Contents 1 Concepts 1.1 Logical form
1.2 Semantics
1.3 Inference
1.4 Logical systems
1.5
2 History 3 Types 3.1 Syllogistic logic 3.2 Propositional logic 3.3 Predicate logic 3.4 Modal logic 3.5 Informal reasoning and dialectic 3.6 Mathematical logic 3.7 Philosophical logic 3.8 Computational logic 3.9 Non-classical logic 4 Controversies 4.1 "Is
5 See also 6 Notes and references 7 Bibliography 8 External links Concepts[edit] “ Upon this first, and in one sense this sole, rule of reason, that in order to learn you must desire to learn, and in so desiring not be satisfied with what you already incline to capably think, there follows one corollary which itself deserves to be inscribed upon every wall of the city of philosophy: Do not block the way of inquiry. ” — Charles Sanders Peirce, "First Rule of Logic" The concept of logical form is central to logic. The validity of an argument is determined by its logical form, not by its content. Traditional Aristotelian syllogistic logic and modern symbolic logic are examples of formal logic.
Logical form[edit]
Main article: Logical form
∀ x ( P ( x ) → Q ( x ) ) displaystyle forall x(P(x)rightarrow Q(x)) in predicate logic, involving the logical connectives for universal quantification and implication rather than just the predicate letter A and using variable arguments P ( x ) displaystyle P(x) where traditional logic uses just the term letter P. With the complexity comes power, and the advent of the predicate calculus inaugurated revolutionary growth of the subject. Semantics[edit]
Main article:
Inference[edit]
Logical systems[edit] Main article: Formal system A formal system is an organization of terms used for the analysis of deduction. It consists of an alphabet, a language over the alphabet to construct sentences, and a rule for deriving sentences. Among the important properties that logical systems can have are: Consistency, which means that no theorem of the system contradicts
another.[10]
Validity, which means that the system's rules of proof never allow a
false inference from true premises.
Completeness, which means that if a formula is true, it can be proven,
i.e. is a theorem of the system.
Soundness, meaning that if any formula is a theorem of the system, it
is true. This is the converse of completeness. (Note that in a
distinct philosophical use of the term, an argument is sound when it
is both valid and its premises are true).[11]
Expressivity, meaning what concepts can be expressed in the system.
Some logical systems do not have all four properties. As an example,
Kurt
a displaystyle a from an observed surprising circumstance b displaystyle b is to surmise that a displaystyle a may be true because then b displaystyle b would be a matter of course.[15] Thus, to abduce a displaystyle a from b displaystyle b involves determining that a displaystyle a is sufficient (or nearly sufficient), but not necessary, for b displaystyle b . While inductive and abductive inference are not part of logic proper, the methodology of logic has been applied to them with some degree of success. For example, the notion of deductive validity (where an inference is deductively valid if and only if there is no possible situation in which all the premises are true but the conclusion false) exists in an analogy to the notion of inductive validity, or "strength", where an inference is inductively strong if and only if its premises give some degree of probability to its conclusion. Whereas the notion of deductive validity can be rigorously stated for systems of formal logic in terms of the well-understood notions of semantics, inductive validity requires us to define a reliable generalization of some set of observations. The task of providing this definition may be approached in various ways, some less formal than others; some of these definitions may use logical association rule induction, while others may use mathematical models of probability such as decision trees. Rival conceptions[edit]
Main article: Conceptions of logic
History[edit]
Main article: History of logic
Aristotle, 384–322 BCE.
Types[edit]
Syllogistic logic[edit]
Main article: Aristotelian logic
A depiction from the 15th century of the square of opposition, which
expresses the fundamental dualities of syllogistic.
The
.mw-parser-output .templatequote overflow:hidden;margin:1em 0;padding:0 40px .mw-parser-output .templatequote .templatequotecite line-height:1.5em;text-align:left;padding-left:1.6em;margin-top:0 I was upset. I had always believed logic was a universal weapon, and now I realized how its validity depended on the way it was employed.[29] Propositional logic[edit] Main article: Propositional calculus A propositional calculus or logic (also a sentential calculus) is a formal system in which formulae representing propositions can be formed by combining atomic propositions using logical connectives, and in which a system of formal proof rules establishes certain formulae as "theorems". An example of a theorem of propositional logic is A → B → A displaystyle Arightarrow Brightarrow A , which says that if A holds, then B implies A. Predicate logic[edit]
Gottlob Frege's
∀ x . F ( x ) displaystyle forall x.F(x) is true.
Main article: Predicate logic
( ∃ x ) ( man ( x ) ∧ ( ∀ y ) ( man ( y ) → ( shaves ( x , y ) ↔ ¬ shaves ( y , y ) ) ) ) displaystyle (exists x)( text man (x)wedge (forall y)( text man (y)rightarrow ( text shaves (x,y)leftrightarrow neg text shaves (y,y)))) , using the non-logical predicate man ( x ) displaystyle text man (x) to indicate that x is a man, and the non-logical relation shaves ( x , y ) displaystyle text shaves (x,y) to indicate that x shaves y; all other symbols of the formulae are
logical, expressing the universal and existential quantifiers,
conjunction, implication, negation and biconditional.
Whilst Aristotelian syllogistic logic specifies a small number of
forms that the relevant part of the involved judgements may take,
predicate logic allows sentences to be analysed into subject and
argument in several additional ways—allowing predicate logic to
solve the problem of multiple generality that had perplexed medieval
logicians.
The development of predicate logic is usually attributed to Gottlob
Frege, who is also credited as one of the founders of analytical
philosophy, but the formulation of predicate logic most often used
today is the first-order logic presented in Principles of Mathematical
Modal logic[edit]
Main article: Modal logic
In languages, modality deals with the phenomenon that sub-parts of a
sentence may have their semantics modified by special verbs or modal
particles. For example, "We go to the games" can be modified to give
"We should go to the games", and "We can go to the games" and perhaps
"We will go to the games". More abstractly, we might say that modality
affects the circumstances in which we take an assertion to be
satisfied. Confusing modality is known as the modal fallacy.
Aristotle's logic is in large parts concerned with the theory of
non-modalized logic. Although, there are passages in his work, such as
the famous sea-battle argument in
Informal reasoning and dialectic[edit]
Main articles:
Mathematical logic[edit]
Main article: Mathematical logic
Philosophical logic[edit]
Main article: Philosophical logic
Computational logic[edit]
Main articles:
Section F.3 on Logics and meanings of programs and F.4 on Mathematical
logic and formal languages as part of the theory of computer science:
this work covers formal semantics of programming languages, as well as
work of formal methods such as Hoare logic;
Non-classical logic[edit]
Main article: Non-classical logic
The logics discussed above are all "bivalent" or "two-valued"; that
is, they are most naturally understood as dividing propositions into
true and false propositions. Non-classical logics are those systems
that reject various rules of Classical logic.
Controversies[edit]
"Is
Implication: Strict or material[edit]
Main article: Paradoxes of material implication
The notion of implication formalized in classical logic does not
comfortably translate into natural language by means of "if ...
then ...", due to a number of problems called the paradoxes of
material implication.
The first class of paradoxes involves counterfactuals, such as If the
moon is made of green cheese, then 2+2=5, which are puzzling because
natural language does not support the principle of explosion.
Eliminating this class of paradoxes was the reason for C.I. Lewis's
formulation of strict implication, which eventually led to more
radically revisionist logics such as relevance logic.
The second class of paradoxes involves redundant premises, falsely
suggesting that we know the succedent because of the antecedent: thus
"if that man gets elected, granny will die" is materially true since
granny is mortal, regardless of the man's election prospects. Such
sentences violate the
Tolerating the impossible[edit]
Main article: Paraconsistent logic
Rejection of logical truth[edit]
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, typically 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.
See also[edit]
Book: Logic Digital electronics – Electronic circuits that utilize
digital signals (also known as digital logic or logic gates)
Fallacies
List of logicians
List of logic journals
List of logic symbols
Notes and references[edit] ^ "possessed of reason, intellectual, dialectical, argumentative", also related to λόγος (logos), "word, thought, idea, argument, account, reason, or principle" (Liddell & Scott 1999; Online Etymology Dictionary 2001). ^ Due to Frege, see the
^
^ a b c Whitehead, Alfred North; Russell, Bertrand (1967). Principia Mathematica to *56. Cambridge University Press. ISBN 978-0-521-62606-4. ^ a b For a more modern treatment, see Hamilton, A.G. (1980). Logic for Mathematicians. Cambridge University Press. ISBN 978-0-521-29291-7. ^ Łukasiewicz, Jan (1957). Aristotle's syllogistic from the standpoint of modern formal logic (2nd ed.). Oxford University Press. p. 7. ISBN 978-0-19-824144-7. ^ Summa Logicae Part II c.4 transl. as Ockam's Theory of Propositions, A. Freddoso and H. Schuurman, St Augustine's Press 1998, p. 96 ^ Arnauld,
^ Locke, 1690. An Essay Concerning Human Understanding, IV. v. 1–8) ^ Bergmann, Merrie; Moor, James; Nelson, Jack (2009). The
^ Internet Encyclopedia of Philosophy, Validity and Soundness ^ Mendelson, Elliott (1964). "Quantification Theory: Completeness Theorems". Introduction to Mathematical Logic. Van Nostrand. ISBN 978-0-412-80830-2. ^ On abductive reasoning, see: Magnani, L. Abduction, Reason, and Science: Processes of Discovery and Explanation. Kluwer Academic Plenum Publishers, New York, 2001. xvii. 205 pages. Hardcover, ISBN 0-306-46514-0. R. Josephson, J. & G. Josephson, S. Abductive Inference: Computation, Philosophy, Technology. Cambridge University Press, New York & Cambridge. viii. 306 pages. Hardcover (1994), ISBN 0-521-43461-0, Paperback (1996), ISBN 0-521-57545-1. Bunt, H. & Black, W. Abduction, Belief and Context in Dialogue: Studies in Computational Pragmatics. (Natural Language Processing, 1.) John Benjamins, Amsterdam & Philadelphia, 2000. vi. 471 pages. Hardcover, ISBN 90-272-4983-0, 1-55619-794-2 ^ See Abduction and Retroduction at Commens Dictionary of Peirce's
Terms, and see Peirce's papers:
"On the
^ Peirce, C.S. (1903), Harvard lectures on pragmatism, Collected Papers v. 5, paragraphs 188–189. ^ Hofweber, T. (2004). "
^ Brandom, Robert (2000). Articulating Reasons. Cambridge, MA: Harvard University Press. ISBN 978-0-674-00158-9. ^ E.g., Kline (1972, p. 53) wrote "A major achievement of Aristotle was the founding of the science of logic". ^ "Aristotle", MTU Department of Chemistry. ^ Jonathan Lear (1986). "
^ Simo Knuuttila (1981). "Reforging the great chain of being: studies of the history of modal theories". Springer Science & Business. p. 71. ISBN 90-277-1125-9 ^ Michael Fisher, Dov M. Gabbay, Lluís Vila (2005). "Handbook of temporal reasoning in artificial intelligence". Elsevier. p. 119. ISBN 0-444-51493-7 ^ Harold Joseph Berman (1983). "
^ The four Catuṣkoṭi logical divisions are formally very close to
the four opposed propositions of the Greek tetralemma, which in turn
are analogous to the four truth values of modern relevance logic Cf.
Belnap (1977); Jayatilleke, K.N., (1967, The logic of four
alternatives, in
^ S.C. Vidyabhusana (1971). A History of Indian Logic: Ancient, Mediaeval, and Modern Schools, pp. 17–21. ^ Kisor Kumar Chakrabarti (June 1976). "Some Comparisons Between
Frege's
^ Jonardon Ganeri (2001). Indian logic: a reader. Routledge. pp. vii, 5, 7. ISBN 978-0-7007-1306-6. ^ "Aristotle". Encyclopædia Britannica. ^ Eco, Umberto (1980). The
^ "History of logic: Arabic logic". Encyclopædia Britannica. ^ Rescher, Nicholas (1978). "Dialectics: A Controversy-Oriented Approach to the Theory of Knowledge". Informal Logic. 1 (#3). ^ Hetherington, Stephen (2006). "Nicholas Rescher: Philosophical Dialectics". Notre Dame Philosophical Reviews (2006.07.16). ^ Rescher, Nicholas (2009). Jacquette, Dale (ed.). Reason, Method,
and Value: A Reader on the
^ Stolyar, Abram A. (1983). Introduction to Elementary Mathematical Logic. Dover Publications. p. 3. ISBN 978-0-486-64561-2. ^ Barnes, Jonathan (1995). The Cambridge Companion to Aristotle. Cambridge University Press. p. 27. ISBN 978-0-521-42294-9. ^
^ Mendelson, Elliott (1964). "Formal Number Theory: Gödel's Incompleteness Theorem". Introduction to Mathematical Logic. Monterey, Calif.: Wadsworth & Brooks/Cole Advanced Books & Software. OCLC 13580200. ^ Barwise (1982) divides the subject of mathematical logic into model theory, proof theory, set theory and recursion theory. ^ Brookshear, J. Glenn (1989). "Computability: Foundations of Recursive Function Theory". Theory of computation: formal languages, automata, and complexity. Redwood City, Calif.: Benjamin/Cummings Pub. Co. ISBN 978-0-8053-0143-4. ^ Brookshear, J. Glenn (1989). "Complexity". Theory of computation: formal languages, automata, and complexity. Redwood City, Calif.: Benjamin/Cummings Pub. Co. ISBN 978-0-8053-0143-4. ^ Goldman, Alvin I. (1986),
^ Demetriou, A.; Efklides, A., eds. (1994), Intelligence, Mind, and Reasoning: Structure and Development, Advances in Psychology, 106, Elsevier, p. 194, ISBN 978-0-08-086760-1 ^ Hegel, G.W.F (1971) [1817].
^ Joseph E. Brenner (3 August 2008).
^ Zegarelli, Mark (2010),
^ Hájek, Petr (2006). "Fuzzy Logic". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. ^ Putnam, H. (1969). "Is
^ Birkhoff, G.; von Neumann, J. (1936). "The
^ Dummett, M. (1978). "Is
^ Priest, Graham (2008). "Dialetheism". In Zalta, Edward N. (ed.). Stanford Encyclopedia of Philosophy. ^ Nietzsche, 1873, On
^ Nietzsche, 1882, The Gay Science. ^ Nietzsche, 1878, Human, All Too Human ^ Babette Babich, Habermas, Nietzsche, and Critical Theory ^ Georg Lukács. "The Destruction of
^ Russell, Bertrand (1945), A History of Western
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