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Movie Title Year Distributor Notes Rev Formats Anal Graveyard 1997 Sin City Anal 1 Blowjob Adventures of Dr. Fellatio 4 1997 Elegant Angel BJOnly Facial 4 DRO Choke The Bitch 1998 Cream Entertainment Facial 1 Cock Smokers 11 1999 Extreme Associates D College Maidens 3 1997 Heatwave Anal Facial 1 DO Depraved And Shaved 2000 Spunky Spur NonSex DRO Dirty Video 6 1997 Al Borda Video Facial 1 Exotica Erotika 5 1997 4-Play Video Facial DO Extreme Fetish 2 2000 Filmco Releasing NonSex DO Fuck Pigs 3 1999 Extreme Associates Facial A2M 1 D Fuck You 1997 Tight Ends Anal DO Gothic Whore 1997 Al Borda Video LezOnly 1 Harem 74 1998 Vivid LezOnly 1 O Haulin' Ass 1997 Tight Ends 1 DRO Hollywood Hardcore 3 1998 Filmwest Productions Facial 1 Kinky Underground 2000 Spunky Spur NonSex Max 17: Dead Or Alive 1997 Legend Video Anal Max Hardcore's Anal Auditions 7 1997 Legend Video Anal Facial 1 Nightshift 3: Gothic 1998 Sin City Facial 2 One Big Orgy: The Wedding 1997 Zane Entertainment Group 1 DO
Perverted Porno Movie 1997 Notorious O Q Balls 3: Bald And Balled 2000 Totally Tasteless 1 O SexHibition 6: Queen Of Pain 1998 Sunshine Films D Shut Up and Blow Me 16 1999 All Good Video BJOnly 1 DO Shut Up and Blow Me 8 1998 All Good Video BJOnly Facial 4 DO Snow White And The Three Dwarfs 2001 Totally Tasteless DO Solo Adventures 3 1997 Al Borda Video MastOnly 1 Stinky Fingers 1999 Cream Entertainment MastOnly D Suck It Out 11 2014 Sinister TV O Sunrise 1997 Al Borda Video Anal 1 DO Tattoo Parlor Pussy 1997 Tight Ends Facial 1 DO Teen Anal 10 2004 Metro DO Todd And 2 Slaves 2000 Spunky Spur NonSex O Trendy Pierced Young Things 1997 Amazing Twisted Desires 2013 Filmco Releasing DRO Virgin Kink 6 1997 Red Board Video NonSex Wizzard of Odds 3 2003 Totally Tasteless DO World's Luckiest Black Man 1998 Vivid 2 O World's Luckiest Man 1997 Vivid Facial 3 DO Xtreme Erotica 2003 Filmco Releasing

Formal semantics Mathematical semantics is the application of mathematics to study the meaning of expressions in a formal language. It has three elements: a mathematical specification of a class of objects via syntax, a mathematical specification of various semantic domains and the relation between the two, which is usually expressed as a function from syntactic objects to semantic ones. This article only addresses the issue of how quantifier elements are interpreted. The syntax of a formula can be given by a syntax tree. A quantifier has a scope, and an occurrence of a variable x is free if it is not within the scope of a quantification for that variable. Thus in {\displaystyle \forall x(\exists yB(x,y))\vee C(y,x)} \forall x (\exists y B(x,y)) \vee C(y,x) the occurrence of both x and y in C(y, x) is free, while the occurrence of x and y in B(y, x) is bound (i.e. non-free). Syntax tree of the formula {\displaystyle \forall x(\exists yB(x,y))\vee C(y,x)} \forall x (\exists y B(x,y)) \vee C(y,x) , illustrating scope and variable capture. Bound and free variable occurrences are colored in red and green, respectively. An interpretation for first-order predicate calculus assumes as given a domain of individuals X. A formula A whose free variables are x1, ..., xn is interpreted as a boolean-valued function F(v1, ..., vn) of n arguments, where each argument ranges over the domain X. Boolean-valued means that the function assumes one of the values T (interpreted as truth) or F (interpreted as falsehood). The interpretation of the formula {\displaystyle \forall x_{n}A(x_{1},\ldots ,x_{n})} \forall x_n A(x_1, \ldots , x_n) is the function G of n-1 arguments such that G(v1, ..., vn-1) = T if and only if F(v1, ..., vn-1, w) = T for every w in X. If F(v1, ..., vn-1, w) = F for at least one value of w, then G(v1, ..., vn-1) = F. Similarly the interpretation of the formula {\displaystyle \exists x_{n}A(x_{1},\ldots ,x_{n})} \exists x_n A(x_1, \ldots , x_n) is the function H of n-1 arguments such that H(v1, ..., vn-1) = T if and only if F(v1, ..., vn-1, w) = T for at least one w and H(v1, ..., vn-1) = F otherwise. The semantics for uniqueness quantification requires first-order predicate calculus with equality. This means there is given a distinguished two-placed predicate "="; the semantics is also modified accordingly so that "=" is always interpreted as the two-place equality relation on X. The interpretation of {\displaystyle \exists !x_{n}A(x_{1},\ldots ,x_{n})} \exists ! x_n A(x_1, \ldots , x_n) then is the function of n-1 arguments, which is the logical and of the interpretations of {\displaystyle \exists x_{n}A(x_{1},\ldots ,x_{n})} \exists x_n A(x_1, \ldots , x_n) {\displaystyle \forall y,z\left\{A(x_{1},\ldots ,x_{n-1},y)\wedge A(x_{1},\ldots ,x_{n-1},z)\implies y=z\right\}.} \forall y,z \left\{ A(x_1, \ldots ,x_{n-1}, y) \wedge A(x_1, \ldots ,x_{n-1}, z) \implies y = z \right\}. Each kind of quantification defines a corresponding closure operator on the set of formulas, by adding, for each free variable x, a quantifier to bind x.[7] For example, the existential closure of the open formula n>2 ? xn+yn=zn is the closed formula ?n ?x ?y ?z (n>2 ? xn+yn=zn); the latter formula, when interpreted over the natural numbers, is known to be false by Fermat's last theorem. As another example, equational axioms, like x+y=y+x, are usually meant to denote their universal closure, like ?x ?y (x+y=y+x) to express commutativity. Paucal, multal and other degree quantifiers See also: Fubini's theorem and measurable None of the quantifiers previously discussed apply to a quantification such as There are many integers n < 100, such that n is divisible by 2 or 3 or 5. One possible interpretation mechanism can be obtained as follows: Suppose that in addition to a semantic domain X, we have given a probability measure P defined on X and cutoff numbers 0 < a = b = 1. If A is a formula with free variables x1,...,xn whose interpretation is the function F of variables v1,...,vn then the interpretation of {\displaystyle \exists ^{\mathrm {many} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})} \exists^{\mathrm{many}} x_n A(x_1, \ldots, x_{n-1}, x_n) is the function of v1,...,vn-1 which is T if and only if {\displaystyle \operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\geq b} \operatorname{P} \{w: F(v_1, \ldots, v_{n-1}, w) = \mathbf{T} \} \geq b and F otherwise. Similarly, the interpretation of {\displaystyle \exists ^{\mathrm {few} }x_{n}A(x_{1},\ldots ,x_{n-1},x_{n})} \exists^{\mathrm{few}} x_n A(x_1, \ldots, x_{n-1}, x_n) is the function of v1,...,vn-1 which is F if and only if {\displaystyle 0<\operatorname {P} \{w:F(v_{1},\ldots ,v_{n-1},w)=\mathbf {T} \}\leq a} 0< \operatorname{P} \{w: F(v_1, \ldots, v_{n-1}, w) = \mathbf{T}\} \leq a and T otherwise.[citation needed] Other quantifiers A few other quantifiers have been proposed over time. In particular, the solution quantifier,[8]:28 noted § (section sign) and read "those". For example, {\displaystyle \left[\S n\in \mathbb {N} \quad n^{2}\leq 4\right]=\{0,1,2\}}{\displaystyle \left[\S n\in \mathbb {N} \quad n^{2}\leq 4\right]=\{0,1,2\}} is read "those n in N such that n2 = 4 are in {0,1,2}." The same construct is expressible in set-builder notation as {\displaystyle \{n\in \mathbb {N} :n^{2}\leq 4\}=\{0,1,2\}.}{\displaystyle \{n\in \mathbb {N} :n^{2}\leq 4\}=\{0,1,2\}.} Contrary to the other quantifiers, § yields a set rather than a formula.[9] Some other quantifiers sometimes used in mathematics include: There are infinitely many elements such that... For all but finitely many elements... (sometimes expressed as "for almost all elements..."). There are uncountably many elements such that... For all but countably many elements... For all elements in a set of positive measure... For all elements except those in a set of measure zero... History Term logic, also called Aristotelian logic, treats quantification in a manner that is closer to natural language, and also less suited to formal analysis. Term logic treated All, Some and No in the 4th century BC, in an account also touching on the alethic modalities. In 1827, George Bentham published his Outline of a new system of logic, with a critical examination of Dr Whately's Elements of Logic, describing the principle of the quantifier, but the book was not widely circulated.[10] Augustus De Morgan (1806-1871) was the first to use "quantifier" in the modern sense. William Hamilton claimed to have coined the terms "quantify" and "quantification", most likely in his Edinburgh lectures c. 1840. Augustus De Morgan confirmed this in 1847, but modern usage began with De Morgan in 1862 where he makes statements such as "We are to take in both all and some-not-all as quantifiers".[11] Gottlob Frege, in his 1879 Begriffsschrift, was the first to employ a quantifier to bind a variable ranging over a domain of discourse and appearing in predicates. He would universally quantify a variable (or relation) by writing the variable over a dimple in an otherwise straight line appearing in his diagrammatic formulas. Frege did not devise an explicit notation for existential quantification, instead employing his equivalent of ~?x~, or contraposition. Frege's treatment of quantification went largely unremarked until Bertrand Russell's 1903 Principles of Mathematics. In work that culminated in Peirce (1885), Charles Sanders Peirce and his student Oscar Howard Mitchell independently invented universal and existential quantifiers, and bound variables. Peirce and Mitchell wrote ?x and Sx where we now write ?x and ?x. Peirce's notation can be found in the writings of Ernst Schröder, Leopold Loewenheim, Thoralf Skolem, and Polish logicians into the 1950s. Most notably, it is the notation of Kurt Gödel's landmark 1930 paper on the completeness of first-order logic, and 1931 paper on the incompleteness of Peano arithmetic. Peirce's approach to quantification also influenced William Ernest Johnson and Giuseppe Peano, who invented yet another notation, namely (x) for the universal quantification of x and (in 1897) ?x for the existential quantification of x. Hence for decades, the canonical notation in philosophy and mathematical logic was (x)P to express "all individuals in the domain of discourse have the property P," and "(?x)P" for "there exists at least one individual in the domain of discourse having the property P." Peano, who was much better known than Peirce, in effect diffused the latter's thinking throughout Europe. Peano's notation was adopted by the Principia Mathematica of Whitehead and Russell, Quine, and Alonzo Church. In 1935, Gentzen introduced the ? symbol, by analogy with Peano's ? symbol. ? did not become canonical until the 1960s. Around 1895, Peirce began developing his existential graphs, whose variables can be seen as tacitly quantified. Whether the shallowest instance of a variable is even or odd determines whether that variable's quantification is universal or existential. (Shallowness is the contrary of depth, which is determined by the nesting of negations.) Peirce's graphical logic has attracted some attention in recent years by those researching heterogeneous reasoning and diagrammatic inference.

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