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160-407: Vagueness is a major topic of research in philosophical logic , where it serves as a potential challenge to classical logic . Work in formal semantics has sought to provide a compositional semantics for vague expressions in natural language. Work in philosophy of language has addressed implications of vagueness for the theory of meaning, while metaphysicists have considered whether reality itself

320-430: A r d ( s a n t a ) {\displaystyle Beard(santa)} " (Santa Claus has a beard) that " ∃ x ( B e a r d ( x ) ) {\displaystyle \exists x(Beard(x))} " (something has a beard) in classical logic but not in free logic. In free logic, often an existence-predicate is used to indicate whether a singular term denotes an object in

480-657: A r k ( t 1 ) ∧ ∃ t 0 ( t 0 < t 1 ∧ l i g h t ( t 0 ) ) ∧ ∃ t 2 ( t 1 < t 2 ∧ l i g h t ( t 2 ) ) {\displaystyle dark(t_{1})\land \exists t_{0}(t_{0}<t_{1}\land light(t_{0}))\land \exists t_{2}(t_{1}<t_{2}\land light(t_{2}))} " . While similar approaches are often seen in physics, logicians usually prefer an autonomous treatment of time in terms of operators. This

640-507: A tautology (from Ancient Greek : ταυτολογία ) is a formula that is true regardless of the interpretation of its component terms , with only the logical constants having a fixed meaning. For example, a formula that states, "the ball is green or the ball is not green," is always true, regardless of what a ball is and regardless of its colour. Tautology is usually, though not always, used to refer to valid formulas of propositional logic . The philosopher Ludwig Wittgenstein first applied

800-421: A borderline case is due to one's ignorance. For example, in the epistemicist view, there is a fact of the matter, for every person, about whether that person is old or not old; some people are ignorant of this fact. One possibility is that one's words and concepts are perfectly precise, but that objects themselves are vague. Consider Peter Unger 's example of a cloud (from his famous 1980 paper, "The Problem of

960-494: A certain city; courts often find such expressions to be too vague, giving municipal inspectors discretion beyond what the law allows. In the US this is known as the vagueness doctrine and in Europe as the principle of legal certainty . Vagueness is primarily a filter of natural human cognition , other tasks of vagueness are derived from that, and they are secondary. The ability to cognition

1120-460: A description of fuzzy or stochastic values of quantities, and fuzzy or stochastic relations (represented by mathematical functions) between quantities. The difference between the non-exact sciences (called descriptive) and the exact sciences is that the former use natural human cognition (with the Russell’s filter of vagueness) and refined natural language, and the exact sciences use cognition based on

1280-408: A discussion of the fundamental concepts of logic. The current article treats philosophical logic in the narrow sense, in which it forms one field of inquiry within the philosophy of logic. An important issue for philosophical logic is the question of how to classify the great variety of non-classical logical systems, many of which are of rather recent origin. One form of classification often found in

1440-426: A formal treatment of notions like "for some" and "for all". They can be used to express whether predicates have an extension at all or whether their extension includes the whole domain. Quantification is only allowed over individual terms but not over predicates, in contrast to higher-order logics. Alethic modal logic has been very influential in logic and philosophy. It provides a logical formalism to express what

1600-870: A given vague concept. Examples include disability (how much loss of vision is required before one is legally blind?), human life (at what point from conception to birth is one a legal human being, protected for instance by laws against murder?), adulthood (most familiarly reflected in legal ages for driving, drinking, voting, consensual sex, etc.), race (how to classify someone of mixed racial heritage), etc. Even such apparently unambiguous concepts such as biological sex can be subject to vagueness problems, not just from transsexuals ' gender transitions but also from certain genetic conditions which can give an individual mixed male and female biological traits (see intersex ). Many scientific concepts are of necessity vague, for instance species in biology cannot be precisely defined, owing to unclear cases such as ring species . Nonetheless,

1760-552: A good familiarity with it is still required since many of the logical systems of direct concern to philosophical logic can be understood either as extensions of classical logic, which accept its fundamental principles and build on top of it, or as modifications of it, rejecting some of its core assumptions. Classical logic was initially created in order to analyze mathematical arguments and was applied to various other fields only afterward. For this reason, it neglects many topics of philosophical importance not relevant to mathematics, like

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1920-402: A group of suitable quantities, find the natural laws that apply in the real world between them, and describe them in mathematical language. We get a mathematical knowledge (cognitive) model of a given part of the real world. A group of selected quantities forms a discrete Newtonian filter (sieve) through which man ‘’look’’ at a given part of the real world. Thus, in exact science, a given part of

2080-436: A language characterizes the conditions under which the sentences of this language are true or false. Formal semantics play a central role in the model-theoretic conception of validity . They are able to provide clear criteria for when an inference is valid or not: an inference is valid if and only if it is truth-preserving, i.e. if whenever its premises are true then its conclusion is also true. Whether they are true or false

2240-402: A little true or very true. It is often used to deal with vague expressions in natural language. For example, saying that "Petr is young" fits better (i.e. is "more true") if "Petr" refers to a three-year-old than if it refers to a 23-year-old. Many-valued logics with a finite number of truth-values can define their logical connectives using truth tables, just like classical logic. The difference

2400-414: A logically precise manner by formally expressing the inferential roles they play in relation to each other. Some theorists understand philosophical logic in a wider sense as the study of the scope and nature of logic in general. On this view, it investigates various philosophical problems raised by logic, including the fundamental concepts of logic. In this wider sense, it can be understood as identical to

2560-404: A method to mathematize the science. Even exact science needs to have a tool, with which it can describe the uncertainty of the results (obtained – knowledge), whether out of necessity or the need to abandon excessive precision. Since that cannot (must not) be an internal vagueness, it can only use linguistically graspable uncertainty (external vagueness). For this purpose it has for its disposal

2720-410: A much simpler expression in higher-order logic than in first-order logic. For example, Peano arithmetic and Zermelo-Fraenkel set theory need an infinite number of axioms to be expressed in first-order logic. But they can be expressed in second-order logic with only a few axioms. But despite this advantage, first-order logic is still much more widely used than higher-order logic. One reason for this

2880-455: A proper subset of the first-order logical validities. In the context of propositional logic, these two terms coincide. A tautology in first-order logic is a sentence that can be obtained by taking a tautology of propositional logic and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable). For example, because A ∨ ¬ A {\displaystyle A\lor \lnot A}

3040-448: A rule of inference. Different systems of logic provide different accounts for when an inference is valid. This means that they use different rules of inference. The traditionally dominant approach to validity is called classical logic. But philosophical logic is concerned with non-classical logic: it studies alternative systems of inference. The motivations for doing so can roughly be divided into two categories. For some, classical logic

3200-471: A second, outer domain for non-existing objects, which is then used to determine the corresponding truth values. Neutral semantics , on the other hand, hold that atomic formulas containing empty terms are neither true nor false. This is often understood as a three-valued logic , i.e. that a third truth value besides true and false is introduced for these cases. Many-valued logics are logics that allow for more than two truth values. They reject one of

3360-400: A sentence is a tautology. This means, in particular, the set of tautologies over a fixed finite or countable alphabet is a decidable set . As an efficient procedure , however, truth tables are constrained by the fact that the number of valuations that must be checked increases as 2 , where k is the number of variables in the formula. This exponential growth in the computation length renders

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3520-428: A sentence is not not true, then it is true, i.e. " ¬ ¬ A → A {\displaystyle \lnot \lnot A\to A} " . Due to these restrictions, many proofs are more complicated and some proofs otherwise accepted become impossible. These modifications of classical logic are motivated by the idea that truth depends on verification through a proof . This has been interpreted in

3680-447: A sentence modified by the ◻ {\displaystyle \Box } -operator is true if it is true in all possible worlds. So the sentence " ◊ W ( s ) {\displaystyle \Diamond W(s)} " (it is possible that Socrates is wise) is true since there is at least one world where Socrates is wise. But " ◻ W ( s ) {\displaystyle \Box W(s)} " (it

3840-457: A simpler variant of the deductive systems employed for first-order logic (see Kleene 1967, Sec 1.9 for one such system). A proof of a tautology in an appropriate deduction system may be much shorter than a complete truth table (a formula with n propositional variables requires a truth table with 2 lines, which quickly becomes infeasible as n increases). Proof systems are also required for the study of intuitionistic propositional logic, in which

4000-694: A tautology (Kleene 1967 p. 27). For example, let S {\displaystyle S} be A ∧ ( B ∨ ¬ B ) {\displaystyle A\land (B\lor \lnot B)} . Then S {\displaystyle S} is not a tautology, because any valuation that makes A {\displaystyle A} false will make S {\displaystyle S} false. But any valuation that makes A {\displaystyle A} true will make S {\displaystyle S} true, because B ∨ ¬ B {\displaystyle B\lor \lnot B}

4160-410: A tautology is in the context of propositional logic. The definition can be extended, however, to sentences in first-order logic . These sentences may contain quantifiers, unlike sentences of propositional logic. In the context of first-order logic, a distinction is maintained between logical validities , sentences that are true in every model, and tautologies (or, tautological validities ), which are

4320-445: A tautology will end in a column with only T , while the truth table for a sentence that is not a tautology will contain a row whose final column is F , and the valuation corresponding to that row is a valuation that does not satisfy the sentence being tested. This method for verifying tautologies is an effective procedure , which means that given unlimited computational resources it can always be used to mechanistically determine whether

4480-412: A true material conditional, its antecedent has to be relevant to the consequent. A difficulty faced for this issue is that relevance usually belongs to the content of the propositions while logic only deals with formal aspects. This problem is partially addressed by the so-called variable sharing principle . It states that antecedent and consequent have to share a propositional variable. This would be

4640-411: A true proposition. So since it is true that "the sun is bigger than the moon", it is possible to infer that "the sun is bigger than the moon or Spain is controlled by space-rabbits". According to the disjunctive syllogism , one can infer that one of these disjuncts is true if the other is false. So if the logical system also contains the negation of this proposition, i.e. that "the sun is not bigger than

4800-428: A vague statement is true if it is true on at least one precisification and false if it is false under at least one precisification. If a vague statement comes out true under one precisification and false under another, it is both true and false. Subvaluationism ultimately amounts to the claim that vagueness is a truly contradictory phenomenon. Of a borderline case of "bald man" it would be both true and false to say that he

4960-521: Is possibly or necessarily true . It constitutes an extension of first-order logic, which by itself is only able to express what is true simpliciter . This extension happens by introducing two new symbols: " ◊ {\displaystyle \Diamond } " for possibility and " ◻ {\displaystyle \Box } " for necessity. These symbols are used to modify propositions. For example, if " W ( s ) {\displaystyle W(s)} " stands for

Vagueness - Misplaced Pages Continue

5120-466: Is a tautology if the formula itself is always true, regardless of which valuation is used for the propositional variables . There are infinitely many tautologies. In many of the following examples A represents the statement "object X is bound", B represents "object X is a book", and C represents "object X is on the shelf". Without a specific referent object X , A → B {\displaystyle A\to B} corresponds to

5280-432: Is a generalization of Aristotelian logic. On this view, classical predicate logic introduces predicates with an empty extension while free logic introduces singular terms of non-existing things. An important problem for free logic consists in how to determine the truth value of expressions containing empty singular terms, i.e. of formulating a formal semantics for free logic. Formal semantics of classical logic can define

5440-541: Is a more restricted version of classical logic. It is more restricted in the sense that certain rules of inference used in classical logic do not constitute valid inferences in it. This concerns specifically the law of excluded middle and the double negation elimination . The law of excluded middle states that for every sentence, either it or its negation are true. Expressed formally: A ∨ ¬ A {\displaystyle A\lor \lnot A} . The law of double negation elimination states that if

5600-419: Is a proposition of logic has got to be in some sense or the other like a tautology. It has got to be something that has some peculiar quality, which I do not know how to define, that belongs to logical propositions but not to others. Here, logical proposition refers to a proposition that is provable using the laws of logic. Many logicians in the early 20th century used the term 'tautology' for any formula that

5760-432: Is a tautology is a finite and mechanical one: one needs only to evaluate the truth value of the formula under each of its possible valuations. One algorithmic method for verifying that every valuation makes the formula to be true is to make a truth table that includes every possible valuation. For example, consider the formula There are 8 possible valuations for the propositional variables A , B , C , represented by

5920-530: Is a tautology is equivalent to verifying that there is no valuation satisfying ¬ S {\displaystyle \lnot S} . The Boolean satisfiability problem is NP-complete , and consequently, tautology is co-NP-complete . It is widely believed that (equivalently for all NP-complete problems) no polynomial-time algorithm can solve the satisfiability problem, although some algorithms perform well on special classes of formulas, or terminate quickly on many instances. The fundamental definition of

6080-414: Is a tautology of propositional logic, ( ∀ x ( x = x ) ) ∨ ( ¬ ∀ x ( x = x ) ) {\displaystyle (\forall x(x=x))\lor (\lnot \forall x(x=x))} is a tautology in first order logic. Similarly, in a first-order language with a unary relation symbols R , S , T , the following sentence is a tautology: It

6240-453: Is a tautology, then S {\displaystyle S} is tautologically implied by every formula. There is a general procedure, the substitution rule , that allows additional tautologies to be constructed from a given tautology (Kleene 1967 sec. 3). Suppose that S is a tautology and for each propositional variable A in S a fixed sentence S A is chosen. Then the sentence obtained by replacing each variable A in S with

6400-464: Is a tautology. Let R {\displaystyle R} be the formula A ∧ C {\displaystyle A\land C} . Then R ⊨ S {\displaystyle R\models S} , because any valuation satisfying R {\displaystyle R} will make A {\displaystyle A} true—and thus makes S {\displaystyle S} true. It follows from

6560-416: Is a theorem (derivable from axioms). An axiomatic system is sound if every theorem is a tautology. The problem of constructing practical algorithms to determine whether sentences with large numbers of propositional variables are tautologies is an area of contemporary research in the area of automated theorem proving . The method of truth tables illustrated above is provably correct – the truth table for

Vagueness - Misplaced Pages Continue

6720-708: Is a way to draw the addressee's attention to the vagueness of the message more explicitly and to quantify the vagueness, thus improving understanding in communication using natural language. But the main vagueness of informal languages is the internal vagueness, and the external vagueness serves only as an auxiliary tool. Formal languages , mathematics, formal logic, programming languages (in principle, they must have zero internal vagueness of interpretation of all language constructs, i.e. they have exact interpretation) can model external vagueness by tools of vagueness and uncertainty representation: fuzzy sets and fuzzy logic, or by stochastic quantities and stochastic functions, as

6880-627: Is allowed not just over individual terms but also over predicates. This way, it is possible to express, for example, whether certain individuals share some or all of their predicates, as in " ∃ Q ( Q ( m a r y ) ∧ Q ( j o h n ) ) {\displaystyle \exists Q(Q(mary)\land Q(john))} " ( there are some qualities that Mary and John share). Because of these changes, higher-order logics have more expressive power than first-order logic. This can be helpful for mathematics in various ways since different mathematical theories have

7040-570: Is also closer to natural languages, which mostly use grammar, e.g. by conjugating verbs, to express the pastness or futurity of events. Epistemic logic is a form of modal logic applied to the field of epistemology . It aims to capture the logic of knowledge and belief . The modal operators expressing knowledge and belief are usually expressed through the symbols " K {\displaystyle K} " and " B {\displaystyle B} " . So if " W ( s ) {\displaystyle W(s)} " stands for

7200-470: Is bald", to "perfect truth", for, say, " Patrick Stewart is bald". In ordinary logics, there are only two truth-values : "true" and "false". The fuzzy perspective differs by introducing an infinite number of truth-values along a spectrum between perfect truth and perfect falsity. Perfect truth may be represented by "1", and perfect falsity by "0". Borderline cases are thought of as having a "truth-value" anywhere between 0 and 1 (for example, 0.6). Advocates of

7360-452: Is bald", where Frank is a borderline case of baldness), but does have consequences for logically complex statements. In particular, the tautologies of sentential logic, such as "Frank is bald or Frank is not bald", will turn out to be supertrue, since on any precisification of baldness, either "Frank is bald" or "Frank is not bald" will be true. Since the presence of borderline cases seems to threaten principles like this one (excluded middle),

7520-431: Is bald, and both true and false to say that he is not bald. A fourth approach, known as "the epistemicist view", has been defended by Timothy Williamson (1994), R. A. Sorensen (1988) and (2001), and Nicholas Rescher (2009). They maintain that vague predicates do, in fact, draw sharp boundaries, but that one cannot know where these boundaries lie. One's confusion about whether some vague word does or does not apply in

7680-430: Is contradictory and contradictions within theories are needed to accurately reflect reality. Without paraconsistent logics, dialetheism would be hopeless since everything would be both true and false. Paraconsistent logics make it possible to keep contradictions local, without exploding the whole system. But even with this adjustment, dialetheism is still highly contested. Another motivation for paraconsistent logic

7840-477: Is correct. It is usually advantageous to have the strongest system possible in order to be able to draw many different inferences. But this brings with it the problem that some of these additional inferences may contradict basic modal intuitions in specific cases. This usually motivates the choice of a more basic system of axioms. Possible worlds semantics is a very influential formal semantics in modal logic that brings with it system S5. A formal semantics of

8000-440: Is happening all the time. These two operators behave in the same way as the operators for possibility and necessity in alethic modal logic. Since the difference between past and future is of central importance to human affairs, these operators are often modified to take this difference into account. Arthur Prior 's tense logic, for example, realizes this idea using four such operators: P {\displaystyle P} (it

8160-595: Is hidden for the other human, he can only guess the amount of it. Informal languages, such as natural language, do not make it possible to distinguish between internal and external vagueness strictly, but only with a vague boundary. Fortunately, however, informal languages use appropriate language constructs making meaning a little uncertain (e.g. indeterminate quantifiers POSSIBLY, SEVERAL, MAYBE, etc.). Such quantifiers allow natural language to use external vagueness more strongly and explicitly, thus allowing internal vagueness to be partially shifted up to external vagueness. It

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8320-425: Is hidden from another person, he can only guess that. We either have to accept internal vagueness, which is human, or we can try to reduce it, or completely eliminate it, which is scientific. Demands on the accuracy of the formulation of scientific knowledge and its communication require minimizing the internal vagueness with which one connotes (vaguely, emotionally and subjectively interprets) linguistic constructs of

8480-406: Is itself unclear on just those cases. Others say that one has an interest in making his or her definitions more precise than ordinary language, or his or her ordinary concepts, themselves allow; they recommend one advances precising definitions . Vagueness is also a problem which arises in law, and in some cases, judges have to arbitrate regarding whether a borderline case does, or does not, satisfy

8640-420: Is made, the overall formula will come out true. For if the first disjunct ( A ∧ B ) {\displaystyle (A\land B)} is not satisfied by a particular valuation, then A or B must be assigned F, which will make one of the following disjunct to be assigned T. In natural language, either both A and B are true or at least one of them is false. A formula of propositional logic

8800-422: Is motivated by a form of metaphysical idealism. Applied to mathematics, it states that mathematical objects exist only to the extent that they are constructed in the mind. Free logic rejects some of the existential presuppositions found in classical logic. In classical logic, every singular term has to denote an object in the domain of quantification. This is usually understood as an ontological commitment to

8960-494: Is necessary that Socrates is wise) is false since Socrates is not wise in every possible world. Possible world semantics has been criticized as a formal semantics of modal logic since it seems to be circular. The reason for this is that possible worlds are themselves defined in modal terms, i.e. as ways how things could have been. In this way, it itself uses modal expressions to determine the truth of sentences containing modal expressions. Deontic logic extends classical logic to

9120-477: Is neither a tautology nor a contradiction is said to be logically contingent . Such a formula can be made either true or false based on the values assigned to its propositional variables. The double turnstile notation ⊨ S {\displaystyle \vDash S} is used to indicate that S is a tautology. Tautology is sometimes symbolized by "V pq ", and contradiction by "O pq ". The tee symbol ⊤ {\displaystyle \top }

9280-516: Is no distinction between a tautology and a logically valid formula. In the context of predicate logic, many authors define a tautology to be a sentence that can be obtained by taking a tautology of propositional logic, and uniformly replacing each propositional variable by a first-order formula (one formula per propositional variable). The set of such formulas is a proper subset of the set of logically valid sentences of predicate logic (i.e., sentences that are true in every model ). The word tautology

9440-425: Is not. The condition for the establishment of an exact science is to find suitable quantities, and this is possible only for a small part of the real world and for specific views of it. In other words, the filter of vagueness makes it possible to vaguely know many; Newton's discrete filter makes it possible to know only little but exactly. Philosophical logic Understood in a narrow sense, philosophical logic

9600-442: Is obtained by replacing A {\displaystyle A} with ∃ x R x {\displaystyle \exists xRx} , B {\displaystyle B} with ¬ ∃ x S x {\displaystyle \lnot \exists xSx} , and C {\displaystyle C} with ∀ x T x {\displaystyle \forall xTx} in

9760-481: Is restricted to singular terms. It can be used to talk about whether a predicate has an extension at all or whether its extension includes the whole domain. This way, propositions like " ∃ x ( A p p l e ( x ) ∧ S w e e t ( x ) ) {\displaystyle \exists x(Apple(x)\land Sweet(x))} " ( there are some apples that are sweet) can be expressed. In higher-order logics, quantification

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9920-418: Is sometimes used to denote an arbitrary tautology, with the dual symbol ⊥ {\displaystyle \bot } ( falsum ) representing an arbitrary contradiction; in any symbolism, a tautology may be substituted for the truth value " true ", as symbolized, for instance, by "1". Tautologies are a key concept in propositional logic , where a tautology is defined as a propositional formula that

10080-415: Is specified by the formal semantics. Possible worlds semantics specifies the truth conditions of sentences expressed in modal logic in terms of possible worlds. A possible world is a complete and consistent way how things could have been. On this view, a sentence modified by the ◊ {\displaystyle \Diamond } -operator is true if it is true in at least one possible world while

10240-496: Is that ought implies can . This means that the agent can only have the obligation to do something if it is possible for the agent to do it. Expressed formally: " O A → ◊ A {\displaystyle OA\to \Diamond A} " . Temporal logic , or tense logic, uses logical mechanisms to express temporal relations. In its most simple form, it contains one operator to express that something happened at one time and another to express that something

10400-401: Is that higher-order logic is incomplete . This means that, for theories formulated in higher-order logic, it is not possible to prove every true sentence pertaining to the theory in question. Another disadvantage is connected to the additional ontological commitments of higher-order logics. It is often held that the usage of the existential quantifier brings with it an ontological commitment to

10560-404: Is that if something is necessary, then it must also be possible. This means that " ◊ A {\displaystyle \Diamond A} " follows from " ◻ A {\displaystyle \Box A} " . There is disagreement about exactly which axioms govern modal logic. The different forms of modal logic are often presented as a nested hierarchy of systems in which

10720-539: Is that these truth tables are more complex since more possible inputs and outputs have to be considered. In Kleene's three-valued logic, for example, the inputs "true" and "undefined" for the conjunction-operator " ∧ {\displaystyle \land } " result in the output "undefined". The inputs "false" and "undefined", on the other hand, result in "false". Paraconsistent logics are logical systems that can deal with contradictions without leading to all-out absurdity. They achieve this by avoiding

10880-437: Is the area of logic that studies the application of logical methods to philosophical problems, often in the form of extended logical systems like modal logic . Some theorists conceive philosophical logic in a wider sense as the study of the scope and nature of logic in general. In this sense, philosophical logic can be seen as identical to the philosophy of logic , which includes additional topics like how to define logic or

11040-430: Is the basic natural equipment of human (and other creatures) allowing him to orient and survive in the real (material) world. The task of cognition is to obtain from the epistemologically incalculable (immensely vast and deep) reality to a human its cognitive (knowledge) model, containing an only finite amount of information . For this purpose, there must be a filter performing selection and thus reduction of information. It

11200-462: Is the case but also what someone believes or knows to be the case. Its rules of inference articulate what follows from the fact that someone has these kinds of mental states . Higher-order logics do not directly apply classical logic to certain new sub-fields within philosophy but generalize it by allowing quantification not just over individuals but also over predicates. Deviant logics , in contrast to these forms of extended logics, reject some of

11360-400: Is the elementary building block of exact science . In exact sciences, it is always precisely defined, either consensually (basic set) or the other derived - International System of Units . And what about the artificial filter that allows Newton to avoid internal vagueness? For every problem of the real world that is to be grasped by Newton’s method of the exact science, it is necessary to choose

11520-440: Is the thesis that there is only one true logic. This can be understood in different ways, for example, that only one of all the suggested logical systems is correct or that the correct logical system is yet to be found as a system underlying and unifying all the different logics. Pluralists, on the other hand, hold that a variety of different logical systems can all be correct at the same time. A closely related problem concerns

11680-404: Is the vagueness with which man perceive and then remember information about the real (material) world. Some information gained with less vagueness, others with greater one, according to the distance from the center (focus) of attention occupied by man during his act of cognition. Human is unable to acquire information other than vague one by his natural vague cognition. It is necessary to distinguish

11840-439: Is to provide a logic for discussions and group beliefs where the group as a whole may have inconsistent beliefs if its different members are in disagreement. Relevance logic is one type of paraconsistent logic. As such, it also avoids the principle of explosion even though this is usually not the main motivation behind relevance logic. Instead, it is usually formulated with the goal of avoiding certain unintuitive applications of

12000-434: Is too far removed from the most fundamental logical intuitions. So not everyone agrees that all the formal systems discussed in this article actually constitute logics , when understood in a strict sense. Classical logic is the dominant form of logic used in most fields. The term refers primarily to propositional logic and first-order logic . Classical logic is not an independent topic within philosophical logic. But

12160-493: Is too narrow: it leaves out many philosophically interesting issues. This can be solved by extending classical logic with additional symbols to give a logically strict treatment of further areas. Others see some flaw with classical logic itself and try to give a rival account of inference. This usually leads to the development of deviant logics, each of which modifies the fundamental principles behind classical logic in order to rectify their alleged flaws. Modern developments in

12320-446: Is true in all possible worlds. Deontic logic pertains to ethics and provides a formal treatment of ethical notions, such as obligation and permission . Temporal logic formalizes temporal relations between propositions. This includes ideas like whether something is true at some time or all the time and whether it is true in the future or in the past. Epistemic logic belongs to epistemology . It can be used to express not just what

12480-483: Is true under any possible Boolean valuation of its propositional variables . A key property of tautologies in propositional logic is that an effective method exists for testing whether a given formula is always satisfied (equiv., whether its negation is unsatisfiable). The definition of tautology can be extended to sentences in predicate logic , which may contain quantifiers —a feature absent from sentences of propositional logic. Indeed, in propositional logic, there

12640-488: Is universally valid, whether a formula of propositional logic or of predicate logic . In this broad sense, a tautology is a formula that is true under all interpretations , or that is logically equivalent to the negation of a contradiction. Tarski and Gödel followed this usage and it appears in textbooks such as that of Lewis and Langford. This broad use of the term is less common today, though some textbooks continue to use it. Modern textbooks more commonly restrict

12800-463: Is used if (i) it introduces new vocabulary but all well-formed formulas of classical logic are also well-formed formulas in it and (ii) even when it is restricted to inferences using only the vocabulary of classical logic, some valid inferences in classical logic are not valid inferences in it. The term "deviant logic" is often used in a sense that includes quasi-deviant logics as well. A philosophical problem raised by this plurality of logics concerns

12960-405: Is vague. The concept of vagueness has philosophical importance. Suppose one wants to come up with a definition of "right" in the moral sense. One wants a definition to cover actions that are clearly right and exclude actions that are clearly wrong, but what does one do with the borderline cases? Surely, there are such cases. Some philosophers say that one should try to come up with a definition that

13120-454: The filter of knowledge is primary, we call it internal vagueness (i.e. intrapsychic). The vagueness of a person's subsequent utterance is secondary vagueness. This utterance (transformation from intrapsychic languages to external communicative languages - it is called a formulation , see the semantic triangle ) cannot reveal all the content of the personal intrapsychic cognitive model with all its inherent vagueness. The vagueness contained in

13280-443: The philosophy of logic , where these topics are discussed. The current article discusses only the narrow conception of philosophical logic. In this sense, it forms one area of the philosophy of logic. Central to philosophical logic is an understanding of what logic is and what role philosophical logics play in it. Logic can be defined as the study of valid inferences. An inference is the step of reasoning in which it moves from

13440-404: The principle of explosion found in classical logic. According to the principle of explosion, anything follows from a contradiction. This is the case because of two rules of inference, which are valid in classical logic: disjunction introduction and disjunctive syllogism . According to the disjunction introduction, any proposition can be introduced in the form of a disjunction when paired with

13600-672: The Many"): it is not clear where the boundary of a cloud lies; for any given bit of water vapor, one can ask whether it is part of the cloud or not, and for many such bits, one will not know how to answer. So perhaps one's term 'cloud' denotes a vague object precisely. This strategy has been poorly received, in part due to Gareth Evans's short paper "Can There Be Vague Objects?" (1978). Evans's argument appears to show that there can be no vague identities (e.g. "Princeton = Princeton Borough"), but as Lewis (1988) makes clear, Evans takes for granted that there are in fact vague identities, and that any proof to

13760-616: The academic literature is due to Susan Haack and distinguishes between classical logic , extended logics, and deviant logics . This classification is based on the idea that classical logic, i.e. propositional logic and first-order logic, formalizes some of the most common logical intuitions. In this sense, it constitutes a basic account of the axioms governing valid inference. Extended logics accept this basic account and extend it to additional areas. This usually happens by adding new vocabulary, for example, to express necessity, obligation, or time. These new symbols are then integrated into

13920-414: The agent has the obligation to do something then they automatically also have the permission to do it. This can be expressed formally through the axiom schema " O A → P A {\displaystyle OA\to PA} " . Another question of interest to philosophical logic concerns the relation between alethic modal logic and deontic logic. An often discussed principle in this respect

14080-427: The area of logic have resulted in a great proliferation of logical systems. This stands in stark contrast to the historical dominance of Aristotelian logic , which was treated as the one canon of logic for over two thousand years. Treatises on modern logic often treat these different systems as a list of separate topics without providing a clear classification of them. However, one classification frequently mentioned in

14240-433: The axiom schema " K A → A {\displaystyle KA\to A} " expresses that whenever something is known, then it is true. This reflects the idea that one can only know what is true, otherwise it is not knowledge but another mental state. Another epistemic intuition about knowledge concerns the fact that when the agent knows something, they also know that they know it. This can be expressed by

14400-566: The axiom schema " K A → K K A {\displaystyle KA\to KKA} " . An additional principle linking knowledge and belief states that knowledge implies belief, i.e. " K A → B A {\displaystyle KA\to BA} " . Dynamic epistemic logic is a distinct form of epistemic logic that focuses on situations in which changes in belief and knowledge happen. Higher-order logics extend first-order logic by including new forms of quantification . In first-order logic, quantification

14560-403: The best theoretical treatment of vagueness is—which is closely related to the problem of the paradox of the heap , a.k.a. sorites paradox—has been the subject of much philosophical debate. One theoretical approach is that of fuzzy logic, developed by American mathematician Lotfi Zadeh . Fuzzy logic proposes a gradual transition between "perfect falsity", for example, the statement " Bill Clinton

14720-539: The case, for example, in " ( p ∧ q ) → q {\displaystyle (p\land q)\to q} " but not in " ( p ∧ q ) → r {\displaystyle (p\land q)\to r} " . A closely related concern of relevance logic is that inferences should follow the same requirement of relevance, i.e. that it is a necessary requirement of valid inferences that their premises are relevant to their conclusion. Tautology (logic) In mathematical logic ,

14880-599: The classical approach to these connectives is that they follow certain laws, like the law of excluded middle , the double negation elimination , the principle of explosion , and the bivalence of truth. This sets classical logic apart from various deviant logics, which deny one or several of these principles. In first-order logic , the propositions themselves are made up of subpropositional parts, like predicates , singular terms , and quantifiers . Singular terms refer to objects and predicates express properties of objects and relations between them. Quantifiers constitute

15040-500: The common law system, vagueness is a possible legal defence against by-laws and other regulations. The legal principle is that delegated power cannot be used more broadly than the delegator intended. Therefore, a regulation may not be so vague as to regulate areas beyond what the law allows. Any such regulation would be "void for vagueness" and unenforceable. This principle is sometimes used to strike down municipal by-laws that forbid "explicit" or "objectionable" contents from being sold in

15200-438: The communication language, and thus improve the accuracy of the message. Various scientific procedures aim to improve the credibility and accuracy of the scientific knowledge obtained. To formulate them, however, it is necessary to build a more precise language, with less (internal) vagueness of message than is common in daily life. This is done by purposefully (branch) constructed terminology allowing to more accurately describe

15360-525: The concept of species can be clearly applied in the vast majority of cases. As this example illustrates, to say that a definition is "vague" is not necessarily a criticism. Consider those animals in Alaska that are the result of breeding huskies and wolves : are they dogs ? It is not clear: they are borderline cases of dogs. This means one's ordinary concept of doghood is not clear enough to let us rule conclusively in this case. The philosophical question of what

15520-823: The conditional to be true, its antecedent has to be relevant to its consequent. The term "philosophical logic" is used by different theorists in slightly different ways. When understood in a narrow sense, as discussed in this article, philosophical logic is the area of philosophy that studies the application of logical methods to philosophical problems. This usually happens in the form of developing new logical systems to either extend classical logic to new areas or to modify it to include certain logical intuitions not properly addressed by classical logic. In this sense, philosophical logic studies various forms of non-classical logics, like modal logic and deontic logic. This way, various fundamental philosophical concepts, like possibility, necessity, obligation, permission, and time, are treated in

15680-545: The contrary cannot be right. Since the proof Evans produces relies on the assumption that terms precisely denote vague objects, the implication is that the assumption is false, and so the vague-objects view is wrong. Still by, for instance, proposing alternative deduction rules involving Leibniz's law or other rules for validity some philosophers are willing to defend ontological vagueness as some kind of metaphysical phenomenon. One has, for example, Peter van Inwagen (1990), Trenton Merricks and Terence Parsons (2000). In

15840-496: The core assumptions of classical logic: the principle of the bivalence of truth. The most simple versions of many-valued logics are three-valued logics: they contain a third truth value. In Stephen Cole Kleene 's three-valued logic, for example, this third truth value is "undefined". According to Nuel Belnap 's four-valued logic, there are four possible truth values: "true", "false", "neither true nor false", and "both true and false". This can be interpreted, for example, as indicating

16000-402: The corresponding rules of inference governing these symbols. In the case of alethic modal logic , these new symbols are used to express not just what is true simpliciter , but also what is possibly or necessarily true . It is often combined with possible worlds semantics, which holds that a proposition is possibly true if it is true in some possible world while it is necessarily true if it

16160-414: The corresponding sentence S A is also a tautology. For example, let S be the tautology: Let S A be C ∨ D {\displaystyle C\lor D} and let S B be C → E {\displaystyle C\to E} . It follows from the substitution rule that the sentence: is also a tautology. An axiomatic system is complete if every tautology

16320-426: The definition that if a formula R {\displaystyle R} is a contradiction, then R {\displaystyle R} tautologically implies every formula, because there is no truth valuation that causes R {\displaystyle R} to be true, and so the definition of tautological implication is trivially satisfied. Similarly, if S {\displaystyle S}

16480-512: The difference between necessity and possibility, between obligation and permission, or between past, present, and future. These and similar topics are given a logical treatment in the different philosophical logics extending classical logic. Classical logic by itself is only concerned with a few basic concepts and the role these concepts play in making valid inferences. The concepts pertaining to propositional logic include propositional connectives, like "and", "or", and "if-then". Characteristic of

16640-595: The distinction between extended and deviant logics is sometimes drawn in a slightly different manner. On this view, a logic is an extension of classical logic if two conditions are fulfilled: (1) all well-formed formulas of classical logic are also well-formed formulas in it and (2) all valid inferences in classical logic are also valid inferences in it. For a deviant logic, on the other hand, (a) its class of well-formed formulas coincides with that of classical logic, while (b) some valid inferences in classical logic are not valid inferences in it. The term quasi-deviant logic

16800-428: The domain or not. But the usage of existence-predicates is controversial. They are often opposed, based on the idea that existence is required if any predicates should apply to the object at all. In this sense, existence cannot itself be a predicate. Karel Lambert , who coined the term "free logic", has suggested that free logic can be understood as a generalization of classical predicate logic just as predicate logic

16960-467: The entities over which this quantifier ranges. In first-order logic, this concerns only individuals, which is usually seen as an unproblematic ontological commitment. In higher-order logic, quantification concerns also properties and relations. This is often interpreted as meaning that higher-order logic brings with it a form of Platonism , i.e. the view that universal properties and relations exist in addition to individuals. Intuitionistic logic

17120-429: The exact sciences do. Principle is: If we admit more vagueness (uncertainty), we can gain more information during cognition. See e.g. possibilities of deterministic and stochastic physic. In other cases cognitive model of certain part of real world may be simplified, such a way, that certain amount of deterministic information is possible to replace by fuzzy or stochastic one. The internal vagueness of one person's message

17280-401: The exact world. And it is already evident that we are on the way to the creation of the scientific method that creates science belonging to the exact world, that is, an exact science is born. It is still necessary to explain how to realize Newton`s exact cognition, that is, the cognition when the knowledge obtained from the real world is part of the exact world. The miraculous bridge between

17440-439: The existence of the named entity. But many names are used in everyday discourse that do not refer to existing entities, like "Santa Claus" or "Pegasus". This threatens to preclude such areas of discourse from a strict logical treatment. Free logic avoids these problems by allowing formulas with non-denoting singular terms. This applies to proper names as well as definite descriptions , and functional expressions. Quantifiers, on

17600-412: The expression " B e a r d ( s a n t a ) {\displaystyle Beard(santa)} " is false. Positive semantics allows that at least some expressions with empty terms are true. This usually includes identity statements, like " s a n t a = s a n t a {\displaystyle santa=santa} " . Some versions introduce

17760-494: The extreme, that is zero , was realized by I. Newton. It is an epochal idea, and it needs to be explained how it can be realized. It follows from the above-mentioned Law of maintaining accuracy of information (optimization of the truthfulness of the message) saying, if we require to eliminate the internal vagueness in the knowledge completely (to zero) then of course it must first be completely eliminated in cognition (the source of information). This means that one (Newton) must avoid

17920-526: The fact that supervaluationism can "rescue" them is seen as a virtue. Subvaluationism is the logical dual of supervaluationism, and has been defended by Dominic Hyde (2008) and Pablo Cobreros (2011). Whereas the supervaluationist characterises truth as 'supertruth', the subvaluationist characterises truth as 'subtruth', or "true on at least some precisifications". Subvaluationism proposes that borderline applications of vague terms are both true and false. It thus has "truth-value gluts". According to this theory,

18080-417: The field of ethics . Of central importance in ethics are the concepts of obligation and permission , i.e. which actions the agent has to do or is allowed to do. Deontic logic usually expresses these ideas with the operators O {\displaystyle O} and P {\displaystyle P} . So if " J ( r ) {\displaystyle J(r)} " stands for

18240-423: The first three columns of the following table. The remaining columns show the truth of subformulas of the formula above, culminating in a column showing the truth value of the original formula under each valuation. Because each row of the final column shows T , the sentence in question is verified to be a tautology. It is also possible to define a deductive system (i.e., proof system) for propositional logic, as

18400-440: The form of a singular term and increasing the arity of one's predicates by one. For example, the tense-logic-sentence " d a r k ∧ P ( l i g h t ) ∧ F ( l i g h t ) {\displaystyle dark\land P(light)\land F(light)} " (it is dark, it was light, and it will be light again) can be translated into pure first-order logic as " d

18560-413: The former case analytic propositions are tautological. Here, analytic proposition refers to an analytic truth , a statement in natural language that is true solely because of the terms involved. In 1884, Gottlob Frege proposed in his Grundlagen that a truth is analytic exactly if it can be derived using logic. However, he maintained a distinction between analytic truths (i.e., truths based only on

18720-576: The fundamental principles of classical logic and are often seen as its rivals. Intuitionistic logic is based on the idea that truth depends on verification through a proof. This leads it to reject certain rules of inference found in classical logic that are not compatible with this assumption. Free logic modifies classical logic in order to avoid existential presuppositions associated with the use of possibly empty singular terms, like names and definite descriptions. Many-valued logics allow additional truth values besides true and false . They thereby reject

18880-441: The fuzzy logic approach have included K. F. Machina (1976) and Dorothy Edgington (1993). Another theoretical approach is known as " supervaluationism ". This approach has been defended by Kit Fine and Rosanna Keefe. Fine argues that borderline applications of vague predicates are neither true nor false, but rather are instances of " truth value gaps". He defends an interesting and sophisticated system of vague semantics, based on

19040-440: The information one has concerning whether a state obtains: information that it does obtain, information that it does not obtain, no information, and conflicting information. One of the most extreme forms of many-valued logic is fuzzy logic. It allows truth to arise in any degree between 0 and 1. 0 corresponds to completely false, 1 corresponds to completely true, and the values in between correspond to truth in some degree, e.g. as

19200-428: The internal cognitive model, i.e. the intrapsychic, stored and processed in human consciousness (and probably also in the unconscious), in hypothetical intrapsychic languages: imaginary, emotional and natural and in their mixture, and then the external model, represented in a suitable external language of communication. Cognition and language (Law of maintaining accuracy of information): Communication language should have

19360-422: The internal vagueness of connotation), tools such as classification schemes are used, such as the taxonomy of organisms by Carl von Linné . This is how descriptive (non-exact) sciences do it. Thus, they use natural human cognition (with the Russell’s filter of vagueness) and refined natural language. There is another continuation of the reduction of internal vagueness. The method of reducing internal vagueness to

19520-751: The intrusion of internal vagueness, that is, to choose some filter of cognition other than vagueness. Thus we pass from the natural human world to the artificial one. We call it the exact world and we will explain why. In the case of natural language, it is not possible to completely remove (nullify) the internal vagueness, but it is possible to build artificial formal languages (mathematics, formal logics, programming languages) that have zero internal vagueness of connotation (so they have an exact interpretation) and cannot have another in principle. (Newton for this purpose has created formal language – theory of flux - theory of flowing – infinitesimal calculus). Languages with zero internal vagueness of their interpretation, i.e.

19680-413: The linguistic utterance (of external communication language) is called external vagueness . Linguistically, only external vagueness can be grasped (modeled). We cannot model internal vagueness; it is part of the intrapsychic model, and this vagueness is contained in (vague, emotional, subjective and variable during time) interpretation of constructs (words, sentences) of informal language . This vagueness

19840-545: The literature is to distinguish between extended logics and deviant logics. Logic itself can be defined as the study of valid inference . Classical logic is the dominant form of logic and articulates rules of inference in accordance with logical intuitions shared by many, like the law of excluded middle , the double negation elimination , and the bivalence of truth. Extended logics are logical systems that are based on classical logic and its rules of inference but extend it to new fields by introducing new logical symbols and

20000-408: The logical mechanism by specifying which new rules of inference apply to them, like that possibility follows from necessity. Deviant logics, on the other hand, reject some of the basic assumptions of classical logic. In this sense, they are not mere extensions of it but are often formulated as rival systems that offer a different account of the laws of logic. Expressed in a more technical language,

20160-480: The material conditional "if all lemons are red then there is a sandstorm inside the Sydney Opera House" is true even though the two propositions are not relevant to each other. The fact that this usage of material conditionals is highly unintuitive is also reflected in informal logic , which categorizes such inferences as fallacies of relevance . Relevance logic tries to avoid these cases by requiring that for

20320-605: The material conditional found in classical logic. Classical logic defines the material conditional in purely truth-functional terms, i.e. " p → q {\displaystyle p\to q} " is false if " p {\displaystyle p} " is true and " q {\displaystyle q} " is false, but otherwise true in every case. According to this formal definition, it does not matter whether " p {\displaystyle p} " and " q {\displaystyle q} " are relevant to each other in any way. For example,

20480-458: The meaning of their linguistic constructions, have the property that all these constructions are understood by every appropriately educated person with absolutely precise, i.e. exact meaning. That is why they are part of the exact world. Thus, we have some language that is able to represent knowledge with zero internal vagueness. But these must first be acquired by adequate cognition, providing cognition also with zero internal vagueness, i.e. also from

20640-662: The meanings of their terms) and tautologies (i.e., statements devoid of content). In his Tractatus Logico-Philosophicus in 1921, Ludwig Wittgenstein proposed that statements that can be deduced by logical deduction are tautological (empty of meaning), as well as being analytic truths. Henri Poincaré had made similar remarks in Science and Hypothesis in 1905. Although Bertrand Russell at first argued against these remarks by Wittgenstein and Poincaré, claiming that mathematical truths were not only non-tautologous but were synthetic , he later spoke in favor of them in 1918: Everything that

20800-445: The method of truth tables cannot be employed because the law of the excluded middle is not assumed. A formula R is said to tautologically imply a formula S if every valuation that causes R to be true also causes S to be true. This situation is denoted R ⊨ S {\displaystyle R\models S} . It is equivalent to the formula R → S {\displaystyle R\to S} being

20960-477: The moon", then it is possible to infer any proposition from this system, like the proposition that "Spain is controlled by space-rabbits". Paraconsistent logics avoid this by using different rules of inference that make inferences in accordance with the principle of explosion invalid. An important motivation for using paraconsistent logics is dialetheism, i.e. the belief that contradictions are not just introduced into theories due to mistakes but that reality itself

21120-401: The most fundamental systems, like system K , include only the most fundamental axioms while other systems, like the popular system S5 , build on top of it by including additional axioms. In this sense, system K is an extension of first-order logic while system S5 is an extension of system K. Important discussions within philosophical logic concern the question of which system of modal logic

21280-408: The notion that a vague predicate might be "made precise" in many alternative ways. This system has the consequence that borderline cases of vague terms yield statements that are neither true, nor false. Given a supervaluationist semantics, one can define the predicate "supertrue" as meaning "true on all precisifications ". This predicate will not change the semantics of atomic statements (e.g. "Frank

21440-735: The operators appearing in them. According to them, for example, one can deduce " F ( R a i n y ( l o n d o n ) ) {\displaystyle F(Rainy(london))} " (it will be rainy in London at some time) from " G ( R a i n y ( l o n d o n ) ) {\displaystyle G(Rainy(london))} " . In more complicated forms of temporal logic, also binary operators linking two propositions are defined, for example, to express that something happens until something else happens. Temporal modal logic can be translated into classical first-order logic by treating time in

21600-533: The other hand, are treated in the usual way as ranging over the domain. This allows for expressions like " ¬ ∃ x ( x = s a n t a ) {\displaystyle \lnot \exists x(x=santa)} " (Santa Claus does not exist) to be true even though they are self-contradictory in classical logic. It also brings with it the consequence that certain valid forms of inference found in classical logic are not valid in free logic. For example, one may infer from " B e

21760-451: The overall formula can be deduced from the truth or falsity of each variable. A valuation is a function that assigns each propositional variable to either T (for truth) or F (for falsity). So by using the propositional variables A and B , the binary connectives ∨ {\displaystyle \lor } and ∧ {\displaystyle \land } representing disjunction and conjunction respectively, and

21920-423: The premises to a conclusion. Often the term "argument" is also used instead. An inference is valid if it is impossible for the premises to be true and the conclusion to be false. In this sense, the truth of the premises ensures the truth of the conclusion. This can be expressed in terms of rules of inference : an inference is valid if its structure, i.e. the way its premises and its conclusion are formed, follows

22080-399: The principle of bivalence of truth. Paraconsistent logics are logical systems able to deal with contradictions. They do so by avoiding the principle of explosion found in classical logic. Relevance logic is a prominent form of paraconsistent logic. It rejects the purely truth-functional interpretation of the material conditional by introducing the additional requirement of relevance: for

22240-598: The proposition "Ramirez goes jogging", then " O J ( r ) {\displaystyle OJ(r)} " means that Ramirez has the obligation to go jogging and " P J ( r ) {\displaystyle PJ(r)} " means that Ramirez has the permission to go jogging. Deontic logic is closely related to alethic modal logic in that the axioms governing the logical behavior of their operators are identical. This means that obligation and permission behave in regards to valid inference just like necessity and possibility do. For this reason, sometimes even

22400-402: The proposition "Socrates is wise", then " ◊ W ( s ) {\displaystyle \Diamond W(s)} " expresses the proposition "it is possible that Socrates is wise". In order to integrate these symbols into the logical formalism, various axioms are added to the existing axioms of first-order logic. They govern the logical behavior of these symbols by determining how

22560-438: The proposition "Socrates is wise", then " K W ( s ) {\displaystyle KW(s)} " expresses the proposition "the agent knows that Socrates is wise" and " B W ( s ) {\displaystyle BW(s)} " expresses the proposition "the agent believes that Socrates is wise". Axioms governing these operators are then formulated to express various epistemic principles. For example,

22720-399: The proposition "all bound things are books". A minimal tautology is a tautology that is not the instance of a shorter tautology. The problem of determining whether a formula is a tautology is fundamental in propositional logic. If there are n variables occurring in a formula then there are 2 distinct valuations for the formula. Therefore, the task of determining whether or not the formula

22880-413: The propositional tautology: ( ( A ∧ B ) → C ) ⇔ ( A → ( B → C ) ) {\displaystyle ((A\land B)\to C)\Leftrightarrow (A\to (B\to C))} . Not all logical validities are tautologies in first-order logic. For example, the sentence: is true in any first-order interpretation, but it corresponds to

23040-415: The question of whether all of these formal systems actually constitute logical systems. This is especially relevant for deviant logics that stray very far from the common logical intuitions associated with classical logic. In this sense, it has been argued, for example, that fuzzy logic is a logic only in name but should be considered a non-logical formal system instead since the idea of degrees of truth

23200-466: The question of whether there can be more than one true logic. Some theorists favor a local approach in which different types of logic are applied to different areas. Early intuitionists, for example, saw intuitionistic logic as the correct logic for mathematics but allowed classical logic in other fields. But others, like Michael Dummett , prefer a global approach by holding that intuitionistic logic should replace classical logic in every area. Monism

23360-431: The real and exact worlds that makes this possible is called a quantity (e.g. electric field intensity, velocity, nitric acid concentration, etc.). It is common to both worlds, because in the exact world it is precisely delineated (every knowing person knows them with no doubts, so exactly), and in the real world it is an elemental measurable probe into that, and thus its elemental measurable representative. The quantity

23520-435: The real world is represented by a group of suitably chosen quantities and mathematically (programming language) described relations between them (more precisely between their names – symbols denoting them). Exact science is a method that allows knowledge about the real world to be acquired and recorded so that it is part of the exact world. That is the method of modeling the real world by means of exact world, in other words,

23680-410: The researched reality and the acquired knowledge about it. People properly educated in the field of terminology know it with little internal vagueness, so they know accurately what the individual terms mean. Basic concepts are always formed on the basis of consensus, the other derived from them by definition, to avoid to Circular definition . To improve the accuracy of research and communication (reducing

23840-509: The same amount of vagueness, as have information gained by cognition (source of the information). That means, language must be tuned to appropriate cognition considering vagueness. This is one of secondary tasks of vagueness. A person is able to speak about his inherently vague knowledge (contained in the intrapsychic cognitive model represented in hypothetical intrapsychic languages) in natural (generally informal language, e.g. Esperanto), of course only vaguely. The vagueness of knowledge caused by

24000-399: The same symbols are used as operators. Just as in alethic modal logic, there is a discussion in philosophical logic concerning which is the right system of axioms for expressing the common intuitions governing deontic inferences. But the arguments and counterexamples here are slightly different since the meanings of these operators differ. For example, a common intuition in ethics is that if

24160-585: The sense that "true" means "verifiable". It was originally only applied to the area of mathematics but has since then been used in other areas as well. On this interpretation, the law of excluded middle would involve the assumption that every mathematical problem has a solution in the form of a proof. In this sense, the intuitionistic rejection of the law of excluded middle is motivated by the rejection of this assumption. This position can also be expressed by stating that there are no unexperienced or verification-transcendent truths. In this sense, intuitionistic logic

24320-443: The term to redundancies of propositional logic in 1921, borrowing from rhetoric , where a tautology is a repetitive statement. In logic, a formula is satisfiable if it is true under at least one interpretation, and thus a tautology is a formula whose negation is unsatisfiable. In other words, it cannot be false. Unsatisfiable statements, both through negation and affirmation, are known formally as contradictions . A formula that

24480-416: The truth of their expressions in terms of their denotation. But this option cannot be applied to all expressions in free logic since not all of them have a denotation. Three general approaches to this issue are often discussed in the literature: negative semantics , positive semantics , and neutral semantics . Negative semantics hold that all atomic formulas containing empty terms are false. On this view,

24640-457: The truth table method useless for formulas with thousands of propositional variables, as contemporary computing hardware cannot execute the algorithm in a feasible time period. The problem of determining whether there is any valuation that makes a formula true is the Boolean satisfiability problem ; the problem of checking tautologies is equivalent to this problem, because verifying that a sentence S

24800-439: The unary connective ¬ {\displaystyle \lnot } representing negation , the following formula can be obtained: ( A ∧ B ) ∨ ( ¬ A ) ∨ ( ¬ B ) {\displaystyle (A\land B)\lor (\lnot A)\lor (\lnot B)} . A valuation here must assign to each of A and B either T or F. But no matter how this assignment

24960-406: The use of 'tautology' to valid sentences of propositional logic, or valid sentences of predicate logic that can be reduced to propositional tautologies by substitution. Propositional logic begins with propositional variables , atomic units that represent concrete propositions. A formula consists of propositional variables connected by logical connectives, built up in such a way that the truth of

25120-725: The use of Newtonian discrete filter and thus the use of quantities, and artificial formal language. Artificial formal language also brings a powerful tool to exact science, which is formal inference (information formal processing) known from mathematics. The above-mentioned tools of exact and non-exact science are general principles, and different branches of science use them in combination with both. They have their parts exact and inexact. Purely exact sciences, such as theoretical physics or mathematics, use natural language as meta-language. Exact science provides most trustworthy knowledge. The question can certainly be raised as to whether all science can be transformed into an exact science. The answer

25280-411: The validity of an inference depends on the fact that these symbols are found in it. They usually include the idea that if a proposition is necessary then its negation is impossible, i.e. that " ◻ A {\displaystyle \Box A} " is equivalent to " ¬ ◊ ¬ A {\displaystyle \lnot \Diamond \lnot A} " . Another such principle

25440-558: Was the case that...), F {\displaystyle F} (it will be the case that...), H {\displaystyle H} (it has always been the case that...), and G {\displaystyle G} (it will always be the case that...). So to express that it will always be rainy in London one could use " G ( R a i n y ( l o n d o n ) ) {\displaystyle G(Rainy(london))} " . Various axioms are used to govern which inferences are valid depending on

25600-603: Was used by the ancient Greeks to describe a statement that was asserted to be true merely by virtue of saying the same thing twice, a pejorative meaning that is still used for rhetorical tautologies . Between 1800 and 1940, the word gained new meaning in logic, and is currently used in mathematical logic to denote a certain type of propositional formula, without the pejorative connotations it originally possessed. In 1800, Immanuel Kant wrote in his book Logic : The identity of concepts in analytical judgments can be either explicit ( explicita ) or non-explicit ( implicita ). In

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