Edit: I want to add the disclaimer that the Principle of Explosion might depend on some axiom that doesn't exist in some body of logic and that perhaps there is an interesting logic in which it doesn't hold.
I hope someone might comment on this post to let me know. Re-edit: Looks like there is a concept called paraconsistent logic which rejects the principle of explosion either by removing the law of excluded middle it seems or some other way.
So if you subscribe to a paraconsistent logic then you can have a universe in which paradoxes exist. There are several paradoxes that already exist. Sign up to join this community.
The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Can a "real" paradox exist? Ask Question. Asked 7 years ago. Active 4 years, 8 months ago. Viewed 3k times. Given a statement, S: "S is not true. Improve this question. Craig Craig 99 1 1 silver badge 4 4 bronze badges. First off, welcome to philosophy.
This is a well-formed question. It's also a pretty common example in the philosophy of language called the "Liar Paradox". I'm actually surprised to see it has not been asked and answered here before. This question is partially overlapping but might be a little hard to follow: philosophy. I feel like there's a very similar question here somewhere already maybe asked by MoziburUllah?
The paradox you mention is a strengthened form of the original liar 'S is false'. The strenghtened liar does not even need the assumption that every sentence is either true or false; it only requires that sentences are either true or non-true. And it's conclusion is also stronger: S is both true and not true. If, on the other hand, the barber goes to the barber, then he is cutting his own hair, in violation of the rule.
These two results present a contradiction. The contradiction is that the barber must cut his own hair and must not cut his own hair. This mind-bending little story is a prototypical example of a paradox. That is, it is a thought experiment where an idea is assumed and a contradiction is logically derived from it. A contradiction is a proposition that is, an assertion that something both is, and that it is not. We can also say that a contradiction is a proposition that is both true and false.
There are no contradictions in the physical world. In the physical universe, a thing cannot both be and not be; or, a proposition is either true or it is false. What would it possibly mean for something to be both true and false?
We cannot imagine the universe with a contradiction. Since contradictions are untenable, there must be something wrong with the assumption from which a contradiction is derived. In a sense, a paradox is a way of showing that a given assumption is not part of reason.
Philosophers, scientists, mathematicians, and logicians have all used paradoxes to demonstrate the validity or otherwise of assumptions, and so to demonstrate limitations to reason.
They assume an idea, and if a contradiction or falsehood is derived from it, then the assumption is wrong. Otherwise, the assumption is logically legitimate. Let us return to our little village and see how we can resolve the paradox.
What is the assumption here? In fact, these ideas are shadows of the true resolution to the barber paradox, which is to realize that the village described with its strict rule cannot exist. The physical universe would not permit the existence of such a village, because it implies a contradiction.
There are many isolated villages with one barber, but in them the rule can be violated in a myriad of ways: a barber from another city might visit, or the barber can be bald, or the barber asks his wife to cut his hair, or the barber cuts his own hair so the rule does not hold , etc. All these different scenarios could happen to ensure that there are no contradictions in the universe.
In short, the assumption of the barber paradox was that the village with this rule exists. This assumption is wrong. There is another paradox that every fan of science fiction knows well. If he is never born, he will never be able to go back in time and kill his bachelor grandfather; so, if he kills his bachelor grandfather, then he will not kill his bachelor grandfather; and if he does not kill his bachelor grandfather, then he can go back and kill his bachelor grandfather.
This again is a contradiction. Indeed, one does not need to be so homicidal to get such a result. All the time traveler has to do is go back in time two minutes before he gets into his time machine and make sure that his earlier self does not enter the machine.
Stopping his earlier self from entering the time machine to stop himself from entering the machine will ensure that he cannot stop himself from entering the machine. Again, a contradiction. Most actions affect other actions. A time traveler has the unique ability to perform an action which affects itself. A paradox comes about if a time traveler performs an action that negates itself. The time travel paradox is resolved with ease: the assumption that traveling backwards in time is possible is wrong; or even if backwards time travel were possible, still the time traveler would not be able to go back and perform an action that contradicts itself.
There is no way that the universe is going to let a time traveler kill his own bachelor grandfather. Those two paradoxes are resolved in the same way. We came to a contradiction by assuming that a certain physical object or process exists. Once we abandon this assumption, we are free of the contradiction. These paradoxes show limitations of the physical world: a certain village cannot exist, or the actions of a time traveler are restricted. There are many similar examples of such physical paradoxes, and they are resolved in the same way.
Contradictions do not exist in the physical universe. However there are places where contradictions do exist: in our minds and language.
This brings us to our second type of paradox. Unlike the universe, the human mind is not a perfect machine. It is full of contradictions, with conflicting desires and predictions. However, one could raise at least two objections against the theory on different grounds.
But impredicativity makes the construction of a model or of an interpretation more difficult and less evident.
First of all, he considers, as a concrete case study, the problem of characterizing the explicitly definable concepts of elementary plane geometry: these can be inductively generated by means of five basic definition principles from two suitably chosen primitive concepts e. More explicitly, one requires closure under the logical operations of negation, conjunction, existential quantification and suitable combinatorial operations of permutation and expansion.
Weyl addressed the problem of generating the admissible properties over a given domain a few years later in Das Kontinuum As in , the set of sets of natural numbers that are definable via admissible operations to which now also a form of iteration is added is denumerable. Weyl apparently followed a relativistic attitude, according to which the extension of the universe of sets and their properties depend on the operations which are accepted to construct sets see also the entry Hermann Weyl.
In the period until , the problem of paradoxes led naturally to and was subsumed under the investigation of logical calculi its final by-product being the Hilbert-Ackermann textbook of This in turn opened the way to the simplification of type theory, to important generalizations of the notion of set, and to an almost final axiomatic elaboration of set theory along the Zermelian route, but also following the new path opened up by Johann von Neumann. The basic logical tool is essentially axiomatic formal analysis.
Do circular objects exist in set theory? Once circular sets are allowed, a strengthening of extensional equality by means of a suitable isomorphism relation bisimulation, in current terminology is needed which essentially corresponds to the isomorphism of the trees picturing the given sets.
Indeed, a set of second kind always contains a set of second kind; hence a set of sets of first kind must be of first kind. For the history of paradoxes, it is important to emphasize that Mirimanoff a gave a generalization of the Burali-Forti antinomy, the paradox of grounded sets. He introduced the notion of ordinal rank for ordinary sets and he noticed that ordinary sets can be arranged in a cumulative hierarchy, indexed by their ranks.
However, the existence of a cumulative structure of ordinary sets is not considered as a ground for excluding extraordinary sets. Mirimanoff pp. This non-mathematical example is suggestive of later developments, i. In Mirimanoff a,b, one can also find the idea of von Neumann ordinal von Neumann , and a form of the replacement axiom is present.
There are two sorts of objects: objects of type II functions, corresponding to classes and objects of type I arguments , linked by the application operation of a function to its arguments. The two domains partly overlap and there are objects of type I—II, corresponding to sets as functions which can also be arguments.
According to Finsler, paradoxes hinge upon circular notions, but circularity does not necessarily lead to contradictions. For the contemporary reader, it is worth mentioning that an original intuition of Finsler b is the use of graph theory for representing circular structures.
This is especially clear from his unpublished lecture notes e. Paradoxes are derived by allowing a suitable form of unrestricted comprehension; the problematic assumption is located in the admissibility of predicates and propositions as objects , i. Variants of the traditional Liar and of the Berry antinomy are introduced.
Both authors rejected ramified type theory RTT, in short and the axiom of reducibility. Their work can be considered a typical outcome of the process that was to yield streamlined versions of logical formalisms. Insofar as paradoxes are concerned, the main problem is to show that RTT is not required for solving the paradoxes. On the other hand, Chwistek proposed a version of the Liar that can be reconstructed in the simple theory of types without the axiom of reducibility, once we are allowed to quantify over all propositions.
Ramsey introduced the by-now standard distinction between logical and epistemological contradictions but see already Peano , and section 3. Quantification over arbitrary types is legitimate and hence types are closed under impredicative comprehension, which is considered necessary for mathematics. Types are intrinsic to logical and mathematical objects and the logical paradoxes are exactly those which require type distinctions to be solved e.
In order to solve the semantical antinomies e. Whatever one we take there is still a way of constructing a symbol to mean in a way not included in our relation.
These ideas foreshadow those of Tarski. For an analysis of semantical antinomies in a ramified context, compare also the later contribution of Church , also reconsidered and criticized in Martino In addition, an effort was made to elaborate new grand logics as a reaction to the logic of Principia Mathematica. This line of thought gave impulse to the elaboration of syntactical methods within combinatory logic and to the rise of recursion theory. The diagnosis of the paradoxes was further enriched by a subtler analysis of purely logical features of paradoxical reasoning: this is especially true for negation and the crucial role of contraction and duplication properties built into the laws of standard implication.
However, self-referential constructions attained an adequate degree of mathematical rigor and became genuine mathematical tools only when non trivial number-theoretic techniques were put to work see the entry recursive functions , for instance in the analysis of syntactical substitution and in providing arithmetical models of formal provability the crucial role of substitution for producing contradictions was already noticed by Russell, although he did not publish this; see Pelham and Urquhart This is to be found in Carnap b, p.
As a matter of fact the lemma has become the standard tool for producing self-referential statements and for transforming the semantical paradoxes into indefinability and formal undecidability results see the entry on self-reference. It is also important to stress that a few years later an analog of the diagonalization lemma the so-called second recursion theorem was discovered by Kleene and was soon to become a basic tool in the foundations of recursion theory and computability theory.
It is evident from the work done in the twenties surveyed above that the problem of finding a formal solution to the semantical paradoxes, such as the Liar and the Richard paradox, remained essentially open. Type-theoretic solutions had not been pursued to the extent of providing a systematic formal analysis of semantical notions like truth or definability. But why would this problem be worth studying from a logical and mathematical point of view? First of all, the analysis of the Liar paradox starts out by specifying a formal requirement to be met in the semantical investigation of truth, i.
This amounts to the celebrated schema T , which can be roughly stated in simplified form as:. The result that Tarski draws from the Liar is that there cannot be any interpreted language which is free from contradictions, obeys the classical laws of logic, and meets the requirements I — III , where. Given these essential obstacles, Tarski provides a structural definition of the basic semantical notions, i.
But this route is only viable for a language which is structurally described, e. For such languages, which are usually closed under quantification and contain formulas with free variables, Tarski elaborates an appropriate notion of satisfaction , which allows him to introduce the notions of definability, denotation, truth, logical consequence.
It is then possible to give a precise version and a proof of the adequacy condition T in a meta-science, whose principles comprise: i general logical axioms, ii special axioms that depend upon the object theory we consider, and iii axioms for dealing with the fundamental properties of the structural notions, i. Given this semantical machinery, Tarski can solve in the negative the problem of the existence of a formal counterpart of a universal language, i.
On the positive side, the concept of truth can be adequately defined for any formalized language L in a language the so-called metalanguage , provided it is of higher order than L. In the twenties and in the early thirties, the orthodox view of logic among mathematical logicians was more or less type- or set theoretic. However, there was an effort to develop new grand logics as substitutes for the logic of Principia Mathematica.
If one looks closely at the development of these systems, one can see that paradoxical constructions have become essential tools for defining objects and proving non-trivial logical mathematical facts. The formal system consists of standard equations on combinators e. Combinatory logic is a theory which analyzes the modes of combinations of formal objects, substitution, and the notions of proposition and propositional function see the entry on combinatory logic for a proper introduction to variants of the formalism and an overview of the properties of related calculi.
For Curry, the root of the paradoxes is found in assuming that combinations of concepts are always propositions. The notion of proposition becomes a theoretical concept, which is decided by the theory. Types are not assigned to the expressions of the formal system at the outset, but are instead inferred by means of the system itself, which has a dual nature: it can derive identities, but also truths.
These ideas foreshadow fundamental developments such as the so-called formulas-as types interpretation see Howard The syntax yields a general notation system for functions, based on an applicative language, where there is one basic category of terms well-formed formulas in his terminology.
Some terms are formally provable or assertable and are classified as true. The basic constants designate logical operations: a kind of restricted formal implication; existential quantifier, conjunction, negation, description operation, and generalized abstraction i.
However, the theories of Curry and Church were almost immediately shown inconsistent in , by Kleene and Rosser, who essentially proved a version of the Richard paradox both systems can provably enumerate their own provably total definable number theoretic functions.
The result was triggered by Church himself in , when he used the Richard paradox to prove a kind of incompleteness theorem with respect to statements asserting the totality of number theoretic functions. In the more technical part of the paper, Curry carefully axiomatizes the main ingredients exploited by Kleene and Rosser and carries out a lot of non-trivial work both on the logical side and the mathematical side e.
Curry notes that a twofold construction is possible. It is interesting to note that the two ways correspond to by-now standard tools, the so-called first fixed point theorem and second fixed point theorem of combinatory logic and lambda calculus Barendregt , p.
Here it is enough to recall that according to him, a remedy would be to formulate within the system the very notion of proposition, and a way to avoid the contradictions would lead to a hierarchy of canonical propositions or to a theory of levels of implication, already adumbrated by Church. Related ideas have been developed since the seventies by Scott , Aczel , Flagg and Myhill , and others. In the s, an alternative route to solve the antinomies emerged. The role of contraction was noticed by Fitch , who observed that, in order to derive the Russell paradox one considers a function of two variables, then one diagonalizes and regards such an object as a new unary propositional function.
One has to wait until the mid eighties to see contraction-free logics used systematically in proof theory and in theoretical computer science see the entry linear logic. Fitch proposed a new approach to the problem of finding consistent combinatory logic systems, which were progressively expanded and refined over many years until Truth and membership are inductively generated by iterating rules that correspond to natural logical closure conditions and can be formalized by means of positive i.
This fact implies that the generation process is cumulative and becomes saturated at a certain point, thus yielding consistent non-trivial interpretations for truth and membership.
Later he was able to strengthen his approach to include forms of negation and implication, insofar as he provided a simultaneous generation of truth and falsehood , and this actually amounts to conceive truth as a partial predicate.
To a certain extent, the ideas of Fitch can be regarded as introducing the view that the basic predicates of truth and membership have to be partial or, if you like, three-valued. His logical analysis leads to the conclusion that the paradoxes involve meaningless statements. No formula built up with the standard connectives can be valid or a tautology, i.
Bochvar describes a version of the extended type-free logical calculus of Hilbert-Ackermann , and, in order to dispose of the paradoxes, he restricts substitution and hence the comprehension schema of the form.
This makes his theory quite expressive e. So the contradiction is ascribed to an error in the theory of definitions, namely to the use of definitions that give rise to an infinite chain of substitutions, without converging to a result. For instance, the syllogism Barbara, usually stated in the form. Nevertheless, his work has inspired work by Aczel and Feferman Lewis and Langford are led to conclusions which are not dissimilar to those of Behmann.
According to them, the paradoxes show that certain expressions do not express propositions. In this case, there is no contradiction, but we become entangled in a vicious regress p. In general, one can create arbitrary complicated cycles and check that they can lead either to contradictions or to infinite regress; but in either case, the expression fails to converge to a definite proposition.
Even after the logics developed by Russell, Zermelo and Tarski had created the theoretical means to get rid of difficulties involved in the notions of class, set, truth, definability, the paradoxes have remained alive. This probably is due to a persistent interest in alternative formal paradigms, to the controversial features and axioms of Principia Mathematica, and to the problematic place that self-reference occupies in mathematical logic.
Moreover, in NF the universal set exists. The consistency problem for NF is still open though partial results are known concerning fragments with bounds on stratification or restriction to extensionality. Remarkably, NF refutes the axiom of choice by a classical theorem of Specker.
Again, a classical result of Specker establishes the existence of a model of NF in a suitable version of simple type theory with a formal counterpart of typical ambiguity. Paradoxes are not that far from NF. ML was defined to avoid certain weaknesses of NF e. Once more, the Lyndon-Rosser result brought about the unexpected presence of a paradox in set theory and the foundations of mathematical logic.
As noticed many years ago by Kreisel and quite aptly recalled by Dean , p. Kreisel , Wang For instance, arithmetizing semantics yields a refinement of the completeness theorem, the so-called arithmetized completeness theorem ART: every recursive consistent theory has a model in which the function symbols are replaced by primitive recursive functions and the predicate symbols are replaced by predicates which are definable with just 2 quantifiers in a version of formal number theory cf.
Hilbert and Bernays , p. By adding an arithmetic sentence Con S expressing the consistency of a set theory S as a new axiom to elementary number theory, one can prove in the resulting system arithmetic translations of all theorems of S.
A by-product of this metamathematical formalization is a de facto unification of set theoretic and semantic theoretic paradoxes, in the sense that paradoxes of either sort become tools for proving incompleteness and undecidability. Typically, a given paradoxical notion is formalized as a predicate in a language of a theory interpreting at least a fragment of number theory Z; one then applies diagonalization, self-reference, etc.
Philosophical motivations are strongly influential in contemporary logical investigation of paradoxes and hence it is natural to wonder what is surviving of the initial Fregean theory of concepts, as based upon an inconsistent principle of abstraction and the logicistic outlook. On the other hand, once we set apart the ideological inspiration of logicism, we might believe that the development of logic and set theory in the 20th century has fully sterilized paradoxes, and that contradictions in logical systems are phenomena of the years of foundational crisis only.
But this is not true: paradoxes have been discovered in logical systems related to computer science. Later, Coquand proved that certain extensions of the calculus C of constructions are inconsistent. On the other side, a general type-free development of the theory of constructions as a foundation for constructive provability in logic and mathematics was originally proposed by Kreisel and Goodman, but it turned out to be affected by an antinomy, which has been recently reconsidered by Dean and Kurokawa Indeed, the role of uniformity is essential in previous investigations.
This has led to the study of so-called hyperuniverses. Beginning with , there was an attempt, due to K. Since Mirimanoff, Finsler and others, logicians have studied universes of set theory where circular sets exist. However, it is only since the early Eighties that a genuine mathematics of non-well-founded sets see set theory: non-wellfounded is being developed.
Using the axiom AFA of anti-foundation, direct self-reference is allowed in set theory, and there exist plenty of sets solving general self-referential equations AFA was introduced by Forti and Honsell in ; for systematic development and history, see Aczel In particular, non-wellfounded sets are applied to the analysis of the paradoxes, to the semantics of natural languages and to theoretical computer science see Barwise and Etchemendy , Barwise and Moss Concerning the issue whether self-reference can be avoided in deriving paradoxes, and hence whether there are genuine contradictions arising from ungroundedness, a positive answer has been given by the semantical paradox of Yablo there are infinitely many agents , etc.
The issue this construction raises, namely whether circularity and self-reference are necessary and sufficient conditions to the appearance of paradoxes, has been further considered in Yablo see Cook for a comprehensive study on this matter, and Halbach and Zhang for a proof without diagonal lemma.
On the other hand, category theory has been used for new approaches to paradoxes since Lawvere The Berry paradox has been related to the incompleteness phenomena also because of work going back to the sixties and the seventies in the so-called Kolmogorov complexity and algorithmic information theory.
In particular, Chaitin has shown in a number of papers how to exploit randomness to prove certain limitations of formal systems see Chaitin It is worth recalling that — again on the side of epistemic logic — self-reference is applicable in order to prove incompleteness in belief models. Although not directly connected with the incompleteness phenomenon, there are indeed several accounts of antinomies affecting what appear to be sets of apparently legitimate, natural postulates involving certain epistemic notions.
Let Leitgeb work as an ultimate example of sources of this sort, presenting a solution to the lottery paradox see Kyburg and epistemic paradoxes that affects principles relating categorical belief to graded belief see also Douven that contains the former source. Tarski notwithstanding, since the hierarchical approach has been somewhat superseded by new ideas that have rendered the ideal of logical and semantical closure in many respect accessible especially by means of the fixed point methods used by Kripke and Martin-Woodruff; see Martin We also mention the approach stemming from Herzberger, Gupta and Belnap see the entry revision theory of truth , that has connections with non-elementary parts of definability theory, set theory and higher recursion theory Welch , , , This has led to the general axiomatic study of revision-theoretic definitions and theories of circular definitions see Bruni , b, , The modal logic in question is built upon an operator that is naturally connected with revision-theoretic construction hence, the name , as explained in Gupta and Standefer The proof theory of this system is studied in Standefer Gaifman about rationality being affected by paradoxes resembling the truth theoretic paradoxes like the liar paradox see Gaifman An extension of this approach to the class of all finite games is presented in Bruni and Sillari This limit rule, which is essential to address the concept of truth, turned out to be the most critical aspect of the revision-theoretic approach to circular concepts from both the complexity, and the conceptual point of view see Campbell-Moore for a recent, new approach to the topic.
The wealth and variety of semantical tools has triggered a sort of experimenting with a number of mixed proposals. An analogous combination, studied from a more proof-theoretic angle, was considered in Standefer Field , has proposed influential solutions of the semantical paradoxes which combine Kripkean and revision theoretic techniques.
Field has consequently developed a theory of truth with a non-classical conditional operator , which allows to express a notion of determinate truth and to state that the Liar is not determinately true.
In the same direction, a considerable attention has been directed in recent literature to the so called revenge problem : typical solutions, say, of the Liar paradox, rely on notions that, if expressible in the object language, give rise to new versions of the paradox. So, the solution is only an illusion. The revenge problem can be instantiated by the so-called Strengthened Liar: informally, once we have a model which makes the Liar sentence L itself neither true nor false, and we can express this very fact, L is after all not true.
But this is the claim made by L, and hence L is true. So the paradox seems to show up again for more details, see the entry liar paradox and the collection of essays contained in Beall The idea is that the Liar paradox does not involve sentences, but specific occurrences of sentences , i.
Besides the model-theoretic side, axiomatic investigations of truth and related paradoxes have become increasingly important since the seminal papers of Friedman and Sheard , Feferman Since the year , this research thread has been intensively studied with various aims, from proof-theoretic analysis to philosophical discussion of minimalism for a survey of the varieties of truth theoretic systems and appropriate references, see the entry on axiomatic theories of truth and the recent monographs of Halbach , Horsten ; see also the papers Feferman , Fujimoto , Leigh and Rathjen Last but not least, the axiomatic study of epistemic notions has greatly benefited from application of techniques used for proving incompleteness and indefinability results since the early sixties: they have yielded negative results Kaplan and Montague , Montague , Thomason and established an interesting link with the surprise test paradox.
The situation may have also been changed by the study of possible world semantics for modal notions, conceived as predicates in Halbach, Leitgeb and Welch However this is open to debate and experimentation: for instance it is argued in Halbach and Welch that the predicate approach to necessity is a viable route — insofar as the expressive power is considered — provided one resorts to languages that involve both a truth predicate and the necessity operator.
A number of solutions have been proposed, which rely on the use of paraconsistent logics Priest or substructural logics see the entry logic: paraconsistent , as well as the entry substructural logics and Mares and Paoli The investigation of semantical and set-theoretic paradoxes in infinite-valued logic—which was pioneered by Mow Shaw-Kwei and Skolem in — has received a new impulse by contributions by Hajek, Shepherdson and Paris , and Hajek , Typically, in these papers basic results from mathematical analysis are applied e.
It is worth mentioning that Leitgeb has given a consistency proof for a probabilistic theory of truth with unrestricted T-schema by making use of the Hahn-Banach Theorem.
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