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 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.
However, agreement on what logic is has remained elusive, and although
the field of universal logic has studied the common structure of
logics, in 2007 Mossakowski et al. commented that "it is embarrassing
that there is no widely acceptable formal definition of 'a logic'".[6]
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:
Consistency, which means that no theorem of the system contradicts another.[11] 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).[12] Some logical systems do not have all four properties. As an example,
Kurt
This section may be confusing or unclear to readers. Please help us clarify the section. There might be a discussion about this on the talk page. (May 2016) (Learn how and when to remove this template message) As the study of argument is of clear importance to the reasons that we
hold things to be true, logic is of essential importance to
rationality. Here we have defined logic to be "the systematic study of
the form of arguments"; the reasoning behind argument is of several
sorts, but only some of these arguments fall under the aegis of logic
proper.
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.[16] 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: Rival conceptions of logic
Aristotle, 384–322 BCE. In Europe, logic was first developed by Aristotle.[19] Aristotelian
logic became widely accepted in science and mathematics and remained
in wide use in the West until the early 19th century.[20]
Aristotle's system of logic was responsible for the introduction of
hypothetical syllogism,[21] temporal modal logic,[22][23] and
inductive logic,[24] as well as influential terms such as terms,
predicables, syllogisms and propositions. In
A depiction from the 15th century of the square of opposition, which expresses the fundamental dualities of syllogistic. Main article: Aristotelian logic
The
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.[30] 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
A simple toggling circuit is expressed using a logic gate and a synchronous register.
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;
Furthermore, computers can be used as tools for logicians. For
example, in symbolic logic and mathematical logic, proofs by humans
can be computer-assisted. Using automated theorem proving, the
machines can find and check proofs, as well as work with proofs too
lengthy to write out by hand.
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.
Book: Logic
List of mathematics articles Outline of mathematics Metalogic Outline of logic Philosophy List of philosophy topics Outline of philosophy Reason Truth Vector logic 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
Magnani, L. "Abduction, Reason, and Science: Processes of Discovery and Explanation". Kluwer Academic Plenum Publishers, New York, 2001. xvii. 205 pages. Hard cover, ISBN 0-306-46514-0. R. Josephson, J. & G. Josephson, S. "Abductive Inference: Computation, Philosophy, Technology" Cambridge University Press, New York & Cambridge (U.K.). viii. 306 pages. Hard cover (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. Hard cover, ISBN 90-272-4983-0, ISBN 1-58619-794-2 Parameter error in isbn : Invalid ISBN. ^ 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). "
Bibliography[edit] Barwise, J. (1982). Handbook of Mathematical Logic. Elsevier.
ISBN 9780080933641.
Belnap, N. (1977). "A useful four-valued logic". In Dunn &
Eppstein, Modern uses of multiple-valued logic. Reidel: Boston.
Bocheński, J. M. (1959). A précis of mathematical logic. Translated
from the French and German editions by Otto Bird. D. Reidel,
Dordrecht, South Holland.
Bocheński, J. M. (1970). A history of formal logic. 2nd Edition.
Translated and edited from the German edition by Ivo Thomas. Chelsea
Publishing, New York.
Brookshear, J. Glenn (1989). Theory of computation: formal languages,
automata, and complexity. Redwood City, Calif.: Benjamin/Cummings Pub.
Co. ISBN 0-8053-0143-7.
Cohen, R.S, and Wartofsky, M.W. (1974). Logical and Epistemological
Studies in Contemporary Physics. Boston Studies in the
External links[edit] Find more aboutLogicat's sister projects Definitions from Wiktionary Media from Wikimedia Commons News from Wikinews Quotations from Wikiquote Texts from Wikisource Textbooks from Wikibooks Learning resources from Wikiversity Library resources about Logic Resources in your library Resources in other libraries
An Introduction to Philosophical Logic, by Paul Newall, aimed at
beginners.
forall x: an introduction to formal logic, by P.D. Magnus, covers
sentential and quantified logic.
Nicholas Rescher. (1964). Introduction to Logic, St. Martin's Press. Essays "Symbolic Logic" and "The Game of Logic", Lewis Carroll, 1896. Math & Logic: The history of formal mathematical, logical, linguistic and methodological ideas. In The Dictionary of the History of Ideas. Online Tools Interactive Syllogistic Machine A web based syllogistic machine for exploring fallacies, figures, terms, and modes of syllogisms.
Translation Tips, by Peter Suber, for translating from English into
logical notation.
Reading lists The London
v t e Logic Outline History Fields
Foundations Abduction Analytic and synthetic propositions Antinomy A priori and a posteriori Deduction Definition Description Induction Inference Logical form Logical consequence Logical truth Name Necessity and sufficiency Meaning Paradox Possible world Presupposition Probability Reason Reference Semantics Statement Strict implication Substitution Syntax Truth Validity Lists topics Mathematical logic Boolean algebra Set theory other Logicians
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