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-- Am I a misogynist?
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Maybe you are in the wrong environment. I know my share of idiot women but I also know very intelligent women. From my experience, the two groups don't frequent the same places.

I don't like unintelligent people either but I notice stupidity in both sexes and do my best to stay clear of those people.

quote:

Originally posted by chimera66 Maybe you are in the wrong environment. I know my share of idiot women but I also know very intelligent women. From my experience, the two groups don't frequent the same places.

american psycho.. couch scene.

quote:

Originally posted by jennypie I'm telling you guys, it's just hormones. Christ, even WOMEN don't understand women. Crazy whores.

No excuse.

http://en.wikipedia.org/wiki/Valley_girl
http://en.wikipedia.org/wiki/Valspeak
Didn't exist before the 70's, and I'd assume hormones were the same back then.

quote:

Originally posted by Taz No excuse. http://en.wikipedia.org/wiki/Valspeak Didn't exist before the 70's, and I'd assume hormones were the same back then.

LOL, well I was referring more to the irrational thought process and uncontrollable emotions and all that other shit.

quote:

Originally posted by jennypie I'm telling you guys, it's just hormones. Christ, even WOMEN don't understand women. Crazy whores.

Actually, understanding women is not that difficult.

Here's how female logic works:

1. There's something a certain woman wants (X), and it implies that there's an intention to acquire it (Y). So, here's the first step:
   
   X → Y
   
   
2. Now, women can't just want something. There's the "charm" factor. So, even though a woman might want something, it must pretend not to want that little something.
   
   X → Y → �Y
   
   
3. But, if they pretend not to want something, they might not get that something that they pretend they don't want, but that in reality they're craving for. In that case, there must be something else that breaks the loop:
   
   X → Y → �Y / �Y = Z
   
   
4. This new variable, though, must be hidden from everyone but herself, because their charm will otherwise be blatantly phony. So, in that case:
   
   X → Y → �Y \ �Y = Z ⊂ K {x ∀ K / Z = �Y}
   
   
5. There are a few more steps, though. Fortunately, it's quite simple. As a fire raging in some mountain glen after long drought--and the dense forest is in a blaze, while the wind carries great tongues of fire in every direction--even so furiously did Achilles rage, wielding his spear as though he were a god, and giving chase to those whom he would slay, till the dark earth ran with blood. Or as one who yokes broad-browed oxen that they may tread barley in a threshing-floor--and it is soon bruised small under the feet of the lowing cattle--even so did the horses of Achilles trample on the shields and bodies of the slain. The axle underneath and the railing that ran round the car were bespattered with clots of blood thrown up by the horses' hoofs, and from the tyres of the wheels; but the son of Peleus pressed on to win still further glory, and his hands were bedrabbled with gore. NOW when they came to the ford of the full-flowing river Xanthus, begotten of immortal Jove, Achilles cut their forces in two: one half he chased over the plain towards the city by the same way that the Achaeans had taken when flying panic-stricken on the preceding day with Hector in full triumph; this way did they fly pell-mell, and Juno sent down a thick mist in front of them to stay them. The other half were hemmed in by the deep silver-eddying stream, and fell into it with a great uproar. The waters resounded, and the banks rang again, as they swam hither and thither with loud cries amid the whirling eddies. As locusts flying to a river before the blast of a grass fire--the flame comes on and on till at last it overtakes them and they huddle into the water--even so was the eddying stream of Xanthus filled with the uproar of men and horses, all struggling in confusion before Achilles.
   
   Forthwith the hero left his spear upon the bank, leaning it against a tamarisk bush, and plunged into the river like a god, armed with his sword only. Fell was his purpose as he hewed the Trojans down on every side. Their dying groans rose hideous as the sword smote them, and the river ran red with blood. As when fish fly scared before a huge dolphin, and fill every nook and corner of some fair haven--for he is sure to eat all he can catch--even so did the Trojans cower under the banks of the mighty river, and when Achilles' arms grew weary with killing them, he drew twelve youths alive out of the water, to sacrifice in revenge for Patroclus son of Menoetius. He drew them out like dazed fawns, bound their hands behind them with the girdles of their own shirts, and gave them over to his men to take back to the ships. Then he sprang into the river, thirsting for still further blood.
   
   There he found Lycaon, son of Priam seed of Dardanus, as he was escaping out of the water; he it was whom he had once taken prisoner when he was in his father's vineyard, having set upon him by night, as he was cutting young shoots from a wild fig-tree to make the wicker sides of a chariot. Achilles then caught him to his sorrow unawares, and sent him by sea to Lemnos, where the son of Jason bought him. But a guest-friend, Eetion of Imbros, freed him with a great sum, and sent him to Arisbe, whence he had escaped and returned to his father's house. He had spent eleven days happily with his friends after he had come from Lemnos, but on the twelfth heaven again delivered him into the hands of Achilles, who was to send him to the house of Hades sorely against his will. He was unarmed when Achilles caught sight of him, and had neither helmet nor shield; nor yet had he any spear, for he had thrown all his armour from him on to the bank, and was sweating with his struggles to get out of the river, so that his strength was now failing him.
   
   Then Achilles said to himself in his surprise, "What marvel do I see here? If this man can come back alive after having been sold over into Lemnos, I shall have the Trojans also whom I have slain rising from the world below. OMG, SHOES!!! Could not even the waters of the grey sea imprison him, as they do many another whether he will or no? This time let him taste my spear, that I may know for certain whether mother earth who can keep even a strong man down, will be able to hold him, or whether thence too he will return."
   
   Thus did he pause and ponder. But Lycaon came up to him dazed and trying hard to embrace his knees, for he would fain live, not die. Achilles thrust at him with his spear, meaning to kill him, but Lycaon ran crouching up to him and caught his knees, whereby the spear passed over his back, and stuck in the ground, hungering though it was for blood. With one hand he caught Achilles' knees as he besought him, and with the other he clutched the spear and would not let it go. Then he said, "Achilles, have mercy upon me and spare me, for I am your suppliant. It was in your tents that I first broke bread on the day when you took me prisoner in the vineyard; after which you sold me away to Lemnos far from my father and my friends, and I brought you the price of a hundred oxen. I have paid three times as much to gain my freedom; it is but twelve days that I have come to Ilius after much suffering, and now cruel fate has again thrown me into your hands. Surely father Jove must hate me, that he has given me over to you a second time.
   
   The term "fuzzy logic" emerged in the development of the theory of fuzzy sets by Lotfi Zadeh (1965). A fuzzy subset A of a (crisp) set X is characterized by assigning to each element x of X the degree of membership of x in A (e.g., X is a group of people, A the fuzzy set of old people in X). Now if X is a set of propositions then its elements may be assigned their degree of truth, which may be �absolutely true,� �absolutely false� or some intermediate truth degree: a proposition may be more true than another proposition. This is obvious in the case of vague (imprecise) propositions like �this person is old� (beautiful, rich, etc.). In the analogy to various definitions of operations on fuzzy sets (intersection, union, complement, �) one may ask how propositions can be combined by connectives (conjunction, disjunction, negation, �) and if the truth degree of a composed proposition is determined by the truth degrees of its components, i.e. if the connectives have their corresponding truth functions (like truth tables of classical logic). Saying �yes� (which is the mainstream of fuzzy logic) one accepts the truth-functional approach; this makes fuzzy logic to something distinctly different from probability theory since the latter is not truth-functional (the probability of conjunction of two propositions is not determined by the probabilities of those propositions).
   
   Two main directions in fuzzy logic have to be distinguished (cf. Zadeh 1994). Fuzzy logic in the broad sense (older, better known, heavily applied but not asking deep logical questions) serves mainly as apparatus for fuzzy control, analysis of vagueness in natural language and several other application domains. It is one of the techniques of soft-computing, i.e. computational methods tolerant to suboptimality and impreciseness (vagueness) and giving quick, simple and sufficiently good solutions. The monographs Novak 1989, Zimmermann 1991, Klir-Yuan 1996, Nguyen 1999 can serve as recommended sources of information.
   
   Fuzzy logic in the narrow sense is symbolic logic with a comparative notion of truth developed fully in the spirit of classical logic (syntax, semantics, axiomatization, truth-preserving deduction, completeness, etc.; both propositional and predicate logic). It is a branch of many-valued logic based on the paradigm of inference under vagueness. This fuzzy logic is a relatively young discipline, both serving as a foundation for the fuzzy logic in a broad sense and of independent logical interest, since it turns out that strictly logical investigation of this kind of logical calculi can go rather far. A basic monograph is Hajek 1998, further recommended monographs are Turunen 1999, Novak et al. 2000; also recent monographs dealing with many-valued logic (not specifically oriented to fuzziness), namely Gottwald 2001, Cignoli et al. 2000a; are highly relevant.
   
   The interested reader will find below some more information on fuzzy connectives and a survey of a logical system called basic fuzzy (propositional and predicate) logic together with three stronger systems � Łukasiewicz, G�del and product logic; a short discussion on paradoxes and fuzzy logic; some comments on other formal systems of fuzzy logic, complexity and, finally, a few remarks on fuzzy computing and bibliography.
   
   The standard set of truth degrees is the real interval [0,1] with its natural ordering ≤ (1 standing for absolute truth, 0 for absolute falsity); but one can work with different domains, finite or infinite, linearly or partially ordered. Truth functions of connectives have to behave classically on the extremal values 0,1.
   
   It is broadly accepted that t-norms (triangular norms) are possible truth functions of conjunction. (A binary operation * on the interval [0,1] is a t-norm if it is commutative, associative, non-decreasing and 1 is its unit element. Minimum (min(x,y) is the most popular t-norm. See the Glossary at the end.) Dually, t-conorms serve as truth functions of disjunction. See Klement et al. 2000 for an extensive theory of t-norms. The truth function of negation has to be non-increasing (and assign 0 to 1 and vice versa); the function 1 − x (Łukasiewicz negation) is the best known candidate.
   
   Implication is sometimes disregarded but is of fundamental importance for fuzzy logic in the narrow sense. A straightforward but logically less interesting possibility is to define implication from conjunction and negation (or disjunction and negation) using the corresponding tautology of classical logic; such implications are called S-implications. More useful and interesting are R-implications: an R-implication is defined as a residuum of a t-norm; denoting the t-norm * and the residuum ⇒ we have x ⇒ y = max{z| x*z ≤ y}. This is well-defined only if the t-norm is left-continuous.
   
   X → Y → �Y \ �Y = Z ⊂ K {x ∀ K / Z = �Y → LOL / !LOL ∃ h4x [4,5]}
   
   
6. Basic fuzzy propositional logic is the logic of continuous t-norms (developed in Hajek 1998). Formulas are built from propositional variables using connectives & (conjunction), → (implication) and truth constant 0 (denoting falsity). Negation � φ is defined as φ → 0. Given a continuous t-norm * (and hence its residuum ⇒) each evaluation e of propositional variables by truth degrees for [0,1] extends uniquely to the evaluation e*(φ) of each formula φ using * and ⇒ as truth functions of & and →.
   A formula φ is a t-tautology or standard BL-tautology if e*(φ) = 1 for each evaluation e and each continuous t-norm *. The following t-tautologies are taken as axioms of the logic BL:
   
   (A1) (φ → ψ) → ((ψ → χ) → (φ → χ))
   (A2) (φ & ψ) → φ
   (A3) (φ & ψ) → (ψ & φ)
   (A4) (φ & (φ → ψ)) → (ψ & (ψ → φ))
   (A5a) (φ → (ψ → χ)) → ((φ & ψ) → χ)
   (A5b) ((φ & ψ) → χ) → (φ → (ψ → χ))
   (A6) ((φ → ψ) → χ) → (((ψ → φ) → χ) → χ)
   (A7) 0 → φ
   
   Modus ponens is the only deduction rule; this gives the usual notion of proof and provability of the logic BL. The standard completeness theorem (Cignoli et al. 2000b) says that a formula φ is a t-tautology iff it is provable in BL. There is a more general semantics of BL, based on algebras called BL-algebras (see Hajek 1998 for definition); each BL-algebra can serve as the algebra of truth functions of BL. The general completeness theorem Hajek 1998 says that a formula φ is provable in BL iff it is a general BL-tautology, i.e., a tautology for each (linearly ordered) BL-algebra L.
   
   Basic fuzzy predicate logic has the same formulas as classical predicate logic (they are built from predicates of arbitrary arity using object variables, connectives &, →, truth constant 0 and quantifiers ∀, ∃. A standard interpretation is given by a non-empty domain M and for each n-ary predicate P by a n-ary fuzzy relation on M, i.e., a mapping assigning to each n-tuple of elements of M a truth value from [0,1] � the degree in which the n-tuple satisfies the atomic formula P(x1,�,xn). Given a continuous t-norm, this defines uniquely (in Tarski style) the truth degree ||φ|| of each closed formula φ given by the interpretation M and t-norm *. (The degree of an universally quantified formula ∀xφ is defined as the infimum of truth degrees of instances of φ; similarly ∃xφ and supremum. See the Glossary at the end of this entry.)
   
   This generalizes in an appropriate manner to a so called safe interpretation over any linearly ordered BL-algebra and definition of the truth value ||φ|| M,L given by the L-interpretation M. A formula is a general BL-tautology in the predicate logic BL∀ if its truth value is 1 in each safe interpretation.
   
   The following BL-tautologies are taken as axioms of BL∀: (a) axioms of the propositional logic BL, and
   
   (∀1) ∀xφ(x) → φ(y)
   (∃1) φ(y) → ∃xφ(x)
   (∀2) ∀x(χ→ψ) → (χ → ∀xψ)
   (∃2) ∀x(φ → χ) → (∃xφ → χ)
   (∀3) ∀x(φ vel χ) → (∀xφ vel χ)
   (where y is substitutable for x into φ and x is not free in χ).
   
   Deduction rules are modus ponens and generalization as in classical logic.
   
   The general completeness theorem says that a formula is provable in the fuzzy predicate logic BL∀ iff it is a general BL-tautology (of predicate logic). This generalizes in a natural way to provability in a theory over BL∀ and truth in all models of the theory; see Hajek 1998 for details. But note that standard BL-tautologies, i.e. formulas true in all standard interpretations w.r.t. all continuous t-norms are not recursively axiomatizable (see Hajek 2001a, Montagna 2001 for the final result).
   
   n classical logic, the liar paradox (sentence asserting its own falsity) relies on the fact that no formula can be equivalent to its own negation. In Łukasiewicz logic this is not the case: if φ has the value 0.5 then its negation �φ has the same value and is equivalent to φ. But one may ask if one can add to (classical) arithmetic a fuzzy truth predicate Tr satisfying, for formulas of this extended language, the disquotation schema
   
   φ ≡ Tr(φ), (where φ denotes the G�del number of φ)
   
   The answer is �yes and no�: you get a theory which is consistent but has no model expanding the standard natural numbers. This is discussed in Hajek et al. 2000; see also Grim et al. 1992.
   
   The Sorites paradox is related to notions like small, many etc.; considering them to be crisp (two-valued) leads to unnatural consequences. We shall sketch a treatment of the notion �small number� in fuzzy logic. (See Goguen 1968-69 for a �classic� analysis.) Without going into detail, imagine a theory inside fuzzy predicate calculus (BL∀ or other) containing crisp arithmetic of natural numbers (as above) and an additional predicate Small with the axioms saying that 0 is small (Small(0)), that Small respects ≤, i.e.,
   
   ∀x,y (x≤y → (Small(y) → Small(x))),
   
   and that for all x, the implication Small(x)→Small(x+1) is almost true; finally that there is a non small number, ∃x�Small(x). The �induction� condition can be expressed in various ways, e.g.,
   
   ∀x At(Small(x) → Small(x+1))
   
   where At is an unary connective �almost true�. Its truth function has to satisfy some natural conditions, in particular u→At(u). You can have At definable, introducing a new propositional constant r that should be interpreted by a truth value near to 1 and defining Atφ to be r→φ, thus the above formula becomes
   
   ∀x(r → (Small(x) → Small(x+1))), or equivalently
   ∀x((Small(x) & r) → Small(x+1)).
   
   You see that the theory admits many interpretations (and hence is consistent). All interpretations satisfy in some sense the following: the truth degree of Small(x+1) is only slightly less than (or equal to) the truth degree of Small(x). Thus the paradox can be handled in the frame of fuzzy logic in an axiomatic way, not enforcing any unique semantics. The semantics need not be numerical and the truth values need not be linearly ordered (there are BL algebras whose order is not linear).
   
   Several other notions can be handled similarly; for example the fuzzy notion probably can be axiomatized as a fuzzy modality. Having a probability on Boolean formulas, define for each such formula φ a new formula Pφ, read �probably φ�, and define the truth value of Pφ to be the probability of φ. One gets a reasonably elegant bridge between fuzziness and probability, with a simple axiom system over Łukasiewicz logic. See Hajek 1998; for an axiomatization of �very true� see Hajek 2001b.
   6. Other systems of fuzzy logic
   
   We mention a few:
   
   * Pavelka's logic. (Łukasiewicz with rational truth constants; see Pavelka 1979, Hajek 1998, Novak et al. 2000; V. Novak systematically develops this logic as a logic with evaluated syntax (working with pairs (formula, truth value)), fuzzy theories (sets of evaluated formulas) and fuzzy modus ponens [from (φ,u), (φ→ψ,v) derive (ψ,u*v) where * is Łukasiewicz t-norm]. This has excellent properties thanks to the fact that Łukasiewicz t-norm is the only continuous t-norm whose residuum is continuous. Expansions of other logics with truth constants were studied in Esteva et al. 2000, and recently in Esteva et al. 2006 and Savicky et al. 2006.
   * Expansions of basic logic BL by aditional connectives. These include logics with an additional involutive negation (Esteva et al. 2000), and logics putting Łukasiewicz and product logic together (Esteva & Godo 1999, Cintula 2001, Cintula 2003, Horcik & Citula 2004).
   * The monoidal t-norm based logic MTL. Introduced in Esteva & Godo 2001 as well as its predicate variant MTL∀. This is a generalization of the logic BL � a logic of left continuous t-norms. It has stronger variants IMTL and ΠMTL generalizing the Łukasiewicz and product logic. These logics are (strongly) complete with respect to corresponding algebras. For results on standard completeness of these logics, see Jenei & Montagna 2002 and (for ΠMTL) Horcik 2005.
   * Fuzzy logics with non-commutative conjunction. (φ&ψ not necessarily equivalent to ψ&φ). For details see di Nola et al. 2002, Hajek 2003, and for standard completeness, Jenei & Montagna 2003.
   * Logics with an additional involutive negation. See Esteva et al. 2000. For a logic putting Łukasiewicz product and G�del logic together (mentioned above), see Esteva et al. 1999 and Cintula 2001.
   
   To close this section let us mention a very general treatment of fuzzy logics in the frame of the so-called weakly implicative logics presented in Cintula 2006 and two recent survey papers: on t-norm based propositional logics Gottwald & Hajek 2005 and on t-norm based predicate logics Cintula and Hajek 2006.
   7. On fuzzy computing
   
   We briefly comment on so-called fuzzy IF-THEN rules as an example of fuzzy logic in a broad sense. They may be understood as partial imprecise knowledge on some crisp function and have (in the simplest case) the form IF x is Ai THEN y is Bi. They should not be immediately understood as implications; think of a table relating values of a (dependent) variable y to values of an (independent variable) x:
   
   x A1 � An
   y B1 � Bn
   
   Ai, Bi may be crisp (concrete numbers) or fuzzy (small, medium, �) It may be understood in two, in general non-equivalent ways:
   
   (1) as a listing of n possibilities, called Mamdani's formula:
   
   MAMD(x,y) ≡
   n
   bigvee
   i=1
   (Ai(x) & Bi(y)).
   
   (where x is A1 and y is B1 or x is A2 and y is B2 or �).
   
   (2) as a conjunction of implications:
   
   RULES(x,y) ≡
   n
   bigwedge
   i=1
   (Ai(x) → Bi(y)).
   
   (if x is A1 then y is B1 and �).
   
   Both MAMD and RULES define a binary fuzzy relation (given the interpretation of Ai's, Bi's and truth functions of connectives). Now given a fuzzy input A*(x) one can consider the image B* of A*(x) under this relation, i.e.,
   
   B*(y) ≡ ∃x(A(x) & R(x,y)),
   
   where R(x,y) is MAMD(x,y) (most frequent case) or RULES(x,y). Thus one gets an operator assigning to each fuzzy input set A* a corresponding fuzzy output B*. Usually this is combined with some fuzzifications converting a crisp input x0 to some fuzzy A*(x) (saying something as "x is similar to x0") and a defuzzification converting the fuzzy image B* to a crisp output y0. Thus one gets a crisp function; its relation to the set of rules may be analyzed. For detailed information on fuzzy control see Driankov et al. 1993. (But be sure not to call minimum "Mamdani implication" � minimum is not an implication at all! For logical analysis, see e.g., Hajek 2000.)
   8. Complexity
   
   For propositional logics it is always a natural question whether a logic is decidable, i.e., whether its set of tautologies is recursive, and if it is, whether it is in co-NP (its complement being non-neterministically computable in polynomial time). Similarly for the set of satisfiable formulas and NP. (Also sets of positive tautologies, i.e. formulas having a positive value in each evaluation and positively satisfiable formulas are discussed.) It has been shown that for our logics tautologies are co-NP-complete (of maximal complexity in co-NP) and satisfiable formulas are NP-complete. See Baaz et al. 2002 and Hanikova 2002 for final results.
   
   The corresponding predicate logics are undecidable (as is the classical predicate logic) but of various degree of undecidability in the sense of so-called arithmetical hierarchy of Σn-sets and Πn-sets. For the reader knowing this hierarchy we mention that for example the set of standard predicate tautologies of G�del logic is Σ1-complete, for Łukasiewicz it is Π2-complete and for product logic it is non-arithmetical (outside the arithmetical hierarchy). Not surprisingly, the set of general predicate tautologies of each of these logics is Σ1-complete (due to completeness theorem). Much more is known; see Hajek 2005 for a survey of known results. Most difficult results on non-arithmeticity were obtained by Montagna 2001 and Montagna 2005.
   9. Glossary
   
   To help the reader not familiar with the basic notions of higher mathematics I comment here on two notions used:
   
   Continuous t-norm. A t-norm is a particular operation x*y with arguments and values in the real unit interval [0,1]. Such an operation is continuous, intuitively speaking, if small changes of the arguments lead only to small changes of the result of the operation. Precisely, for each ε > 0 there is a δ > 0 such that wherever |x1 − x2| < δ and |y1 − y2| < δ then |(x1*y1) − (x2*y2)| < ε.
   
   Infimum and supremum of a subset of the real unit interval [0,1]. Let A be a set of truth values, hence a subset of [0,1]. A truth value x is a lower bound of A if x ≤ y for each element y of A; it is the infimum of A if it is the largest lower bound (notation: x = inf(A)). Clearly, if A has a least element then this element is its infimum; but if A has no least element then its infimum is not its element. For example if A is the set of all positive truth values (x > 0) then inf(A)=0. Dually, x is an upper bound of A if x ≥ y for all y in A; the supremum of A is its least upper bound.
   
   1. Motivations for hybrid logic
   
   In the standard Kripke semantics for modal logic, truth is relative to points in a set. Thus, a propositional symbol might have different truth-values relative to different points. Usually, these points are taken to represent possible worlds, times, space, knowledge, states in a computer, or something else. This allows us to formalize natural language statements whose truth-values are relative to for example times, like the statement
   
   it is raining
   
   which has clearly different truth-values at different times. Now, certain natural language statements are true at exactly one time, possible world, or something else. An example is the statement
   
   it is five o'clock 15 March 2006
   
   which is true at the time five o'clock 15 March 2006, but false at all other times. The first kind of natural language statements can be formalized in ordinary modal logic, but the second kind cannot.
   
   A major motivation for hybrid logic is to add further expressive power to ordinary modal logic with the aim of being able to formalize the second kind of statements. This is obtained by adding to ordinary modal logic a second sort of propositional symbols called nominals such that in the Kripke semantics each nominal is true relative to exactly one point. A natural language statement of the second kind (like the example statement with the time five o'clock 15 March 2006) is then formalized using a nominal, not an ordinary propositional symbol (which would be used to formalize the example statement with rainy weather). The fact that a nominal is true relative to exactly one point implies that a nominal can be considered a term referring to a point, for example, if a is a nominal that stands for "it is five o'clock 15 March 2006", then the nominal a can be considered a term referring to the time five o'clock 15 March 2006. Thus, in hybrid logic a term is a specific sort of propositional symbol whereas in first-order logic it is an argument to a predicate.
   
   Most hybrid logics involve further additional machinery than nominals. There is a number of options for adding further machinery; here we shall consider so-called satisfaction operators. The motivation for adding satisfaction operators is to be able to formalize a statement being true at a particular time, possible world, or something else. For example, we want to be able to formalize that the statement "it is raining" is true at the time five o'clock 15 March 2006, that is, that
   
   at five o'clock 15 March 2006, it is raining.
   
   This is formalized by the formula a:p where the nominal a stands for "it is five o'clock 15 March 2006" as above and where p is an ordinary propositional symbol that stands for "it is raining". It is the part a: of the formula a:p that is called a satisfaction operator. In general. if a is a nominal and φ is an arbitrary formula, then a new formula a:φ called a satisfaction statement can be built (some authors use the notation @a instead of a:). The satisfaction statement a:φ expresses that the formula φ is true relative to one particular point, namely the point to which the nominal a refers.
   
   To sum up, we have now added further expressive power to ordinary modal logic in the form of nominals and satisfaction operators. Informally, the nominal a has the truth-condition
   
   a is true relative to a point w
   if and only if
   the reference of a is identical to w
   
   and the satisfaction statement a:φ has the truth-condition
   
   a:φ is true relative to a point w
   if and only if
   φ is true relative to the reference of a
   
   Note that actually the point w does not matter in the truth-condition for a:φ since the satisfaction operator a: moves the point of evaluation to the reference of a whatever the identity of w.
   
   It is remarkable that nominals together with satisfaction operators allow us to express that two points are identical: If the nominals a and b refer to the points w and v, then the formula a:b expresses that w and v are identical. The following line of reasoning shows why.
   
   a:b is true relative to a point w
   if and only if
   b is true relative to the reference of a
   if and only if
   b is true relative to w
   if and only if
   the reference of b is identical to w
   if and only if
   v is identical to w
   
   The identity relation on a set has the well-known properties reflexivity, symmetry, and transitivity, which is reflected in the fact that the formulas
   
   a:a
   a:b → b:a
   (a:b & b:c) → a:c
   
   are valid formulas of hybrid logic. Also the formula
   
   (a:b & a:φ) → b:φ
   
   is valid. This is the rule of replacement.
   
   Beside nominals and satisfaction operators, in what follows we shall consider the so-called binders ∀ and ↓ allowing us to build formulas ∀aφ and ↓aφ. The binders bind nominals to points in two different ways: The ∀ binder quantifies over points analogous to the standard first-order universal quantifier, that is, ∀aφ is true relative to w if and only if whatever point the nominal a refers to, it is the case that φ is true relative to w. The ↓ binder binds a nominal to the point of evaluation, that is, ↓aφ is true relative to w if and only if φ is true relative to w when a refers to w. It turns out that the ↓ binder is definable in terms of ∀ (as shown below).
   2. Formal semantics
   
   The language we consider is the language of ordinary modal logic built over ordinary propositional symbols p, q, r, � as well as nominals a, b, c, � and extended with satisfaction operators and binders. We take the propositional connectives & and � to be primitive; other propositional connectives are defined as usual. Similarly, we take the modal operator □ to be primitive and define the modal operator ◊ as �□�. As the name suggests, binders bind nominals and the notions of free and bound occurrences of nominals are defined analogously to first-order logic. Satisfaction operators do not bind nominals, that is, the free nominal occurrences in a formula a:φ are the free nominal occurrences in φ together with the occurrence of a. We let φ[c/a] be the formula φ where the nominal c has been substituted for all free occurrences of the nominal a. It is assumed that the nominal a does not occur free in φ within the scope of ∀c or ↓c.
   
   We now define models and frames. A model for hybrid logic is a triple (W, R, V) where W is a non-empty set, R is a binary relation on W, and V is a function that to each pair consisting of an element of W and an ordinary propositional symbol assigns an element of the set {0,1}. The pair (W, R) is called a frame. Thus, models and frames are the same as in ordinary modal logic. The elements of W are called worlds and the relation R is called the accessibility relation. The model (W, R, V) is said to be based on the frame (W, R).
   
   An assignment for a model M = (W, R, V) is a function g that to each nominal assigns an element of W. An assignment g′ is an a-variant of g if g′ agrees with g on all nominals save possibly a. The relation M, g, w models φ is defined by induction, where g is an assignment, w is an element of W, and φ is a formula.
   
   M, g, w models p iff V(w, p)=1
   M, g, w models a iff w = g(a)
   M, g, w models φ & ψ iff M, g, w models φ and M, g, w models ψ
   M, g, w models � φ iff not M, g, w models φ
   M, g, w models □φ iff for any element v of W such that wRv, it is the case that M, g, w models φ
   M, g, w models a:φ iff M, g, g(a) models φ
   M, g, w models ∀aφ iff for any a-variant g′ of g, it is the case that M, g′, w models φ
   M, g, w models ↓aφ iff M, g′, w models φ where g′ is the a-variant of g such that g′(a) = w.
   
   A formula φ is said to be true at w if M, g, w models φ; otherwise it is said to be false at w. By convention M, g models φ means M, g, w models φ for every element w of W and M models φ means M, g models φ for every assignment g. A formula φ is valid in a frame if and only if M models φ for any model M that is based on the frame in question. A formula φ is valid in a class of frames F if and only if φ is valid in any frame in F. A formula φ is valid if and only if φ is valid in the class of all frames. The definition of satisfiability is left to the reader.
   
   Note that the binder ↓ is definable in terms of ∀ as the formula ↓aφ ↔ ∀a(a → φ) is valid in any frame.
   
   The fact that hybridizing ordinary modal logic actually does give more expressive power can for example be seen by considering the formula ↓c□�c. It is straightforward to check that this formula is valid in a frame if and only if the frame is irreflexive. Thus, irreflexivity can be expressed by a hybrid-logical formula, but it is well known that it cannot be expressed by any formula of ordinary modal logic. Irreflexivity can actually be expressed just by adding nominals to ordinary modal logic, namely by the formula c→□�c. Other examples of properties expressible in hybrid logic, but not in ordinary modal logic, are asymmetry (expressed by c→□�◊c), antisymmetry (expressed by c→□(◊c→c)), and universality (expressed by ◊c).
   3. Translations
   
   Hybrid logic can be translated into first-order logic with equality, and (a fragment of) first-order logic with equality can be translated back into (a fragment of) hybrid logic. The first-order language under consideration has a 1-place predicate symbol p* corresponding to each ordinary propositional symbol p of modal logic, a 2-place predicate symbol R, and a 2-place predicate symbol =. Of course, the predicate symbol p* will be interpreted such that it relativises the interpretation of the corresponding modal propositional symbol p to worlds, the predicate symbol R will be interpreted using the accessibility relation, and the predicate symbol = will be interpreted using the identity relation on worlds. We let a, b, c, � range over first-order variables. The language does not have constant or function symbols. We shall identify first-order variables with nominals of hybrid logic.
   
   We first translate hybrid logic into first-order logic with equality. Given two new first-order variables a and b, the translations STa and STb are defined by mutual recursion. We just give the translation STa.
   
   STa(p) = p*(a)
   STa(c) = a=c
   STa(φ & ψ) = STa(φ) & STa(ψ)
   STa(�φ) = �STa(φ)
   STa(□φ) = ∀b(R(a, b) → STb(φ))
   STa(c:φ) = STa(φ)[c/a]
   STa(↓cφ) = STa(φ)[a/c]
   STa(∀cφ) = ∀cSTa(φ)
   
   The definition of STb is obtained by exchanging a and b. The translation is an extension of the well-known standard translation from modal logic into first-order logic. As an example, we demonstrate step by step how the hybrid-logical formula ↓c□�c is translated into a first-order formula:
   
   STa(↓c□�c) = STa(□�c)[a/c]
   = ∀b(R(a, b) → STb(�c))[a/c]
   = ∀b(R(a, b) → �STb(c))[a/c]
   = ∀b(R(a, b) → �b=c)[a/c]
   = ∀b(R(a, b) → �b=a).
   
   The resulting first-order formula is equivalent to �R(a, a) which shows that ↓c□�c indeed does correspond to the accessibility relation being irreflexive, cf. above.
   
   First-order logic with equality can be translated back into hybrid logic by the translation HT given below.
   
   HT(p*(a)) = a:p
   HT(R(a, c)) = a:◊c
   HT(a=c) = a:c
   HT(φ & ψ) = HT(φ) & HT(ψ)
   HT(�φ) = �HT(φ)
   HT(∀aφ) = ∀aHT(φ)
   
   Note that the hybrid-logical binder ∀ is needed. The history of the above mentioned observations goes back to the work of Arthur N. Prior, we shall return to that later.
   
   Similarly, the so-called bounded fragment of first-order logic can be translated into the hybrid logic but here only the binder ↓ is needed, as pointed out in the paper Areces, Blackburn, and Marx (2001). The bounded fragment is the fragment of first-order logic with the property that quantifiers only occur as in the formula ∀c(R(a, c) → φ), where it is required that the variables a and c are different. A translation from the bounded fragment to the hybrid logic without the ∀ binder can be obtained by replacing the last clause in the translation HT above by
   
   HT(∀c(R(a, c) → φ)) = a:□↓cHT(φ).
   
   In Areces, Blackburn, and Marx (2001) a number of independent semantic characterisations of the bounded fragment are given.
   
   The translations given above are truth-preserving. To state this formally, one makes use of the well-known observation that models and assignments for hybrid logic can be considered as models and assignments for first-order logic and vice versa. These truth-preservation results are straightforward to formulate and we leave the details to the reader. Thus, the hybrid logic with the binder ∀ has the same expressive power as first-order logic with equality and the hybrid logic without the binder ∀ (but with the binder ↓) has the same expressive power as the bounded fragment of first-order logic (note that the translation STa(φ) of any formula φ without the binder ∀ is in the bounded fragment).
   4. Arthur N. Prior and hybrid logic
   
   The history of hybrid logic goes back to Arthur N. Prior's hybrid tense logic, which is a hybridized version of ordinary tense logic. With the aim of investigating this further, we shall give a formal definition of hybrid tense logic: The language of hybrid tense logic is simply the language of hybrid logic defined above except that there are two modal operators, namely G and H, instead of the single modal operator □. The two new modal operators are called tense operators. The semantics of hybrid tense logic is the semantics of hybrid logic, cf. earlier, with the clause for □ replaced by clauses for the tense operators G and H.
   
   M, g, w models Gφ iff for any element v of W such that wRv, it is the case that M, g, w models φ
   M, g, w models Hφ iff for any element v of W such that vRw, it is the case that M, g, w models φ
   
   Thus, there are now two modal operators, namely one that �looks forwards� along the accessibility relation and one that �looks backwards�. In tense logic the elements of the set W are called moments or instants and the relation R is called the earlier-later relation.
   
   Of course, it is straightforward to modify the translations STa and HT above such that translations are obtained between hybrid tense logic (including the ∀ binder) and first-order logic with equality. The first-order logic under consideration is what Prior called first-order earlier-later logic. Given the translations, it follows that Prior's first-order earlier-later logic has the same expressive power as hybrid tense logic.
   
   Now, Prior introduced hybrid tense logic in connection with what he called four grades of tense-logical involvement. The motivation for his four grades of tense-logical involvement was philosophical. The four grades were presented in the book Prior (1968), Chapter XI (also Chapter XI in the new edition Prior (2003)). Moreover, see Prior (1967), Chapter V.6 and Appendix B.3-4. For a more general discussion, see the posthumously published book Prior and Fine (1977). The stages progress from what can be regarded as pure first-order earlier-later logic to what can be regarded as pure tense logic; the goal being to be able to consider the tense logic of the fourth stage as encompassing the earlier-later logic of the first stage. In other words, the goal was to be able to translate the first-order logic of the earlier-later relation into tense logic. It was with this goal in mind Prior introduced so-called instant-propositions:
   
   What I shall call the third grade of tense-logical involvement consists in treating the instant-variables a, b, c, etc. as also representing propositions. (Prior 1968, p. 122-123)
   
   In the context of modal logic, Prior called such propositions possible-world-propositions. Of course, this is what we here call nominals. Prior also introduced the binder ∀ and what we here call satisfaction operators (he used the notation T(a, φ) instead of a:φ for satisfaction operators). In fact, Prior's third grade tense logic is identical to the hybrid tense logic as defined above. The ↓ binder was introduced much later. Thus, Prior obtained the expressive power of his first-order earlier-later logic by adding to ordinary tense logic further expressive power in the form of nominals, satisfaction operators, and the binder ∀. So from a technical point of view he clearly reached his goal.
   
   However, from a philosophical point of view it has been debated whether or not the ontological import of his third grade tense logic is the same as the ontological import of the first-order earlier-later logic. For example, the ∀ binder is by some authors considered a direct analogy to the first-order ∀ quantifier, and therefore suspect; see for example the paper Sylvan (1996) in the collection Copeland (1996). Also a number of other papers in this collection are relevant. See Bra�ner (2002) for a discussion of Prior's fourth grade tense logic. See also �hrstr�m and Hasle (1993) as well as �hrstr�m and Hasle (2006). Moreover, see the forthcoming paper Blackburn (2007).
   5. The development of hybrid logic since Prior
   
   The first completely formal definition of hybrid logic was given in Robert Bull's paper Bull (1970) which appeared in a special issue of the journal Theoria in memory of Prior. Bull introduces a third sort of propositional symbols where a propositional symbol is assumed to be true exactly at one branch ("course of events") in a branching time model. This idea of sorting propositional symbols according to restrictions on their interpretations has later been developed further by a number of authors, see Section 5 of the paper Blackburn and Tzakova (1999) for a discussion.
   
   The hybrid logical machinery originally invented by Prior in the late 1960s was reinvented in the 1980s by Solomon Passy and Tinko Tinchev from Bulgaria, see Passy and Tinchev (1985) as well as Passy and Tinchev (1991). Rather than ordinary modal logic, this work took place in connection with the much more expressive Propositional Dynamic Logic.
   
   A major contribution in the 1990s was the introduction of the ↓ binder by Valentin Goranko, see the papers Goranko (1994) and Goranko (1996). Since then, hybrid logic with the ↓ binder has been extensively studied by a number of people, notably Patrick Blackburn and his collaborators, see for example the paper Areces, Blackburn, and Marx (2001) on model-theoretic aspects of this logic. A recent very comprehensive study of the model-theory of hybrid logic is the Ph.D. thesis of ten Cate (2004).
   
   Also the weaker hybrid logic obtained by omitting both of the binders ↓ and ∀ has been the subject of extensive exploration. It turns out that this binder-free logic and a number of variants of it are decidable. In the paper Areces, Blackburn, and Marx (1999), a number of complexity results are given for hybrid modal and tense logics over various classes of frames, for example arbitrary, transitive, linear, and branching. It is remarkable that the satisfiability problem of the binder-free hybrid logic over arbitrary frames is decidable in PSPACE, which is the same as the complexity of deciding satisfiability in ordinary modal logic. Thus, hybridizing ordinary modal logic gives more expressive power, but the complexity stays the same. Some work has been carried out on simulating nominals inside modal logic, see Kracht and Wolter (1997).
   
   It is remarkable that first-order hybrid logic offers precisely the features needed to prove interpolation theorems: While interpolation fails in a number of well-known first-order modal logics, their hybridized counterparts have this property, see Areces, Blackburn, and Marx (2003) as well as Blackburn and Marx (2003). The first paper gives a model-theoretic proof of interpolation whereas the second paper gives an algorithm for calculating interpolants based on a tableau system.
   
   It should also be mentioned that logics similar to hybrid logics play a central role within the area of description logic, which is a family of logics used for knowledge representation in Artificial Intelligence, see the paper Blackburn and Tzakova (1998) and Carlos Areces' Ph.D. thesis (2000).
   6. Proof methods for hybrid logic
   
   A number of papers have dealt with axioms for hybrid logic, for example Blackburn (1993) and Blackburn and Tzakova (1999) The latter paper gives an axiom system for hybrid logic and shows the remarkable result that if the axiom system is extended with a set of additional axioms which are pure formulas (that is, formulas where all propositional symbols are nominals), then the extended axiom system is complete with respect to the class of frames validating the axioms in question. Pure formulas correspond to first-order conditions on the accessibility relation (cf. the translation STa above), so axiom systems for new hybrid logics with first-order conditions on the accessibility relation can be obtained in a uniform way simply by adding axioms as appropriate. So, if for example the formula ↓c□�c is added as an axiom, then the resulting system is complete with respect to irreflexive frames, cf. earlier. The Ph.D. thesis ten Cate (2004) investigates orthodox proof-rules (which are proof-rules without side-conditions) in axiom systems, and it is shown that if one requires extended completeness using pure formulas, then unorthodox proof-rules are indispensable in axiom systems for binder-free hybrid logic, but an axiom system can be given only involving orthodox proof-rules for the stronger hybrid logic including the ↓ binder. See also Bra�ner (2006) which gives another axiom system for hybrid logic as well as axiom systems for intuitionistic and paraconsistent hybrid logic.
   
   Tableau, Gentzen, and natural deduction style proof-theory for hybrid logic work very well compared to ordinary modal logic. Usually, when a modal tableau, Gentzen, or natural deduction system is given, it is for one particuler modal logic and it has turned out to be problematic to formulate such systems for modal logics in a uniform way without introducing metalinguistic machinery. This can be remedied by hybridization, that is, hybridization of modal logics enables the formulation of uniform tableau, Gentzen, and natural deduction systems for wide classes of logics. The paper Blackburn (2000) introduces a tableau system for hybrid logic that has this desirable feature: Analogous to the axiom system of Blackburn and Tzakova (1999), completeness is preserved if the tableau system is extended with a set of pure axioms, that is, a set of pure formulas that are allowed to be added to a tableau during the tableau construction. It should be mentioned that the tableau system of Blackburn (2000) is the basis for a decision procedure for the binder-free fragment of hybrid logic given in Bolander and Bra�ner (2006).
   
   Natural deduction style proof-theory of hybrid logic has been explored in the papers Bra�ner (2004a), Bra�ner (2004b), Bra�ner (2005a), and Bra�ner (2005b). The paper Bra�ner (2004a) also gives a Gentzen system for hybrid logic. These natural deduction and Gentzen systems can be extended with additional proof-rules corresponding to first-order conditions on the accessibility relations expressed by so-called geometric theories (this is of course analogous to extending tableau and axiom systems with pure axioms). See also Bra�ner and de Paiva (2006) where a natural deduction system is given for intuitionistic hybrid logic. In the context of situation theory, Gentzen and natural deduction systems for logics similar to hybrid logics were explored already in the early 1990s by Jerry Seligman, see the overview in Seligman (2001).
   
   Work in resolution calculi and model checking is in the beginning, see Areces, de Rijke, and de Nivelle (2001) as well as Franceschet and de Rijke (2006).
   
   Since the mid 1990s, the work on hybrid logic has flourished. For a detailed overview, see the forthcoming handbook chapter Areces and ten Cate (2006).We refer the reader to the publications in the bibliography for further references. Moreover, see the internet resources below.
   
   
   X → Y → �Y \ �Y = Z ⊂ K {x ∀ K / Z = �Y → LOL / !LOL ∃ h4x [4,5]} → X → Y → �Y \ �Y = Z ⊂ K {x ∀ K / Z = �Y → LOL / !LOL ∃ h4x [3,16]}
   
   
7. 1. The history of provability logic
   
   Two strands of research have led to the birth of provability logic. The first one stems from a paper by K. G�del (1933), where he introduces a translation from intuitionistic propositional logic into modal logic (more precisely, into the system nowadays called S4), and briefly mentions that provability can be viewed as a modal operator. Even earlier, C.I. Lewis started the modern study of modal logic by introducing strict implication as a kind of deducibility, where he may have meant deducibility in a formal system like Principia Mathematica, but this is not clear from his writings.
   
   The other strand starts from research in meta-mathematics: what can mathematical theories say about themselves by encoding interesting properties? In 1952, L. Henkin posed a deceptively simple question inspired by G�del's incompleteness theorems. In order to formulate Henkin's question, some more background is needed. As a reminder, G�del's first incompleteness theorem states that, for a sufficiently strong formal theory like Peano Arithmetic, any sentence asserting its own unprovability is in fact unprovable. On the other hand, from "outside" the formal theory, one can see that such a sentence is true in the standard model, pointing to an important distinction between truth and provability.
   
   More formally, let l-corner-quoteAr-corner-quote denote the G�del number of arithmetical formula A, the result of assigning a numerical code to A. Let Prov be the formalized provability predicate for Peano Arithmetic, which is of the form ∃p Proof (p,x). Here, Proof is the formalized proof predicate of Peano Arithmetic, and Proof(p,x) stands for G�del number p codes a correct proof from the axioms of Peano Arithmetic of the formula with G�del number x. (For a more precise formulation, see Smorynski (1985), Davis (1958)). Now, suppose that Peano Arithmetic proves A ↔negProvl-corner-quoteAr-corner-quote, then by G�del's result, A is not provable in Peano Arithmetic, and thus it is true, for in fact the self-referential sentence A states �I am not provable�.
   
   Henkin on the other hand wanted to know whether anything could be said about sentences asserting their own provability: supposing that Peano Arithmetic proves B ↔Prov(l-corner-quoteBr-corner-quote), what does this imply about B? Three years later, M. L�b took up the challenge and answered Henkin's question in a surprising way. Even though all sentences provable in Peano Arithmetic are indeed true about the natural numbers, L�b showed that the formalized version of this fact, Prov(l-corner-quoteBr-corner-quote) → B, can only be proved in Peano Arithmetic in the trivial case that Peano Arithmetic already proves B itself. This result, now called L�b's theorem, immediately answers Henkin's question. (For a proof of L�b's theorem, see section 4.) L�b also showed a formalized version of his theorem, namely that Peano Arithmetic proves
   
   Prov(l-corner-quoteProv(l-corner-quoteBr-corner-quote) → Br-corner-quote)→ Prov(l-corner-quoteBr-corner-quote).
   
   In the same paper, L�b formulated three conditions on the provability predicate of Peano Arithmetic, that form a useful modification of the complicated conditions that Hilbert and Bernays introduced in 1939 for their proof of G�del's second incompleteness theorem. In the following, derivability of A from Peano Arithmetic is denoted by PAprovesA:
   
   1. If PAprovesA, then PAprovesProv(l-corner-quoteAr-corner-quote);
   2. PAprovesProv(l-corner-quoteA→Br-corner-quote) →(Prov(l-corner-quoteAr-corner-quote) → Prov(l-corner-quoteBr-corner-quote));
   3. PAprovesProv(l-corner-quoteAr-corner-quote) → Prov(l-corner-quoteProv(l-corner-quoteAr-corner-quote)r-corner-quote).
   
   These L�b conditions, as they are called nowadays, seem to cry out for a modal logical investigation, where the modality □ stands for provability in PA. Ironically, the first time that the formalized version of L�b's theorem was stated as the modal principle
   
   □(□ A →A) → □ A
   
   was in a paper by Smiley in 1963 about the logical basis of ethics, which did not consider arithmetic at all. More relevant investigations, however, only seriously started almost twenty years after publication of L�b's paper. The early seventies saw the rapid development of propositional provability logic, where several researchers in different countries independently proved the most important results, discussed in sections 2, 3, and 4. Propositional provability logic turned out to capture exactly what many formal theories of arithmetic can say by propositional means about their own provability predicate. Recently, researchers have investigated the boundaries of this approach and have proposed several interesting more expressive extensions of provability logic (see section 5).
   2. The axiom system of propositional provability logic
   
   The logical language of propositional provability logic contains, in addition to propositional atoms and the usual truth-functional operators as well as the contradiction symbol ⊥, a modal operator □ with intended meaning �is provable in T,� where T is a sufficiently strong formal theory, let us say Peano Arithmetic (see section 4). ◊ A is an abbreviation for � □� A. Thus, the language is the same as that of modal systems such as K and S4 presented in the entry modal logic.
   2.1 Axioms and rules
   
   Propositional provability logic is often called GL, after G�del and L�b. (Alternative names found in the literature for equivalent systems are L, KW, K4W, and PrL). The logic GL results from adding the following axiom to the basic modal logic K:
   
   (GL) □(□ A →A) → □ A.
   
   As a reminder, because GL extends K, it contains all formulas having the form of a propositional tautology. For the same reason, GL contains the
   
   Distribution Axiom: □(A→ B) → (□A→ □B).
   
   Furthermore, it is closed under the Modus Ponens Rule allowing to derive B from A →B and B, and the Generalization Rule, which says that if A is a theorem of GL, then so is □A.
   
   The notion GL proves A denotes provability of a modal formula A in propositional provability logic. It is not difficult to see that the modal axiom □A→ □ □A (known as axiom 4 of modal logic) is indeed provable in GL. To prove this, one uses the substitution Awedge□A for A in the axiom (GL), and applies the Distribution Axiom and the Generalization Rule as well as some propositional logic. Unless explicitly stated otherwise, in the sequel �provability logic� stands for the system GL of propositional provability logic.
   2.2 The fixed point theorem
   
   The main "modal" result about provability logic is the fixed point theorem, which D. de Jongh and G. Sambin independently proved in 1975. Even though it is formulated and proved by strictly modal methods, the fixed point theorem still has great arithmetical significance. It says essentially that self-reference is not really necessary, in the following sense. Suppose that all occurrences of the propositional variable p in a given formula A(p) are under the scope of the provability operator, for example A(p) = � □p, or A(p) = □(p →q). Then there is a formula B in which p does not appear, such that all propositional variables that occur in B already appear in A(p), and such that
   
   GL proves B ↔ A(B).
   
   This formula B is called a fixed point of A(p). Moreover, the fixed point is unique, or more accurately, if there is another formula C such that GL proves C ↔ A(C), then we must have GL proves B↔C. Most proofs in the literature give an algorithm by which one can compute the fixed point (see Smorynski 1985, Boolos 1993).
   
   For example, suppose that A(p) = � □p. Then the fixed point produced by such an algorithm is � □⊥, and indeed one can prove that
   
   GL proves � □ ⊥ ↔ � □(� □⊥).
   
   If this is read arithmetically, the direction from left to right is just the formalized version of G�del's second incompleteness theorem: if a sufficiently strong formal theory T like Peano Arithmetic does not prove a contradiction, then it is not provable in T that T does not prove a contradiction. Thus, sufficiently strong consistent arithmetical theories cannot prove their own consistency. We will turn to study the relation between provability logic and arithmetic more precisely in section 4, but to that end another �modal� aspect of GL needs to be provided first: semantics.
   3. Possible worlds semantics
   
   Provability logic has suitable possible worlds semantics, just like many other modal logics. As a reminder, a possible worlds model (or Kripke model) is a triple M = langl10W,R,Vrangl10, where W is a set of possible worlds, R is a binary accessibility relation on W, and V is a valuation that assigns a truth value to each propositional variable for each world in W. The pair F = langl10W,Rrangl10 is called the frame of this model. The notion of truth of a formula A in a model M at a world W, notation M,w models A, is defined inductively. Let us repeat only the most interesting clause, the one for the provability operator □:
   
   M,wmodels □A iff for every w′, if wRw′, then M,w′ modelsA.
   
   See the entry modal logic for more information about possible worlds semantics in general.
   3.1 Characterization and modal soundness
   
   The modal logic K is valid in all Kripke models. Its extension GL however, is not: we need to restrict the class of possible worlds models to a more suitable one. Let us say that a formula A is valid in frame F, notation FmodelsA, iff A is true in all worlds in Kripke models M based on F. It turns our that the new axiom (GL) of provability logic corresponds to a condition on frames, as follows:
   
   For all frames F = , Fmodels □(□ p→p) → □ p iff R is transitive and conversely well-founded.
   
   Here, transitivity is the well-known property that for all worlds w1, w2, w3 in W, if w1Rw2 and w2Rw3, then w1Rw3. A relation is conversely well-founded iff there are no infinite ascending sequences, that is sequences of the form w1Rw2Rw3 R�. Note that conversely well-founded frames are also irreflexive, for if wRw, this gives rise to an infinite ascending sequence wRwRwR�.
   
   The above correspondence result immediately shows that GL is modally sound with respect to the class of possible worlds models on transitive conversely well-founded frames, because all axioms and rules of GL are valid on such models. The question is whether completeness also holds: for example, the formula □A→ □ □A, which is valid on all transitive frames, is indeed provable in GL, as was mentioned in Section 1. But are all valid formulas provable in GL?
   3.2 Modal completeness
   
   Unaware of the arithmetical significance of GL, K. Segerberg proved in 1971 that GL is indeed complete with respect to transitive conversely well-founded frames; D. de Jongh and S. Kripke independently proved this result as well. Segerberg showed that GL is complete even with respect to the more restricted class of finite transitive irreflexive trees, a fact which later turned out to be very useful for Solovay's proof of the arithmetical completeness theorem (see Section 4).
   
   The modal soundness and completeness theorems immediately give rise to a decision procedure to check for any modal formula A whether A follows from GL or not. Looking at the procedure a bit more precisely, it can be shown that GL is decidable in the computational complexity class PSPACE, like the well-known modal logics K, T and S4. This means that there is a Turing machine that, given a formula A as input, answers whether A follows from GL or not; the size of the memory that the Turing machine needs for its computation is only polynomial in the length of A.
   
   To give some more perspective on complexity, the class P of functions computable in an amount of time polynomial in the length of the input, is included in PSPACE, which in turn is included in the class EXPTIME of functions computable in exponential time. It remains a famous open problem whether these two inclusions are strict, although many complexity theorists believe that they are. Some other well-known modal logics, like epistemic logic with common knowledge, are decidable in EXPTIME, thus they may be more complex than GL, depending on the open problems.
   3.3 Failure of strong completeness
   
   Many well-known modal logics S are not only complete with respect to an appropriate class of frames, but even strongly complete in the sense that for all (finite or infinite) sets Γ and all formulas A:
   
   If, on appropriate S-frames, A is true in all worlds in which all formulas of Γ are true, then Γ proves A in the logic S.
   
   This condition holds for systems like K, M, K4, S4, and S5. If restricted to finite sets Γ, the above condition corresponds to completeness.
   
   Strong completeness does not hold for provability logic, however, because semantic compactness fails. Semantic compactness is the property that for each infinite set Γ of formulas,
   
   If every finite subset Δ of Γ has a model on an appropriate S-frame, then Γ also has a model on an appropriate S-frame.
   
   So, in the end:
   
   X = {◊ p0, □(p0→◊ p1), □(p1→ ◊ p2), □(p2 →◊ p3),�, □(pn→◊ pn+s),�} → X
   

Ughhhh, I'm glad I'm a robot.

If you want to meet lots of intelligent people, hang out around ones who have a genuine, strong interest in topics that have a steep learning curve and leave little room for bullshitting.

long post is loooooooong

also, i understood part 4 and then didn't read part 5, but still
gogo symbolic logic!

I say just study at a university. You'll find plenty of higher intelligence people there.

quote:

Originally posted by MrJiveBoJingles If you want to meet lots of intelligent people, hang out around ones who have a genuine, strong interest in topics that have a steep learning curve and leave little room for bullshitting.

While I am not necessarily seeking to discredit those who are diligent in their chosen studies, I find people with the ability to bullshit well really quite interesting. sometimes, even moreso than those who are genuine, as their social accessibility is quite higher. Not that social shut-ins aren't interesting, either...

I dunno.

Echo, it doesn't seem to me like you are truly afraid of women in some latent way. Of course, my pride will not allow me to draw similarities between our considerations, as I am also quite irritated by the dispositions of most women I meet, however, I am also of the mind that most guys are annoying as well. I do find that I am annoyed by more women than I am men... if anything, it's not fear, though - I just tend to hold women to a higher standard than men, in some odd sort of way. Post-feminist hegemony and all that.

quote:

Originally posted by Halcyon+On+On While I am not necessarily seeking to discredit those who are diligent in their chosen studies, I find people with the ability to bullshit well really quite interesting. sometimes, even moreso than those who are genuine, as their social accessibility is quite higher. Not that social shut-ins aren't interesting, either...

Skilled bullshit artists can be very intelligent, of course. It takes a nimble brain to keep straight a ton of lies, exaggerations, and deliberately-created false impressions.

I probably should have specified honest intelligent people.

quote:

Originally posted by mezzir i understood part 4 and then didn't read part 5

Me too. tl;dr

quote:

Originally posted by Arbiter Most people are annoying and stupid; my only concern would be that you seem to only notice it in women.

+54348769

quote:

Originally posted by Theresa +54348769

-54348769

null

quote:

Originally posted by MrJiveBoJingles If you want to meet lots of intelligent people, hang out around ones who have a genuine, strong interest in topics that have a steep learning curve and leave little room for bullshitting.

aka women who rarely shave the moustaches or legs

Hey, not all women with interests in math and hard sciences are beastly.

quote:

Originally posted by Lira Gargantuan post about Achilles, fuzzy logic, hybrid logic, modal logic et al

> ladder theory.

i tend to kick people in the shin when they start the valley girl routine.

quote:

Originally posted by Nrg2Nfinit aka women who rarely shave the moustaches or legs

or lesbians

quote:

Originally posted by Lira That's exactly why I think she should do it

lol it's just done by way too many people imo now. Many of whom don't have the bodies to support showing it off, like the amount of women with lower back tat's now, most of which just look the same. it's like cheap branding

quote:

Originally posted by Lira Me too. tl;dr

wow, I think you won the internet with that other post.

quote:

Originally posted by jennypie Ughhhh, I'm glad I'm a robot.

How does one order a jennypie-model robot?

...price isn't an issue either

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