Beliefs supporting other beliefs: Truthmaker semantics for logic-neutral inference in hyperintensional webs of belief
(Peter Verdée, UCLouvain)
We present a new way to look at the web of beliefs of an epistemic agent (e.g. a scientist or a group of scientists). The web of belief, a notion first introduced by Quine, is the holistic and interconnected collection of all beliefs of an agent, including more fundamental beliefs, such as logical and metaphysical principles. Some beliefs in this web are more central than others, which makes them harder to revise (since this requires revising a lot of beliefs depending on it), but in principle all elements of the web can be revised. In our approach we distinguish structural and objectual beliefs. The former determine the internal structure of descendent parts of the web itself (logical, epistemological and metaphysical beliefs) while the latter rather concern the external world, including abstract objects (scientific, political and ethical beliefs). This is not a sharp distinction; some beliefs may have both functions and the transition may be gradual.
We argue that the objectual beliefs in such a web should not be seen as sentential, in the sense of entities that depend on the specific way in which they are formulated, nor as propositional, in the traditional sense of a proposition being a set of possible worlds. The most important argument against the latter is the fact that agents do not have the competence to (and should not be expected to) recognize logically equivalent beliefs. Instead one could see believing as an attitude towards hyperintensions and towards relations between hyperintensions. Hyperintensions are ways to understand meanings in a more fine-grained manner than intensions, so that, unlike in the case of intensions, logically equivalent sentences do not necessarily express the same hyperintension. We offer a semantic approach to hyperintensions. For that purpose, we use a variant of truthmaker semantics as proposed by Kit Fine, in which hyperintensions are seen as sets of possible states. States are, much like situations, incomplete and possibly inconsistent parts of (im)possible worlds.
What is radically new in our approach is that the states are defined in a logic neutral way. Because logical beliefs are themselves part of the web of belief, the web of belief does not at all require sticking to classical logic, nor does it require the believer to single out one logic as the correct logic. Different logics may be acceptable in different reasoning contexts, as long as the logical beliefs clearly determine in each context which logic should be used. We devise a way to define an L-truthmaker semantics for each propositional Tarskian formal logic L (i.e. monotonic, transitive and reflexive formal consequence relations). In L-reasoning contexts, a belief is identified with the set of states that, according to the L-truthmaker semantics, verify a specific sentence expressing the belief.
Within the set of objectual beliefs we distinguish restrictive beliefs from normal beliefs. Normal beliefs can be seen as (attitudes towards) sets of states, but restrictive beliefs are on top of that also (attitudes towards) relations among sets of states. They determine how the web holds the normal beliefs together. Concrete examples are the beliefs that are used as the axioms in axiomatically structured sets of beliefs, lawlike beliefs, and, more generally, general beliefs agents tend to explicitly remember instead of derive from other beliefs in concrete cases. The appropriate semantic relation for the latter is not the traditional relation of states verifying beliefs (as for normal beliefs). Rather, we define a new relation of “support” between sets of restrictive beliefs and sets of states, interpreted as follows: a set of restrictive beliefs supports a set of states iff whatever is made true by all members of the set of states can be (relevantly) L-inferred from the combination of “restrictive beliefs”. A logic neutral non-transitive approach to relevant implication will be used to formally specify the relation of relevant L-inference.
The upshot of this new approach to the web of belief is that a formal method is proposed to individuate beliefs and provide the structure of the web, without imposing any specific logic that would hold for all webs of belief and that therefore would overrule or be incompatible with logical beliefs inside a web.
Click for slides.
Obs: this talk has been exceptionally recorded at the Department of Philosophy of the University of Bergen (Norway) in joint work with the Bergen Logic Group.
The present recording has been realized with support from the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES, Brazil).
We argue that the objectual beliefs in such a web should not be seen as sentential, in the sense of entities that depend on the specific way in which they are formulated, nor as propositional, in the traditional sense of a proposition being a set of possible worlds. The most important argument against the latter is the fact that agents do not have the competence to (and should not be expected to) recognize logically equivalent beliefs. Instead one could see believing as an attitude towards hyperintensions and towards relations between hyperintensions. Hyperintensions are ways to understand meanings in a more fine-grained manner than intensions, so that, unlike in the case of intensions, logically equivalent sentences do not necessarily express the same hyperintension. We offer a semantic approach to hyperintensions. For that purpose, we use a variant of truthmaker semantics as proposed by Kit Fine, in which hyperintensions are seen as sets of possible states. States are, much like situations, incomplete and possibly inconsistent parts of (im)possible worlds.
What is radically new in our approach is that the states are defined in a logic neutral way. Because logical beliefs are themselves part of the web of belief, the web of belief does not at all require sticking to classical logic, nor does it require the believer to single out one logic as the correct logic. Different logics may be acceptable in different reasoning contexts, as long as the logical beliefs clearly determine in each context which logic should be used. We devise a way to define an L-truthmaker semantics for each propositional Tarskian formal logic L (i.e. monotonic, transitive and reflexive formal consequence relations). In L-reasoning contexts, a belief is identified with the set of states that, according to the L-truthmaker semantics, verify a specific sentence expressing the belief.
Within the set of objectual beliefs we distinguish restrictive beliefs from normal beliefs. Normal beliefs can be seen as (attitudes towards) sets of states, but restrictive beliefs are on top of that also (attitudes towards) relations among sets of states. They determine how the web holds the normal beliefs together. Concrete examples are the beliefs that are used as the axioms in axiomatically structured sets of beliefs, lawlike beliefs, and, more generally, general beliefs agents tend to explicitly remember instead of derive from other beliefs in concrete cases. The appropriate semantic relation for the latter is not the traditional relation of states verifying beliefs (as for normal beliefs). Rather, we define a new relation of “support” between sets of restrictive beliefs and sets of states, interpreted as follows: a set of restrictive beliefs supports a set of states iff whatever is made true by all members of the set of states can be (relevantly) L-inferred from the combination of “restrictive beliefs”. A logic neutral non-transitive approach to relevant implication will be used to formally specify the relation of relevant L-inference.
The upshot of this new approach to the web of belief is that a formal method is proposed to individuate beliefs and provide the structure of the web, without imposing any specific logic that would hold for all webs of belief and that therefore would overrule or be incompatible with logical beliefs inside a web.
Click for slides.
Obs: this talk has been exceptionally recorded at the Department of Philosophy of the University of Bergen (Norway) in joint work with the Bergen Logic Group.
The present recording has been realized with support from the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES, Brazil).
