Frode A . Bjørdal - Academia.edu (original) (raw)
Events by Frode A . Bjørdal
This was the 4th world congress organized about the square of opposition after very succesful pre... more This was the 4th world congress organized about the square of opposition after very succesful previous editions in Montreux, Switzerland 2007, Corté, Corsica 2010, Beirut, Lebanon, 2012. An interdisciplinary event gathering logicians, philosophers, mathematicians, semioticians, theologians, cognitivists, artists and computer scientists.
http://www.square-of-opposition.org/square2014.html
Papers by Frode A . Bjørdal
Logic and Logical Philosophy, Jun 13, 2013
Mathematical Reviews, 2019
This paper analyzes the use of diagrams in mathematical reasoning and puts forward examples where... more This paper analyzes the use of diagrams in mathematical reasoning and puts forward examples where diagrams are useful. It is not suggested that diagrammatic reasoning is sometimes necessary to justify mathematical results, but rather that such reasoning in some cases may be useful in discovering new theorems or in their communication. The topic could to a large extent have been treated by psychological research, it seems. Some typos interfere with the presentation. In the references, the title of the article and the page numbers of the article by Barwise and Etchemendy have been replaced by the information for the article by De Cruz
arXiv (Cornell University), Dec 22, 2022
Springer eBooks, 2015
The main purpose of this article is to discuss and clarify philosophical issues in connection wit... more The main purpose of this article is to discuss and clarify philosophical issues in connection with the librationst set theory £. We take technical results in and upon £ for granted though make references to them as that is useful. We defend the view that £ is neither an inconsistent nor a contradictory system and point out that it is neither paraconsistent nor dialetheist; in contrast we consider £ a bialethic and paracoherent theory.
Kantiansk anskuelse og høyere aritmetikk av Herman Ruge Jervell Time Out of Time: Towards an Expa... more Kantiansk anskuelse og høyere aritmetikk av Herman Ruge Jervell Time Out of Time: Towards an Expanded, Non-trivial and Unitary Theory of Time from Recent Advances in the Hadronic Sciences av Stein Erik Johansen (hilsen til jubilanten side 217) 219 Et strukturalistisk syn på naturlige tall og deres representerbarhet av Frode Kjosavik Naturen i tenkningen av Telje Kvilhaug Tidens mote: En kamp mellom produsenter og lwnsumenter av Tlygve Lavik Kants absolutte rom i GegelUJell og rommet som (I priori i Kri/ik En sammenstilling av Anita Leirfall VI Potensialitet i naturen av Svein Anders Noer Lie Discourse on the Method versus Orientation in Thinking av Francoise Monnoyeur Den som kun tar spøg for spøg, og alvor kun alvorligt. .. Om humor og alvor i en blodig hverdag av Anton Myhra
Librationist accounts of paradox and other mathematical phenomena Librationism's name was coined ... more Librationist accounts of paradox and other mathematical phenomena Librationism's name was coined from "libration" partly because of its invocation of shifts in perspectives in its treatment of paradoxes. It's a semiformal theory of sorts; all sorts are properties, but not all properties are sorts. Iterative sorts as those in the least Jensen-closure of are sets, but not all sorts are sets in this iterative sense. All conditions on a variable engender a sort, so librationism includes universal sorts and other non-wellfounded sorts; it's a nonextensional theory. Librationism reminds of paraconsistent approaches, but unlike these keeps all theorems of classical logic and never contradicts any of those. For paradoxical sentences such as the one stating that Russell's sort (of all and only sorts that are not self-membered) is a member of itself, librationism proves it, while it also proves its negation. Inference rules are novel, so librationism doesn't prove the conjunction. The librationist perspectives are immune to revenge paradoxes. The semantics is based on a semi inductive Herzberger process, and focuses on one designated model; so librationism is negation complete. This also facilitates an account of Curry's paradox. Librationism is very strong. A fixed point construction shows it's fully impredicative and interprets ID < + Bar-Induction. Related constructions will lift this significantly. Recent progress suggests an Arithmetical Program may be viable as suggested in Bjørdal (2011). Surprisingly, Cantor's entirely valid arguments for uncountable infinites do not prove the existence of uncountable infinities librationistically, but instead serve to show that the assumption that certain sorts are not paradoxical must be given up.
We present a strategy to avoid versions of Humean and counterfactual skepticism based upon a deon... more We present a strategy to avoid versions of Humean and counterfactual skepticism based upon a deontologist theory of justification, a partial guideline for how to side step Gettier problems for certain statements and the assumption that certain statements are compelling. As an upshot the threats of Humean skeptical arguments disappear for some subjects and classes of statements
The purpose of the following is to come to more clarity concerning what entities we need commit t... more The purpose of the following is to come to more clarity concerning what entities we need commit to in order to provide adequate semantics for modal logics and related logics for e.g. counterfactuals. The theses defended have as consequence that we do not need possible worlds as first order objects, and certainly not pluralities so utterly implausibly suggested by David Lewis. As there is a plethora of points of view to make sense of what "possible worlds" are, e.g. modal fictionalism, it seems pertinent to point out certain simplifications that can be made in order to avoid some befuddlements which are prone to put philosophical discourse in disrepute. The view defended here is that we only need valuation-attributes, which may be regarded as states or properties of propositions (sentences) and, in the quantificational case, also of ordered pairs formed from elements of and sets from a given domain of discourse and linguistic items. We first point out that we may presuppose a kindred semantics for classical propositional logic by letting valuations, or valuation-attributes, be states or properties of propositional variables of the formal language. For V a valuation and p a propositional variable, we have Vp or not Vp. For the recursive clauses we presuppose Vp iff not Vp and V(pq) iff Vp and Vq. It is straightforward to verify that is a tautology iff V for any valuation-attribute fulfilling the imposed constraints. A slight advantage with this approach over typical approaches is that we need not postulate functions that have truth or falsity, or more conventionally e.g. 0 or 1, as values when applied to formulas. The slight advantage lies in the fact that the denotations of "truth" and "falsity" remain obscure and the invocation of e.g. "0" and "1" seem arbitrary, whereas copulation as presupposed in the suggested framework here does not involve the postulation of such arbitrary or obscure entities. We move on to modal logics and first consider the propositional case: Let Greek letters , , in the following stand for formulas in the object language of some modal propositional logic. Standardly, a formula is seen as valid on a frame <W,R> iff V(,w)=1 for all models <W,R,V> based on <W,R> and every wW. W is thought of as a set of possible worlds, and RW 2 is the accessibility relation on W. Instead of V(,w)=1, many prefer to write wV. This latter notation suggests that the "possible worlds" may be seen as functions from the value assignment V of a modal model to a compounded valuation-property wV. This hints that we instead of postulating a set of possible worlds, may restrict ourselves to what we call an evaluation frame <E,R>, where the evaluation E is a set of valuationattributes (often we just write valuations) V, V'... and RE 2 the accessibility relation on E. We think of the evaluation frames as our models, and are interested in validity relative to various models, i.e. evaluation frames, with various restrictions on R. In order to remind that we are making use of the evaluation semantics and not a standard possible worlds semantics we continually use the term "evaluation frame" instead of "model". Notice that what we henceforth think of as valuations in the evaluation semantics can neither be identified with possible worlds nor with valuation-assignments (often just called valuations) relative to frames. Instead, what we take as valuations in the evaluation semantics is rather some sort of hybrid, if you like.
The purpose of the following is to come to more clarity concerning what entities we need commit t... more The purpose of the following is to come to more clarity concerning what entities we need commit to in order to provide adequate semantics for modal logics and related logics for e.g. counterfactuals. The thesis defended is that we do not need possible worlds as first order objects, and certainly not pluralities so utterly implausibly suggested by David Lewis. As there is a plethora of points of view to make sense of what "possible worlds" are, e.g. modal fictionalism, it seems pertinent to point out certain simplifications that can be made in order to avoid some befuddlements which are prone to put philosophical discourse in disrepute. The view defended here is that we only need valuation-attributes, which may be regarded as properties of propositions (sentences) and, in the quantificational case, also of ordered pairs formed from elements of and sets from a given domain of discourse and linguistic items. The propositional case: Let Greek letters , , stand for formulas in the object language of some modal propositional logic. Standardly, a formula is seen as valid on a frame <W,R> iff V(,w)=1 for all models <W,R,V> based on <W,R> and every wW. W is thought of as a set of possible worlds, and RW 2 is the accessibility relation on W. Instead of V(,w)=1, many prefer to write wV. This latter notation suggests that the "possible worlds" may be seen as functions from the value assignment V of a modal model to a compounded valuation-property wV. This hints that we instead of postulating a set of possible worlds, may restrict ourselves to what we call an evaluation frame <E, R>, where the evaluation E is a set of valuation-attributes (often we just write valuations) V, V'... and RE 2 the accessibility relation on E. We think of the evaluation frames as our models, and are interested in validity relative to various models, i.e. evaluation frames, with various restrictions on R. In order to make clear that we are making use of the evaluation semantics and not a standard possible worlds semantics we continually use the term "evaluation frame" instead of "model". Notice that what we henceforth think of as valuations in the evaluation semantics can neither be identified with possible worlds nor with valuation-assignments (often just called valuations) relative to frames. Instead, what we take as valuations in the evaluation semantics is rather some sort of hybrid, if you like.
arXiv (Cornell University), Jul 15, 2014
A metaphysical system engendered by a third order quantified modal logic S5 plus impredicative co... more A metaphysical system engendered by a third order quantified modal logic S5 plus impredicative comprehension principles is used to isolate a third order predicate , and by being able to impredicatively take a second order predicate G to hold of an individual just if the individual necessarily has all second order properties which are we in Section 2 derive the thesis (40) that all properties are or some individual is G. In Section 3 theorems 1 to 3 suggest a sufficient kinship to Gödelian ontological arguments so as to think of thesis (40) in terms of divine property and Godly being; divine replaces positive with Gödel and others. Thesis (40), the sacred thesis, supports the ontological argument that God exists because some property is not divine. In Section 4 a fixed point analysis is used as diagnosis so that atheists may settle for the minimal fixed point. Theorem 3 shows it consistent to postulate theistic fixed points, and a monotheistic result follows if one assumes theism and that it is divine to be identical with a deity. Theorem 4 (the Monotheorem) states that if Gg and it is divine to be identical with g, then necessarily all objects which are G are identical with g. The impredicative origin of suggests weakened Gaunilolike objections that offer related theses for other second order properties and their associated diverse presumptive individual bearers. Nevertheless, in the last section we finesse these Gaunilo-like objections by adopting what we call an apathiatheistic opinion which suggest that the best concepts 'God' allow thorough indifference as to whether God exists or not.
The Review of Symbolic Logic, 2010
We show that a paraconsistent set theory proposed in Weber (2010) is strong enough to provide a q... more We show that a paraconsistent set theory proposed in Weber (2010) is strong enough to provide a quite classical nonprimitive notion of identity, so that the relation is an equivalence relation and also obeys full substitutivity: a = b → (F(a) → F(b)). With this as background it is shown that the proposed theory also proves ∀x(x ≠ x). While not by itself showing that the proposed system is trivial in the sense of proving all statements, it is argued that this outcome makes the system inadequate.
With the avoidance of Russell’s paradox and its cognates as one paramount motivation, and the avo... more With the avoidance of Russell’s paradox and its cognates as one paramount motivation, and the avoidance of ungrounded mathematical objects as another, the twentieth century from early on saw the initiation of various foundational theories which altogether avoided an invocation of infinite power sets. This is famously the case in the predicativist tradition going back to Herman Weyl’s Das Kontinuum, and further investigated later principally by Solomon Feferman, but also by others. This was clearly also an important motivational aspect of the perhaps less rigorously formulated original intent of Luitzen Brouwer’s intuitionist program, and it is presently manifest much more precisely within parts of the intuitionist tradition in that Per Martin Lof’s constructive Type Theory lacks an analogue of the infinitary power-set operation, as does Peter Aczel’s constructive set theory CZF. The reverse mathematics program initiated by Harvey Friedman seemed to have established that only a very ...
This was the 4th world congress organized about the square of opposition after very succesful pre... more This was the 4th world congress organized about the square of opposition after very succesful previous editions in Montreux, Switzerland 2007, Corté, Corsica 2010, Beirut, Lebanon, 2012. An interdisciplinary event gathering logicians, philosophers, mathematicians, semioticians, theologians, cognitivists, artists and computer scientists.
http://www.square-of-opposition.org/square2014.html
Logic and Logical Philosophy, Jun 13, 2013
Mathematical Reviews, 2019
This paper analyzes the use of diagrams in mathematical reasoning and puts forward examples where... more This paper analyzes the use of diagrams in mathematical reasoning and puts forward examples where diagrams are useful. It is not suggested that diagrammatic reasoning is sometimes necessary to justify mathematical results, but rather that such reasoning in some cases may be useful in discovering new theorems or in their communication. The topic could to a large extent have been treated by psychological research, it seems. Some typos interfere with the presentation. In the references, the title of the article and the page numbers of the article by Barwise and Etchemendy have been replaced by the information for the article by De Cruz
arXiv (Cornell University), Dec 22, 2022
Springer eBooks, 2015
The main purpose of this article is to discuss and clarify philosophical issues in connection wit... more The main purpose of this article is to discuss and clarify philosophical issues in connection with the librationst set theory £. We take technical results in and upon £ for granted though make references to them as that is useful. We defend the view that £ is neither an inconsistent nor a contradictory system and point out that it is neither paraconsistent nor dialetheist; in contrast we consider £ a bialethic and paracoherent theory.
Kantiansk anskuelse og høyere aritmetikk av Herman Ruge Jervell Time Out of Time: Towards an Expa... more Kantiansk anskuelse og høyere aritmetikk av Herman Ruge Jervell Time Out of Time: Towards an Expanded, Non-trivial and Unitary Theory of Time from Recent Advances in the Hadronic Sciences av Stein Erik Johansen (hilsen til jubilanten side 217) 219 Et strukturalistisk syn på naturlige tall og deres representerbarhet av Frode Kjosavik Naturen i tenkningen av Telje Kvilhaug Tidens mote: En kamp mellom produsenter og lwnsumenter av Tlygve Lavik Kants absolutte rom i GegelUJell og rommet som (I priori i Kri/ik En sammenstilling av Anita Leirfall VI Potensialitet i naturen av Svein Anders Noer Lie Discourse on the Method versus Orientation in Thinking av Francoise Monnoyeur Den som kun tar spøg for spøg, og alvor kun alvorligt. .. Om humor og alvor i en blodig hverdag av Anton Myhra
Librationist accounts of paradox and other mathematical phenomena Librationism's name was coined ... more Librationist accounts of paradox and other mathematical phenomena Librationism's name was coined from "libration" partly because of its invocation of shifts in perspectives in its treatment of paradoxes. It's a semiformal theory of sorts; all sorts are properties, but not all properties are sorts. Iterative sorts as those in the least Jensen-closure of are sets, but not all sorts are sets in this iterative sense. All conditions on a variable engender a sort, so librationism includes universal sorts and other non-wellfounded sorts; it's a nonextensional theory. Librationism reminds of paraconsistent approaches, but unlike these keeps all theorems of classical logic and never contradicts any of those. For paradoxical sentences such as the one stating that Russell's sort (of all and only sorts that are not self-membered) is a member of itself, librationism proves it, while it also proves its negation. Inference rules are novel, so librationism doesn't prove the conjunction. The librationist perspectives are immune to revenge paradoxes. The semantics is based on a semi inductive Herzberger process, and focuses on one designated model; so librationism is negation complete. This also facilitates an account of Curry's paradox. Librationism is very strong. A fixed point construction shows it's fully impredicative and interprets ID < + Bar-Induction. Related constructions will lift this significantly. Recent progress suggests an Arithmetical Program may be viable as suggested in Bjørdal (2011). Surprisingly, Cantor's entirely valid arguments for uncountable infinites do not prove the existence of uncountable infinities librationistically, but instead serve to show that the assumption that certain sorts are not paradoxical must be given up.
We present a strategy to avoid versions of Humean and counterfactual skepticism based upon a deon... more We present a strategy to avoid versions of Humean and counterfactual skepticism based upon a deontologist theory of justification, a partial guideline for how to side step Gettier problems for certain statements and the assumption that certain statements are compelling. As an upshot the threats of Humean skeptical arguments disappear for some subjects and classes of statements
The purpose of the following is to come to more clarity concerning what entities we need commit t... more The purpose of the following is to come to more clarity concerning what entities we need commit to in order to provide adequate semantics for modal logics and related logics for e.g. counterfactuals. The theses defended have as consequence that we do not need possible worlds as first order objects, and certainly not pluralities so utterly implausibly suggested by David Lewis. As there is a plethora of points of view to make sense of what "possible worlds" are, e.g. modal fictionalism, it seems pertinent to point out certain simplifications that can be made in order to avoid some befuddlements which are prone to put philosophical discourse in disrepute. The view defended here is that we only need valuation-attributes, which may be regarded as states or properties of propositions (sentences) and, in the quantificational case, also of ordered pairs formed from elements of and sets from a given domain of discourse and linguistic items. We first point out that we may presuppose a kindred semantics for classical propositional logic by letting valuations, or valuation-attributes, be states or properties of propositional variables of the formal language. For V a valuation and p a propositional variable, we have Vp or not Vp. For the recursive clauses we presuppose Vp iff not Vp and V(pq) iff Vp and Vq. It is straightforward to verify that is a tautology iff V for any valuation-attribute fulfilling the imposed constraints. A slight advantage with this approach over typical approaches is that we need not postulate functions that have truth or falsity, or more conventionally e.g. 0 or 1, as values when applied to formulas. The slight advantage lies in the fact that the denotations of "truth" and "falsity" remain obscure and the invocation of e.g. "0" and "1" seem arbitrary, whereas copulation as presupposed in the suggested framework here does not involve the postulation of such arbitrary or obscure entities. We move on to modal logics and first consider the propositional case: Let Greek letters , , in the following stand for formulas in the object language of some modal propositional logic. Standardly, a formula is seen as valid on a frame <W,R> iff V(,w)=1 for all models <W,R,V> based on <W,R> and every wW. W is thought of as a set of possible worlds, and RW 2 is the accessibility relation on W. Instead of V(,w)=1, many prefer to write wV. This latter notation suggests that the "possible worlds" may be seen as functions from the value assignment V of a modal model to a compounded valuation-property wV. This hints that we instead of postulating a set of possible worlds, may restrict ourselves to what we call an evaluation frame <E,R>, where the evaluation E is a set of valuationattributes (often we just write valuations) V, V'... and RE 2 the accessibility relation on E. We think of the evaluation frames as our models, and are interested in validity relative to various models, i.e. evaluation frames, with various restrictions on R. In order to remind that we are making use of the evaluation semantics and not a standard possible worlds semantics we continually use the term "evaluation frame" instead of "model". Notice that what we henceforth think of as valuations in the evaluation semantics can neither be identified with possible worlds nor with valuation-assignments (often just called valuations) relative to frames. Instead, what we take as valuations in the evaluation semantics is rather some sort of hybrid, if you like.
The purpose of the following is to come to more clarity concerning what entities we need commit t... more The purpose of the following is to come to more clarity concerning what entities we need commit to in order to provide adequate semantics for modal logics and related logics for e.g. counterfactuals. The thesis defended is that we do not need possible worlds as first order objects, and certainly not pluralities so utterly implausibly suggested by David Lewis. As there is a plethora of points of view to make sense of what "possible worlds" are, e.g. modal fictionalism, it seems pertinent to point out certain simplifications that can be made in order to avoid some befuddlements which are prone to put philosophical discourse in disrepute. The view defended here is that we only need valuation-attributes, which may be regarded as properties of propositions (sentences) and, in the quantificational case, also of ordered pairs formed from elements of and sets from a given domain of discourse and linguistic items. The propositional case: Let Greek letters , , stand for formulas in the object language of some modal propositional logic. Standardly, a formula is seen as valid on a frame <W,R> iff V(,w)=1 for all models <W,R,V> based on <W,R> and every wW. W is thought of as a set of possible worlds, and RW 2 is the accessibility relation on W. Instead of V(,w)=1, many prefer to write wV. This latter notation suggests that the "possible worlds" may be seen as functions from the value assignment V of a modal model to a compounded valuation-property wV. This hints that we instead of postulating a set of possible worlds, may restrict ourselves to what we call an evaluation frame <E, R>, where the evaluation E is a set of valuation-attributes (often we just write valuations) V, V'... and RE 2 the accessibility relation on E. We think of the evaluation frames as our models, and are interested in validity relative to various models, i.e. evaluation frames, with various restrictions on R. In order to make clear that we are making use of the evaluation semantics and not a standard possible worlds semantics we continually use the term "evaluation frame" instead of "model". Notice that what we henceforth think of as valuations in the evaluation semantics can neither be identified with possible worlds nor with valuation-assignments (often just called valuations) relative to frames. Instead, what we take as valuations in the evaluation semantics is rather some sort of hybrid, if you like.
arXiv (Cornell University), Jul 15, 2014
A metaphysical system engendered by a third order quantified modal logic S5 plus impredicative co... more A metaphysical system engendered by a third order quantified modal logic S5 plus impredicative comprehension principles is used to isolate a third order predicate , and by being able to impredicatively take a second order predicate G to hold of an individual just if the individual necessarily has all second order properties which are we in Section 2 derive the thesis (40) that all properties are or some individual is G. In Section 3 theorems 1 to 3 suggest a sufficient kinship to Gödelian ontological arguments so as to think of thesis (40) in terms of divine property and Godly being; divine replaces positive with Gödel and others. Thesis (40), the sacred thesis, supports the ontological argument that God exists because some property is not divine. In Section 4 a fixed point analysis is used as diagnosis so that atheists may settle for the minimal fixed point. Theorem 3 shows it consistent to postulate theistic fixed points, and a monotheistic result follows if one assumes theism and that it is divine to be identical with a deity. Theorem 4 (the Monotheorem) states that if Gg and it is divine to be identical with g, then necessarily all objects which are G are identical with g. The impredicative origin of suggests weakened Gaunilolike objections that offer related theses for other second order properties and their associated diverse presumptive individual bearers. Nevertheless, in the last section we finesse these Gaunilo-like objections by adopting what we call an apathiatheistic opinion which suggest that the best concepts 'God' allow thorough indifference as to whether God exists or not.
The Review of Symbolic Logic, 2010
We show that a paraconsistent set theory proposed in Weber (2010) is strong enough to provide a q... more We show that a paraconsistent set theory proposed in Weber (2010) is strong enough to provide a quite classical nonprimitive notion of identity, so that the relation is an equivalence relation and also obeys full substitutivity: a = b → (F(a) → F(b)). With this as background it is shown that the proposed theory also proves ∀x(x ≠ x). While not by itself showing that the proposed system is trivial in the sense of proving all statements, it is argued that this outcome makes the system inadequate.
With the avoidance of Russell’s paradox and its cognates as one paramount motivation, and the avo... more With the avoidance of Russell’s paradox and its cognates as one paramount motivation, and the avoidance of ungrounded mathematical objects as another, the twentieth century from early on saw the initiation of various foundational theories which altogether avoided an invocation of infinite power sets. This is famously the case in the predicativist tradition going back to Herman Weyl’s Das Kontinuum, and further investigated later principally by Solomon Feferman, but also by others. This was clearly also an important motivational aspect of the perhaps less rigorously formulated original intent of Luitzen Brouwer’s intuitionist program, and it is presently manifest much more precisely within parts of the intuitionist tradition in that Per Martin Lof’s constructive Type Theory lacks an analogue of the infinitary power-set operation, as does Peter Aczel’s constructive set theory CZF. The reverse mathematics program initiated by Harvey Friedman seemed to have established that only a very ...
Logic and Logical Philosophy
A metaphysical system engendered by a third order quantified modal logic S5 plus impredicative co... more A metaphysical system engendered by a third order quantified modal logic S5 plus impredicative comprehension principles is used to isolate a third order predicate , and by being able to impredicatively take a second order predicate G to hold of an individual just if the individual necessarily has all second order properties which are we in Section 2 derive the thesis (40) that all properties are or some individual is G. In Section 3 theorems 1 to 3 suggest a sufficient kinship to Gödelian ontological arguments so as to think of thesis (40) in terms of divine property and Godly being; divine replaces positive with Gödel and others. Thesis (40), the sacred thesis, supports the ontological argument that God exists because some property is not divine. In Section 4 a fixed point analysis is used as diagnosis so that atheists may settle for the minimal fixed point. Theorem 3 shows it consistent to postulate theistic fixed points, and a monotheistic result follows if one assumes theism and that it is divine to be identical with a deity. Theorem 4 (the Monotheorem) states that if Gg and it is divine to be identical with g, then necessarily all objects which are G are identical with g. The impredicative origin of suggests weakened Gaunilolike objections that offer related theses for other second order properties and their associated diverse presumptive individual bearers. Nevertheless, in the last section we finesse these Gaunilo-like objections by adopting what we call an apathiatheistic opinion which suggest that the best concepts 'God' allow thorough indifference as to whether God exists or not.