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Excluded Middle  Logic Texts  Read III 108 Similarity metrics/the conditionally excluded middle/Read: the conditionally excluded middle: one or the other member of a pair of conditional sentences must be true.  That equals the assumption that there is always a single most similar world.  Stalnaker pro  LewisVsStalnaker: e.g. Bizet/Verdi: all combinations are wrong  Stalnaker: instead of the only similar one at least one similar  LewisVs: The amount of the possible worlds in the Lewis 2 m + e is large, whereby e decreases suitably; it has no limit.  Solution/Lewis: instead of the selection function: similarity relation: he proposes that "if A, then B" is then true in w if there is either no "A or nonB" world, or some "A" and "B" world that is more similar than any "A and nonB" world. >Similarity metrics.  III 110 VerdiExample: where there is no unique, most similar world, the "would" condition sentences are false because there is no similar world for any of the most appropriate similar worlds in which they are fellow country men, in which Bizet has a different nationality.  Example: if you get an A, you will receive a scholarship: will be true if there is a more similar world in which you get both for each world in which you get an A and not a scholarship.  ((s) without conditional sentence of the excluded middle). >Similarity.  III 263 Law of the excluded middle/constructivism/Read: Constructivists often present socalled "weak counterexamples" against the excluded middle  if a is a real number, "a = 0" is not decidable. Consequently, the constructivist cannot claim that all real numbers are either identical to zero or not.  But this is more of a question of representation. >Constructivism, >Presentation. 
Logic Texts Me I Albert Menne Folgerichtig Denken Darmstadt 1988 HH II HoyningenHuene Formale Logik, Stuttgart 1998 Re III Stephen Read Philosophie der Logik Hamburg 1997 Sal IV Wesley C. Salmon Logic, Englewood Cliffs, New Jersey 1973  German: Logik Stuttgart 1983 Sai V R.M.Sainsbury Paradoxes, Cambridge/New York/Melbourne 1995  German: Paradoxien Stuttgart 2001 Re III St. Read Thinking About Logic: An Introduction to the Philosophy of Logic. 1995 Oxford University Press German Edition: Philosophie der Logik Hamburg 1997 
Excluded Middle  Bigelow  I 117 Conditional of the excluded middle/conditionally excluded middle/Lewis/Bigelow/Pargetter: could be considered as an axiom: (A would be > would be b) v (a would be > would be ~ b) Lewis: Thesis: this is not always true. StalnakerVsLewis: (1968^{(1)}, 1981^{(2)}) defends the conditional sentence from the excluded middle against Lewis:  I 118 We must consider cases of the following kind: there is a temptation to say that it can be wrong to assert: "If I had gone to the movies yesterday, I would have watched The Fly." And it can also be wrong to say: "If I had gone to the movies yesterday, I would not have watched The Fly." ((s) Do not omit the front link for the second time!) Bigelow/Pargetter: we might rather say: "If I had gone to the movies yesterday, I could have watched The Fly (or not)." Logical form: (a would be > could be b) u (a would be > could be ~ b). That is, we deny something of the form (a would be > would be b) And we also deny something of the form (A would be > would be ~ b). So we deny both sides. Therefore, it seems that we must deny the conditionally excluded middle. Conditionally excluded middle/Pargetter: these were intuitive reasons for his rejection. Now we must also consider some of its formal consequences: Problem: would it be accepted, the difference between "would" and "could" would collapse. Would/could/Bigelow/Pargetter: normally it is clear that (a would be > would be b) entails a would be > could b) ((s) "would" implies "could"). Problem/Bigelow/Pargetter: if we accept the conditional sentence of the excluded middle (conditionally excluded middle), the inverse implication is also valid! For (a would be > could be b) is by definition ~ (a would be > would be ~ b) and this is the negation of one of the two disjuncts in the conditionally excluded middle. Then we must assert the other disjoint, thus the assumption of (a would be > could be b) implies that (a would be > would be b).  I 119 According to this "would have been" and "could have been" would be equivalent and we do not want that. 1. Stalnaker, R. C. (1968) A theory of conditional. In: Studies in logical theory (Ed. N.Rescher), pp. 98112. Osxofrd: Blackwell Publishers. 2. Stalnaker, R. C. (1981) A defense of conditional excluded middle. In: Ifs (Ed. Wl.L.Harper, R. Stalnaker and G.Pearce), pp. 87104. Dordrecht. Reidel. 
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 
Similarity Metrics  Logic Texts  Read III 104ff Similarity Metrics/Stalnaker: smallest possible revision  i.e. the most similar world. Selection function: f(A, w)  "If you get a one, you will receive a scholarship" is true if the world in which you receive a scholarship is most similar to the world in which you are getting a one. Possible world view: deviates from the probability function if the forelink is wrong".  Because all combinations can be realized in a possible world. >Possible world, >Similarity. III 105 Similarity Metrics/possible world/conditional sentence/Read: some classical logical principles fail here: e.g. contraposition that "if B, then notA" follows from "if A, then not B"  the similar world in which it rains, can be very well one in which it rains only lightly. But the most similar world in which it rains violently cannot be one in which it does not rain at all. >Conditional. III 106 Another principle that fails: the reinforcement of the ifsentence: "If A, then B. So if A and C, then B."  For example, when I put sugar in my tea, it will taste good. So when I put sugar and diesel oil in my tea, it will taste good. In the most similar world in which I put diesel oil like sugar in my tea, it tastes horrible  further: the results of the conditionality principle are invalid:  If A, then B. So if A and C, then B  and if A, then B. If B, then C. So if A, then C  Reason: the conditional sentence has become a modal connection. We must know that these statements are strong enough in any appropriate modal sense  to ensure that the most similar A and C world is the most similar Aworld, we must know that C is true everywhere. III 108 Similarity metrics/the conditionally excluded middle/Read: the sentence of the conditionally excluded middle: one or other member of a pair of conditional sentences must be true. This corresponds to the assumption that there is always a single most similar world.  (Stalnaker pro). >Conditionally excluded middle. LewisVsStalnaker: e.g. Bizet/Verdi  All combinations are false. Stalnaker: instead of the only similar one at least one similar world. LewisVs: the set of possible world in which Lewis is 2 m + e tall, whereby e decreases appropriately, this has no boundary. Solution/Lewis: instead of the selection function: similarity relation: Lewis proposes, that "if A then B" is true in w if there is either no "A or nonB"world, or any "A and B"world that is more similar than any "A and notB"World. III 110 Bizet/Verdiexample: where there is no unique most similar world, the "would" conditional sentences are wrong because there is no most similar world for any of the most appropriate similar worlds in which they are compartiots, where Bizet has a different nationality. >Bizet/Verdi case. E.g. If you get a one, you will receive a scholarship: will be true, if there is for every world in which you get a one and do not receive scholarship, is a more similar world in which you get both (without conditional sentence of the excluded middle). III 115 Similarity metrics/similarity analysis/possible world/ReadVsLewis: problem: e.g. (assuming John is in Alaska) If John is not in Turkey, then he is not in Paris.  This conditional sentence is true according to the "similarity statement", because it only asks, whether the thensentence is true in the most similar world. >Conditional, >Counterfactual conditional. 
Logic Texts Me I Albert Menne Folgerichtig Denken Darmstadt 1988 HH II HoyningenHuene Formale Logik, Stuttgart 1998 Re III Stephen Read Philosophie der Logik Hamburg 1997 Sal IV Wesley C. Salmon Logic, Englewood Cliffs, New Jersey 1973  German: Logik Stuttgart 1983 Sai V R.M.Sainsbury Paradoxes, Cambridge/New York/Melbourne 1995  German: Paradoxien Stuttgart 2001 Re III St. Read Thinking About Logic: An Introduction to the Philosophy of Logic. 1995 Oxford University Press German Edition: Philosophie der Logik Hamburg 1997 
Similarity Metrics  Bigelow  I 129 Counterfactual Conditional/Valuation/Valuation Function/Valuation Rules/Bigelow/Pargetter: V9 If a = (ß would be > weould be γ) then V (a) is the set of all possible worlds w ε w so that there is a possible world u where β is true and γ is true and every possible world v in which ß is true and γ is false, is less accessible from w than u. ((s) > similarity metrics.) Resemblance/possible worlds/similarity metrics/counterfactual conditional/Bigelow/Pargetter: rule V9 states that a counterfactual conditional (β would be > would be γ) is true in a possible world if the next ßworlds are all γworlds. For example, Violet says: "If I were a blackbird, I would sing" this is true in the actual world, because the next worlds where Violet is a blackbird are possible worlds where she sings. Similarity metrics/Bigelow/Pargetter: the possible worlds in which Violet is a blackbird on the tree or on the mailbox can be possibly at the same distance from the actual world. Connection/Possible worlds: the question of whether such "connections" can exist is discussed in connection with the conditionally excluded middle (see above). Complexity/Bigelow/Pargetter: the complexity of V) is due to the desired generality.  I 130 Resemblance/similarity metrics/counterfactual conditional/proximity/possible worlds/Bigelow/Pargetter: we want a possible world, in which the fore link and back link of the counterfactual conditional are both true, to be closer than one in which only the fore links are true, and the back links are false.  I 209 Possible World/Variant/Bigelow/Pargetter: we could also specify individuals by describing their position in the course of their existence. Through an infinite sequence of quadruples. There are many variants, including more economical ones. We can combine all the positions of a particle into one function. This is also possible for other properties that we attribute to a particle. So we can combine a particle not only with numbers, but also with whole functions. Function: these functions could describe the changes of the particle. Book/Bigelow/Pargetter: a book for such a described world could be a Hilbert space. But a book is not a world yet! A book for the actual world would consist of two components: 1. a world property, or a maximum specific structural universal 2. to something that instantiates this universal, that is the world itself. This applies to the actual world! Other possible worlds correspond to a universal, but this is not instantiated, so there is no world here. Representation/Bigelow/Pargetter: now the numbers representing these world properties could seem all too abstract.  I 210 But they are not! They represent the proportions in which the properties of the parts that we have chosen as units are related to each other. Crossworldrelations/world properties/property theory/Bigelow/Pargetter: now it seems as if our theory is making a surprising turn: it seems to provide a measure for the distance between possible worlds that we have been unable to gain so far. And that measure would not be arbitrary! Accessibility: could we get it under control with this? (see below) If the possible worlds contain the same individuals, it is even easy to construct a similarity metrics for them. If the individuals are different, it is more difficult. 
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 
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Cond. Excl. Middle  Lewis, D.  V 329 Def Principle of the Conditionally Excluded Middle/Stalnaker/Lewis: thesis: either X w > w Y or X w > w ~Y is valid in every world. From this it follows that if Y,Y",...are a partition and X is possible, then X w > w Y,X w > w Y",..., is also a partition. Then the conjunctions of full patterns are a partition, because for each option A, the counterfactual conditional A w > w S, A w > w S"... are a partition. There are two objections to the principle of the conditional sentence of the excluded middle: 1. Vs: it makes arbitrary selections, it says that the way things are when there is a wrong but possible proposition X is no less specific than the way things actually are. 2. Vs: more serious objection: V 330/331 Example, suppose the actor thinks that the real world could very well be indeterministic, where many things come out by chance. Explanation/(s): A w>w B: If A were the case, B would be the case. 
