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History of logic | A Wisdom Archive on History of logic | | History of logic A selection of articles related to History of logic | |
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History of logic
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| ARTICLES RELATED TO History of logic | | | | | |  History of logic: Encyclopedia II - Proof theory - Kinds of proof calculusThe three most well known proof calculi are:
The Hilbert-style calculi
The natural deduction calculus
The sequent calculus
To say these are proof calculi, rather than proof systems, is to say they are flexible frameworks for the study of many kinds of logical consequence relations. Each of these can formalise propositional or predicate logics of either the classical or intuitionistic flavour, or almost any modal logic studied, many substructural logics, such as relevance logic or lin ...
See also:Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography Read more here: » Proof theory: Encyclopedia II - Proof theory - Kinds of proof calculus |
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| | | | | History of logic: Encyclopedia II - Kripke semantics - Semantics of intuitionistic logic Kripke semantics for the intuitionistic logic follows the same principles as the semantics of modal logic, but it uses a different definition of satisfaction.
An intuitionistic Kripke model is a triple <W,≤,>, where <W,≤> is a transitive and reflexive Kripke frame (i.e., the accessibility relation is a preorder), and satisfies the following conditions:
if p is a propositional variable, w ≤ u, and w p, then u p ( ...
See also:Kripke semantics, Kripke semantics - Semantics of modal logic, Kripke semantics - Basic definitions, Kripke semantics - Correspondence and completeness, Kripke semantics - Canonical models, Kripke semantics - Finite model property, Kripke semantics - Polymodal logics, Kripke semantics - Semantics of intuitionistic logic, Kripke semantics - Intuitionistic first-order logic, Kripke semantics - Kripke-Joyal semantics, Kripke semantics - Model constructions, Kripke semantics - History and terminology Read more here: » Kripke semantics: Encyclopedia II - Kripke semantics - Semantics of intuitionistic logic |
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| | | | | History of logic: Encyclopedia II - Epimenides paradox - Logical analysis If we define "liar" to mean that every statement made by a liar is false (so that Epimenides' statement amounts to "Anything said by a Cretan is false"), then the statement "All Cretans are liars," if uttered by the Cretan Epimenides, cannot be consistently true. (And, as will be noted below, according to one interpretation it also cannot be consistently false, either.)
The conjunction of "Epimenides said all Cretans are liars" and "Epimenides is a Cretan" would, if true, imply that a Cretan has truthfully asserted that no Cretan has ...
See also:Epimenides paradox, Epimenides paradox - Logical analysis, Epimenides paradox - History, Epimenides paradox - Sources Read more here: » Epimenides paradox: Encyclopedia II - Epimenides paradox - Logical analysis |
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| | | | | | | History of logic: Encyclopedia II - Kripke semantics - Model constructions As in the classical model theory, there are methods for constructing a new Kripke model from other models.
The natural homomorphisms in Kripke semantics are called p-morphisms (which is short for pseudo-epimorphism, but the latter term is rarely used). A p-morphism of Kripke frames <W,R> and <W’,R’> is a mapping f:W → W’ such that
f preserves the accessibility relation, i.e., u R v implies f(u) R’ ...
See also:Kripke semantics, Kripke semantics - Semantics of modal logic, Kripke semantics - Basic definitions, Kripke semantics - Correspondence and completeness, Kripke semantics - Canonical models, Kripke semantics - Finite model property, Kripke semantics - Polymodal logics, Kripke semantics - Semantics of intuitionistic logic, Kripke semantics - Intuitionistic first-order logic, Kripke semantics - Kripke-Joyal semantics, Kripke semantics - Model constructions, Kripke semantics - History and terminology Read more here: » Kripke semantics: Encyclopedia II - Kripke semantics - Model constructions |
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| | | | | History of logic: Encyclopedia II - Kripke semantics - History and terminology Kripke semantics does not originate with Kripke, but instead the idea of giving semantics in the style given above, that is based on valuations made that are relative to nodes, predates Kripke by a long margin:
Carnap seems to have been the first to have the idea that one can give a possible world semantics for the modalities of necessity and possibility by means of giving the valuation function a parameter that ranges over Leibnizian possible worlds. Bayart develops this idea further, but neither gave recursive definitio ...
See also:Kripke semantics, Kripke semantics - Semantics of modal logic, Kripke semantics - Basic definitions, Kripke semantics - Correspondence and completeness, Kripke semantics - Canonical models, Kripke semantics - Finite model property, Kripke semantics - Polymodal logics, Kripke semantics - Semantics of intuitionistic logic, Kripke semantics - Intuitionistic first-order logic, Kripke semantics - Kripke-Joyal semantics, Kripke semantics - Model constructions, Kripke semantics - History and terminology Read more here: » Kripke semantics: Encyclopedia II - Kripke semantics - History and terminology |
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| | | | | | | | History of logic: Encyclopedia II - Proof theory - History Although the formalisation of logic was much advanced by the work of such figures as Gottlob Frege, Peano, Russell and Dedekind, conventionally the story of modern proof theory is seen as being established by David Hilbert, who initiated what is called Hilbert's program in the Foundations of mathematics. Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem seemed to bring Hilbert's problem of reducing all mathematics to a finitist formal system, then his incompleteness theorems showed that was unattainable. All of this work was carried out with the pr ...
See also:Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography Read more here: » Proof theory: Encyclopedia II - Proof theory - History |
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| | | | | | | History of logic: Encyclopedia II - Proof theory - Formal and informal proof However, the proofs used in everyday mathematical practice are almost never like the formal proofs in proof theory. They are rather like high-level sketches that would allow an expert to reconstruct a formal proof at least in principle, and given enough time and patience. For most mathematicians, writing a fully formal proof would have all the drawbacks of programming in machine code.
Formal proofs are constructed, with the help of computers, in automated theorem proving. Significantly, these proofs can be checked automatically by com ...
See also:Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography Read more here: » Proof theory: Encyclopedia II - Proof theory - Formal and informal proof |
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| | | | | History of logic: Encyclopedia II - Proof theory - Consistency proofs Main article: Consistency proof
As we have discussed, the spur for the mathematical investigation of proofs in formal theories was Hilbert's program. The central idea of this program was that if we could give finitary proofs of consistency for all the sophisticated formal theories needed by mathematicians, then we could ground these theories by means of a metamathematical argument, which shows that all of their purely universal assertions (more technically their provable Pi-0-1 sentences) are finitarily true; once so grounded w ...
See also:Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography Read more here: » Proof theory: Encyclopedia II - Proof theory - Consistency proofs |
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| | | | | History of logic: Encyclopedia II - Proof theory - Structural proof theory Main article: Structural proof theory
Structural proof theory is the subdiscipline of proof theory that studies proof calculi that support a notion of analytic proof. The notion of analytic proof was introduced by Gentzen for the sequent calculus; there the analytic proofs are those that are cut-free. His natural deduction calculus also supports a notion of analytic proof, as shown by Dag Prawitz; the definition is slightly more complex, we say the analytic proofs are the normal forms, which are related to the notion of normal form in term rewriting. More exotic proof calculi such as Jean-Yves Gira ...
See also:Proof theory, Proof theory - History, Proof theory - Formal and informal proof, Proof theory - Kinds of proof calculus, Proof theory - Consistency proofs, Proof theory - Structural proof theory, Proof theory - Tableau systems, Proof theory - Ordinal analysis, Proof theory - Substructural logics, Proof theory - Selected bibliography Read more here: » Proof theory: Encyclopedia II - Proof theory - Structural proof theory |
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| | | | | History of logic: Encyclopedia II - Ordinal fraction - Data modelling Logical concepts are modelled mathematically by sets or by conditions. Concatenation of two concepts is modelled by the intersection of the two sets or as the logical conjunction of the two conditions. The set of good girls is the intersection between the set of good persons and the set of girls. The condition (X is a good girl) is the conjunction: (X is good) AND (X is a girl).
Any concept has alternatives. An alternative to 'girl' may be 'boy', and an alternative to 'good' may be 'bad'. So the following example model this conceptual miniuniverse:
01 girl
02 boy
10 ...
See also:Ordinal fraction, Ordinal fraction - Positional notation, Ordinal fraction - Logical conditions, Ordinal fraction - Ordinal fraction arithmetic, Ordinal fraction - Addition, Ordinal fraction - Comparison, Ordinal fraction - Subtraction, Ordinal fraction - Multiplication, Ordinal fraction - Division, Ordinal fraction - Algebraic rules, Ordinal fraction - Data modelling, Ordinal fraction - History, Ordinal fraction - Reference Read more here: » Ordinal fraction: Encyclopedia II - Ordinal fraction - Data modelling |
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| | | | | History of logic: Encyclopedia II - Church–Turing thesis - Success of the thesis Since that time, many other formalisms for describing effective computability have been proposed, including recursive functions, the lambda calculus, register machines, Post systems, combinatory logic, and Markov algorithms. All these systems have been shown to compute the same functions as Turing machines; systems like this are called Turing-complete. Because all these different attempts of formalizing the concept of algorithm have yielded equivalent results, it is now generally assumed that the Church–Turing thesis is correct. However, t ...
See also:Church–Turing thesis, Church–Turing thesis - Church–Turing thesis, Church–Turing thesis - History, Church–Turing thesis - Success of the thesis, Church–Turing thesis - Philosophical implications, Church–Turing thesis - Reference Read more here: » Church–Turing thesis: Encyclopedia II - Church–Turing thesis - Success of the thesis |
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| | | | | | History of logic: Encyclopedia II - Karnaugh map - Race hazards Karnaugh maps are useful for detecting and eliminating race hazards.
In the above example, a potential race condition exists when C and D are both 0, A is a 1, and B changes from a 0 to a 1 (moving from the green state to the blue state).
For this case, the output is defined to remain unchanged at 1, but because this transition is not covered by a specific term in the equation, a potential for a glitch (a momentary transition of the output to 0) exists. This is very easy to spot using a Karnaugh map, because a race condition may exist when moving between any pair of adjacent, but d ...
See also:Karnaugh map, Karnaugh map - History and nomenclature, Karnaugh map - Usage in boolean logic, Karnaugh map - Example, Karnaugh map - Race hazards, Karnaugh map - When not to use K-maps Read more here: » Karnaugh map: Encyclopedia II - Karnaugh map - Race hazards |
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