehead published Principia Mathematica, a landmark work that sought to define mathematics with first-order logic.

By the 1950s, with the work of many logicians including Hilbert, Emile Post, Alfred Tarski and Kurt Gödel, most of the major results in first-order logic had been proved, and in the 1960s Saul Kripke added a completeness proof for modal logic.

Most logical systems are bivalent; that is, they admit only two truth-values, true or false. However, there is a fair amount of work done on trivalent systems of logic, specifically paraconsistent logic and relevance logic (a subset of paraconsistent logic). These logics were developed in response to the paradoxes that classical logic can create, namely, that anything can follow from a falsehood and the principle of explosion.

Formal logic requires, since Frege, a distinction between the object language and the metalanguage. The metalanguage is ordinarily a natural language like English or German, while the object language is a symbolic language with an alphabet limited to variable meanings and a symbolic system for describing the relationships between the variables.

Logical reasoning

See also: Logical reasoning

Logical reasoning is a form of rational thinking that focuses on drawing conclusions from information using structural thinking and a rigorous approach. It involves analyzing premises and assumptions to see if they sufficently support a conclusion and ensuring the conclusion is reasonable based on the given evidence. It's a vital skill for Evidence-based thinking, critical thinking, decision making, problem solving, analytical thinking and systems thinking in various contexts, including legal reasoning.[2]

Informal Logic

See also: Rhetoric

The art of rhetoric is sometimes called "informal logic", and the classic fallacies are often described as "logical fallacies", though, strictly speaking, most of them are simply emotionally effective ways to build an invalid argument. Generally speaking, "informal logic" consists of "common sense" and a collection of other rather loose rules that people employ while making most decisions and even in debate. It is unstructured, and depends largely on one's view of "the reasonable". The thresholds of what is "reasonable" and what is not are inexact and subject to change with the receipt of sufficient contrary evidence – and again, what constitutes "sufficiency" in this context might vary from person to person.

Deduction and Induction

Deductive logic is characterized by certainty: in a valid argument, when the premises are true, the conclusion must be true. Take a classic argument of deductive logic:

P1: All men are mortal
P2: Socrates is a man.
C: Socrates is mortal.

The conclusion is "contained" in the premises. In this sense the conclusion can be said to "follow" from the premises. The conclusion, "Socrates is mortal", is also less informative than the premises, which imply not just that Socrates is mortal but that all men are also mortal. As a result, deductive logic is not thought to add to knowledge, merely to clarify it.

Inductive logic is ampliative, but is famously less certain. In a strong inductive argument, even when the premises are true, it is still possible for the conclusion to be false. An example of an inductive argument is:

P1: The first student is wearing red
P2: The second student is wearing red
P(n): The nth student is wearing red
Conclusion: All students are wearing red

Given the amount of evidence, it is reasonable to inductively conclude that all students are wearing red; however, one's senses could fail or a student could be hiding who is wearing blue. Given this, the conclusion is never certainly true and can only be highly probably true. The event of moving from "n examples are this way" to a universal statement that "all examples are this way" is commonly known as the problem of induction.

Uses of logic in other disciplines

Logic is a necessary discipline in philosophy, because it deals with how we study and interact with propositions. Logic and mathematics are also closely connected, and much of mathematics can be reduced to first-order logic, though Gödel's Incompleteness Theorems shows that not all of mathematics can be so reduced.

  • In computer science, logic dictates how a machine will follow a set of instructions, including how to test its "world," evaluate it, and act according to that evaluation, except that instead of establishing propositions, a computer following a program is usually choosing between and among different commands to execute.
  • Logic is also used extensively in theology, and especially the study of the Bible
  • Logic, and especially formal logic, informs the discipline of critical thinking – which, by no coincidence, takes its name from the Greek word for a judge. Indeed, judges and juries in courts of law must apply logic to evaluate the arguments and evidence and arrive at their decisions.

Propositional logic

Propositional logic is the study of logical propositions. It is sometimes called zeroth-order logic because it does not deal with subjects or predicates but is specifically the study of the truth values of logical propositions and the logical connectives used to make compound propositions. Lewis Carroll wrote a book on it for young students.

Logical arguments for the existence of God

See: Logical arguments for the existence of God

Recommended reading

  • Aristotle, "Organon"
  • Gottlob Frege, "Begriffsschrift" and "The Foundations of Arithmetic"
  • Bertrand Russell and Albert N Whitehead, "Principia Mathematica"
  • Bas van Fraassen and J C Beall, "Possibility and Paradox"
  • Geoffrey Hunter, "Metalogic"
  • Patrick Hurley, "A Concise Introduction to Logic, 12th Edition"

See also

References

  1. ↑ Laws of Thought, Encyclopedia Britannica
  2. ↑
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