Succedent house is an astrological term for the houses that follow (i.e., succeed) the angular houses in an Astrological chart. “Succedent” derives from the Latin succedens meaning "subsequent" or "succeeding". Since the angular houses are the first, fourth, seventh and tenth houses, the succedent houses are the second, fifth, eight and eleventh houses.
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Propositional calculus is commonly organized as a Hilbert system, whose operations are just those of Boolean algebra and whose theorems are Boolean tautologies, those Boolean terms equal to the Boolean constant 1. Another form is sequent calculus, which has two sorts, propositions as in ordinary propositional calculus, and pairs of lists of propositions called sequents, such as A∨B, A∧C,... \vdash A, B→C,.... The two halves of a sequent are called the antecedent and the succedent respectively. The customary metavariable denoting an antecedent or part thereof is Γ, and for a succedent Δ; thus Γ,A \vdash Δ would denote a sequent whose succedent is a list Δ and whose antecedent is a list Γ with an additional proposition A appended after it. The antecedent is interpreted as the conjunction of its propositions, the succedent as the disjunction of its propositions, and the sequent itself as the entailment of the succedent by the antecedent.
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The general notion of sequent introduced here can be specialized in various ways. A sequent is said to be an intuitionistic sequent if there is at most one formula in the succedent (although multi-succedent calculi for intuitionistic logic are also possible). More precisely, the restriction of the general sequent calculus to single-succedent-formula sequents, with the same inference rules as for general sequents, constitutes an intuitionistic sequent calculus. (This restricted sequent calculus is denoted LJ.) Similarly, one can obtain calculi for dual-intuitionistic logic (a type of paraconsistent logic) by requiring that sequents be singular in the antecedent.
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Hypersequent calculi based on intuitionistic or single-succedent sequents have been used successfully to capture a large class of intermediate logics, i.e., extensions of intuitionistic propositional logic. Since the hypersequents in this setting are based on single-succedent sequents, they have the following form: : \Gamma_1 \Rightarrow A_1 \mid \dots \mid \Gamma_n \Rightarrow A_n The standard formula interpretation for such an hypersequent is : (\bigwedge\Gamma_1 \to A_1) \lor \dots \lor (\bigwedge\Gamma_n \to A_n) Most hypersequent calculi for intermediate logics include the single-succedent versions of the propositional rules given above, a selection of the structural rules. The characteristics of a particular intermediate logic are mostly captured using a number of additional structural rules. E.g.
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Motivation for systems of connexive logic not only comes from considerations about a content connection between antecedent and succedent in valid inferences or implications, but also from more instrumental considerations.
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In astrology, a cadent house is the last house of each quadrant of the zodiac. A quadrant begins with an Angular house, (the house in which a chart angle lies) proceeds to a Succedent house and ends with a Cadent house. There are four quadrants in an astrological chart, providing four Angular, four Succedent, and four Cadent houses. The Greeks called the cadent houses apoklima, which literally means "falling" or "decline," because the houses were seen to be falling away from the strength of the angular houses, which were considered to be most influential because of their perpendicular and oppositive relationships to the Ascendant.
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This means that, when all formulae in Γ hold, then at least one formula in Σ also has to be true. If the succedent is empty, this is interpreted as falsity, i.e. \Gamma\vdash means that Γ proves falsity and is thus inconsistent. On the other hand an empty antecedent is assumed to be true, i.e.
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Because the angular houses are the most powerful places in the chart (Lilly says "Planets in angles do more forcibly show their effects"), succedent houses—which are less powerful than the angular but more powerful than the cadent houses—also have a quality of appertaining to the angular houses, much as a representative or underling of a powerful person. In this way, the second house, for example, which succeeds the first house of the body and personality, tends to signify the things that belong to the person. In a similar fashion, the eighth house, which follows the seventh house of the partner or spouse or “other person,” represents the belongings of the other person or partner. Succedent houses, as a whole, have a stable, unchanging, fixed quality, deriving from their central position in each quadrant of the chart.
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The eighth is the one unfortunate house in the succedent group. This connotation derives from the lack of relationship of the eighth house with the house of the Ascendant. Because there is no astrological aspect with the Ascendant, the eighth house cannot "see" the Ascendant from its position. For this reason it has a rather tenuous relationship with the health and power of the first house.
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Since every formula in the antecedent (the left side) must be true to conclude the truth of at least one formula in the succedent (the right side), adding formulas to either side results in a weaker sequent, while removing them from either side gives a stronger one. This is one of the symmetry advantages which follows from the use of disjunctive semantics on the right hand side of the assertion symbol, whereas conjunctive semantics is adhered to on the left hand side.
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The word apoklima also carries a denotation of degeneration and decline. Our English word "cadent" comes from the Latin translation of apoklima and is the source of our word "cadet," which originally meant a lesser branch of the family, or the younger son. Cadent houses are therefore usually considered by astrologers as less fertile and productive places by their nature than either angular or succedent houses, and the planets located in them are seen as generally less powerful and comfortable. This view of cadency is universally found in ancient sources.
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Finally, sequent calculus generalizes the form of a natural deduction judgment to : A_1, \ldots, A_n \vdash B_1, \ldots, B_k, a syntactic object called a sequent. The formulas on left-hand side of the turnstile are called the antecedent, and the formulas on right-hand side are called the succedent or consequent; together they are called cedents or sequents. Again, A_i and B_i are formulae, and n and k are nonnegative integers, that is, the left-hand-side or the right-hand-side (or neither or both) may be empty. As in natural deduction, theorems are those B where \vdash B is the conclusion of a valid proof.
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The structural rules deserve some additional discussion. Weakening (W) allows the addition of arbitrary elements to a sequence. Intuitively, this is allowed in the antecedent because we can always restrict the scope of our proof (if all cars have wheels, then it's safe to say that all black cars have wheels); and in the succedent because we can always allow for alternative conclusions (if all cars have wheels, then it's safe to say that all cars have either wheels or wings). Contraction (C) and Permutation (P) assure that neither the order (P) nor the multiplicity of occurrences (C) of elements of the sequences matters.
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There are four sets of trikonasthanas (Trines). The first set comprises the 1st house or the lagna, the 5th house (which is also a succedent house or a panapara) and the 9th house (which is also a cadent house or an apoklima), which three are the trinal houses of righteousness and form the Dharma- trikona. The second set comprises the 2nd house (which is a succeedent house), the 6th house (which is a cadent house and an upachayasthana) and the 10th house (which is a quadrant i.e. a kendra, and an upachayasthana), which three are the trinal houses for material possessions and form the Artha-trikona.
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