Alan Ross Anderson and Jr. Nuel D. Belnap. Entailment: The Logic of Relevance and Necessity. Princeton University Press, Princeton, NJ, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....literals. Identifying generic answers with literals containing variables from the original question can be viewed as an issue of relevance: the literals that are relevant to a generic description of the arbitrary individuals mentioned in the answer literals are those that mention the individuals. Anderson and Belnap [1975, pp.32 33] discuss relevance in a logic of entailment as depending on variable sharing: So we propose as a necessary, but by no means sufficient, condition for the relevance of A to B in the pure calculus of entailment, that A and B must share a variable. The notion of variable sharing has to ....

Alan Ross Anderson and Jr. Nuel D. Belnap. Entailment: The Logic of Relevance and Necessity. Princeton University Press, Princeton, NJ, 1975.

....is not. That is, I endorse classical reasoning, in the sense that I think that I will never step from truth to untruth in taking the step from premises to a conclusion in a classically valid argument. 2 The relation I will call relevant logical consequence agrees with first degree entailment [1, 12, 29, 30]. First degree entailment rejects the inference of explosion and its dual, which infers B # #B from A. Obviously, since I endorse classical reasoning, I endorse relevant reasoning in the same sense relevantly valid arguments also never take us from truth to untruth. However, I think that ....

ALAN ROSS ANDERSON AND NUEL D. BELNAP. Entailment: The Logic of Relevance and Necessity, volume 1. Princeton University Press, Princeton, 1975.

....[81] and its descendant Relevance Logic [94] Read s Relevant Logic [224] and Troelstra s Lectures on Linear Logic [264] are more narrowly focussed than this essay, concentrating on one or other of the many relevant and substructural logics. The Anderson Belnap two volume Entailment [10, 11] is a goldmine of historical detail in the tradition of relevance logic, but it contains little about other important traditions in substructural logics. My Introduction to Substructural Logics [234] has a similar scope to this chapter, in that it covers the broad sweep of substructural logics: ....

....conclude A C from 13 If something is entailed by a necessity, it too is necessary. If A entails B then if we cannot have A false, we cannot have B false either. 14 The interested reader is referred to Ackermann s paper (in German) 2] or to Anderson, Belnap and Dunn s sympathetic summary [11, 44 46] (in English) 15 The choice of counterexample as a thesis connecting implication and negation in place of reductio (as in Orlov) is of no matter. The two are equivalent in the presence of contraposition and double negation rules. Showing this is a gentle exercise in axiom chopping. Greg ....

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ALAN ROSS ANDERSON, NUEL D. BELNAP, AND J. MICHAEL DUNN. Entailment: The Logic of Relevance and Necessity, volume 2. Princeton University Press, Princeton, 1992.

....[81] and its descendant Relevance Logic [94] Read s Relevant Logic [224] and Troelstra s Lectures on Linear Logic [264] are more narrowly focussed than this essay, concentrating on one or other of the many relevant and substructural logics. The Anderson Belnap two volume Entailment [10, 11] is a goldmine of historical detail in the tradition of relevance logic, but it contains little about other important traditions in substructural logics. My Introduction to Substructural Logics [234] has a similar scope to this chapter, in that it covers the broad sweep of substructural logics: ....

....as a proponent of substructural logics, as his aim was not to do without contraction, but to give an axiomatic account of material implication. 5 The names are mine, and not Orlov s. I have attempted to give each axiom or rule its common name (see for example Anderson and Belnap s Entailment [10] for a list of axioms and their names) In this case, contraposed reductio is my name, as the axiom A (A A) is a rarely seen axiom, but it is a contraposed form of (A A) A, which is commonly known as reductio. Greg Restall, Greg.Restall mq.edu.au June 23, 2001 ....

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ALAN ROSS ANDERSON AND NUEL D. BELNAP. Entailment: The Logic of Relevance and Necessity, volume 1. Princeton University Press, Princeton, 1975.

....acceptance and rejection, while not taking R theorems to be rationally mandatory to accept. in particular, an agent guided by this policy would not be thereby obliged to accept each proposition of the form B # #B. Such an approach is surely not what Anderson and Belnap intended in Entailment [1, 2], but it is not foreign to the enterprise of relevant logic. 11 In just the same way, nothing in the adoption of R requires that we reject f and the contradictions which entail it but it is completely natural to do so. The burden, if there is any at all to be borne, is on the one who fails ....

ALAN ROSS ANDERSON, NUEL D. BELNAP, AND J. MICHAEL DUNN. Entailment: The Logic of Relevance and Necessity, volume 2. Princeton University Press, Princeton, 1992.

....acceptance and rejection, while not taking R theorems to be rationally mandatory to accept. in particular, an agent guided by this policy would not be thereby obliged to accept each proposition of the form B # #B. Such an approach is surely not what Anderson and Belnap intended in Entailment [1, 2], but it is not foreign to the enterprise of relevant logic. 11 In just the same way, nothing in the adoption of R requires that we reject f and the contradictions which entail it but it is completely natural to do so. The burden, if there is any at all to be borne, is on the one who fails ....

ALAN ROSS ANDERSON AND NUEL D. BELNAP. Entailment: The Logic of Relevance and Necessity, volume 1. Princeton University Press, Princeton, 1975.

....is not. That is, I endorse classical reasoning, in the sense that I think that I will never step from truth to untruth in taking the step from premises to a conclusion in a classically valid argument. 2 The relation I will call relevant logical consequence agrees with first degree entailment [1, 12, 29, 30]. First degree entailment rejects the inferences of explosion and and its dual, which infers B # #B from A. Obviously, since I endorse classical reasoning, I endorse relevant reasoning in the same sense relevantly valid arguments also never take us from truth to untruth. However, I think that ....

ALAN ROSS ANDERSON AND NUEL D. BELNAP. Entailment: The Logic of Relevance and Necessity, volume 1. Princeton University Press, Princeton, 1975.

....Man did not go down the left fork. Without hesitating to 1 check for the scent down the right fork (and thus, without trying to find any independent evidence or warrant for B) the Dog continues her pursuit down the right fork in the track, inferring B, the claim that the Man has gone that way [1, 2, 5]. On this account of the Dogs behaviour, the Dog is reasoning, and indeed, she seems to be reasoning well. She has used the inference disjunctive syllogism, which has the following form: A# B #A (DS) B Proponents of relevant logics reject disjunctive syllogism, because when you combine it ....

ALAN ROSS ANDERSON, NUEL D. BELNAP, AND J. MICHAEL DUNN. Entailment: The Logic of Relevance and Necessity, volume 2. Princeton University Press, Princeton, 1992.

....acceptance and rejection, while not taking R theorems to be rationally mandatory to accept. in particular, an agent guided by this policy would not be thereby obliged to accept each proposition of the form B # #B. Such an approach is surely not what Anderson and Belnap intended in Entailment [1, 2], but it is not 8 This is surely more than a historical accident, given the primacy of implication as an assertoric record of valid inference. However, there is nothing mandatory in valid inference which makes implication the only sensible record of validity. We could just as well take ....

ALAN ROSS ANDERSON, NUEL D. BELNAP, AND J. MICHAEL DUNN. Entailment: The Logic of Relevance and Necessity, volume 2. Princeton University Press, Princeton, 1992.

....acceptance and rejection, while not taking R theorems to be rationally mandatory to accept. in particular, an agent guided by this policy would not be thereby obliged to accept each proposition of the form B # #B. Such an approach is surely not what Anderson and Belnap intended in Entailment [1, 2], but it is not 8 This is surely more than a historical accident, given the primacy of implication as an assertoric record of valid inference. However, there is nothing mandatory in valid inference which makes implication the only sensible record of validity. We could just as well take ....

ALAN ROSS ANDERSON AND NUEL D. BELNAP. Entailment: The Logic of Relevance and Necessity, volume 1. Princeton University Press, Princeton, 1975.

....2 Theta it follows that B 2 Theta, since A; A oe B B. The other part is more complex. It involves showing that if A oe B 62 Gamma then there is a prime Theta Gamma such that A 2 Theta and B 62 Theta. We use the pair extension lemma to do this. Lemma 4 (Pair Extension (Belnap, Gabbay) [1]) If is a suitable consequence relation on a well ordered language, then if Sigma 6 Delta, then there are Sigma 0 Sigma and Delta 0 Delta such that Sigma 0 6 Delta 0 , and Sigma 0 Delta 0 is the whole language. For our details it is sufficient for me to assure you ....

Alan Ross Anderson, Nuel D. Belnap, and J. Michael Dunn. Entailment: The Logic of Relevance and Necessity, volume 2. Princeton University Press, Princeton, 1992.

....first degree entailment in the context of the kind of logical coherence exhibited by the class of infons supported by a situation. The Routley Meyer semantics for relevant logics is another case where we have points supporting some kind of information, which has a degree of logical coherence [2, 13, 35, 36]. The semantic structures associated with relevant logics are interesting in a number of ways. Firstly, as relevant logics reject the inferences A A B, and from A B B, but they keep the intuition that entailment on a frame is defined by taking A B to be valid just when for every point x ....

Alan Ross Anderson, Nuel D. Belnap, and J. Michael Dunn. Entailment: The Logic of Relevance and Necessity, Volume 2. Princeton University Press, Princeton, 1992.

.... symbols: E x (E x ) A) I argued in my paper Truthmakers, Entailment and Necessity [14] that if we wish to use this account of truthmaking, we ought understand the entailment connective ) in such a claim as a relevant entailment, in the tradition of Anderson and Belnap and their co workers [1, 2, 8, 11]. Furthermore, I proposed a number of theses about truthmaking as necessitation. The most controversial of these is the disjunction thesis: x makes a disjunction A B true if and only if it makes one of the disjuncts (A or B) true. That paper left one important task unfinished. I did not explain ....

Alan Ross Anderson, Nuel D. Belnap, and J. Michael Dunn. Entailment: The Logic of Relevance and Necessity, volume 2. Princeton University Press, Princeton, 1992.

.... symbols: E x (E x ) A) I argued in my paper Truthmakers, Entailment and Necessity [14] that if we wish to use this account of truthmaking, we ought understand the entailment connective ) in such a claim as a relevant entailment, in the tradition of Anderson and Belnap and their co workers [1, 2, 8, 11]. Furthermore, I proposed a number of theses about truthmaking as necessitation. The most controversial of these is the disjunction thesis: x makes a disjunction A B true if and only if it makes one of the disjuncts (A or B) true. That paper left one important task unfinished. I did not explain ....

Alan Ross Anderson and Nuel D. Belnap. Entailment: The Logic of Relevance and Necessity, volume 1. Princeton University Press, Princeton, 1975.

....proof theory. Instead, we will take our semantic considerations as primary, and see what this means for validity. As a result, the results vary from original work in relevant logics. This paper is not an apologetic for the particular formal systems E (the focus of the magisterial Entailment [1] [2]) or R (the focus of Dunn s excellent survey article on relevant logics [7] The situation is similar to that of classical modal logic given the appearance of possible worlds semantics. Considerations about the structure of possible worlds could dictate logical investigations, and as a result, ....

....on W . This is approach is new option in the debate about the validity of the disjunctive syllogism. Though it is hinted at in a number of places in the literature, such as the Routleys original paper [23] The extended discussion given by Belnap and Dunn [11] reprinted as Section 80 in [2]) is an analysis of different positions possible in regards to disjunctive syllogism, but it does not consider this position. The closest position to ours is the option that Belnap and Dunn dub I m all right Jack. According to this option, we can reason as follows. The argument from A and AB to ....

[Article contains additional citation context not shown here]

Alan Ross Anderson, Nuel D. Belnap, and J. Michael Dunn. Entailment: The Logic of Relevance and Necessity, Volume 2. Princeton University Press, Princeton, 1992.

....the proof theory. Instead, we will take our semantic considerations as primary, and see what this means for validity. As a result, the results vary from original work in relevant logics. This paper is not an apologetic for the particular formal systems E (the focus of the magisterial Entailment [1], 2] or R (the focus of Dunn s excellent survey article on relevant logics [7] The situation is similar to that of classical modal logic given the appearance of possible worlds semantics. Considerations about the structure of possible worlds could dictate logical investigations, and as a ....

....that grass being purple doesn t. One way to understand this distinction is to note that there there are (undoubtedly inconsistent) states involving my trisecting an angle and my fame, without also involving grass being purple. Much more could be said here, but it would only rework old discussions [1], 7] 22] which are much better than anything I could attempt here. The main point is that even if A and A 0 have the same truth value in all possible worlds we do not necessarily have A B if and only if A 0 B, and nor do we have B A if and only if B A 0 . To evaluate ....

Alan Ross Anderson and Nuel D. Belnap. Entailment: The Logic of Relevance and Necessity, Volume 1. Princeton University Press, Princeton, 1975.

....reasoning (where each entire construction is invariant) is pinned between two different semantic interpretations of classical reasoning: one requiring no invariance, and the other requiring invariance at every level of the type hierarchy. ffl Characterize useful concepts of relevant implication [1] in realizability models. For example, in Lauchli realizability models a candidate definition is ff relevantly implies fi if and only if there is a function from P (ff) to P (fi) that is an isomorphism on each orbit of P (ff) invariance requires only a homomorphism) Or, in relational ....

Allan Ross Anderson and Nuel D. Belnap. Entailment: the Logic of Relevance and Necessity, volume I. Princeton University Press, Princeton NJ, 1975.

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ALAN ROSS ANDERSON, NUEL D. BELNAP, AND J. MICHAEL DUNN. Entailment: The Logic of Relevance and Necessity, volume 2. Princeton University Press, Princeton, 1992.

No context found.

ALAN ROSS ANDERSON, NUEL D. BELNAP, AND J. MICHAEL DUNN. Entailment: The Logic of Relevance and Necessity, volume 2. Princeton University Press, Princeton, 1992.

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