by Brian Bosse, Copyright February 25, 2008, all rights reserved. 68 views
This thread will be an brief explication of Middle Knowledge (MK) followed by a possible objection and response. MK is an attempt to reconcile divine providence with libertarian freedom. The issue is rather accute. The full range of the issue spans those that deny libertarian freedom in light of divine providence (most, but not all Calvinists) to those that deny traditional attributes of God in keeping with libertarian freedom (open theists), and everything in between. The MK solution proposes to maintain the traditional understanding of God together with the view that humans have libertarian freedom. MK lends itself to be understood within the framework of Possible World Semantics, and as such I will begin with a very brief introduction to the logic of possible world semantics.
Possible World Semantics
The idea of a possible world is simply a world made up of a set of possible events. Our actual world is one such world in the set of all possible worlds. The only events that are precluded are events that are logically incoherent – whether this incoherence is relative to the event itself (a square circle), or incoherent relative to the events around it (a possible world with no oxygen cannot have water). Here are some possible worlds:
Possible World 1 (our actual world): All events as they have happened up to this point.
Possible World 2: All events as they have happened up to this point with the exception of me wearing a red shirt instead of a blue shirt on February 25, 2008.
Possible World 3: A world where Cesar did not cross the Rubicon.
Possible World 4: A world where Germany won World War II.
A logic was developed in an attempt to capture the ideas of 'possibility' and 'necessity' within possible world semantics. This logic is called 'Modal Logic'. In propositional logic, there are functions that assign truth-values to atomic sentences, and functions that assign truth-values to more complex sentences built up from these atomic sentences using the sentential connectives: ¬, →, ↔, ∧, ∨. In modal semantics, a set W of possible worlds is introduced where these truth-value functions assign a truth-value to each proposition for each of the possible worlds in W. It is possible for particular propositions to be assigned different truth-values in different possible worlds. For instance, in some possible world it is true that Germany won World War II; whereas, in another possible world, it false that Germany won World War II. This makes truth-value relative to a particular possible world. We can now introduce the modal operators of 'necessity' and 'possibility' that make up modal logic.
Modal Operators
□p = 'p' is necessarily true. For 'p' to be necessary (□p), then 'p' is true in all possible worlds.
◊p = 'p' is possibly true. For 'p' to possible (◊p), then 'p' is true in at least one possible world.
p = 'p' is actually true. For 'p' to be actual (p), then 'p' is true in the real world.
It should be noted that we can define both □ and ◊ in terms of each other as follows:
Rule N: □p ↔ ¬◊¬p. That is to say, 'p' is necessarily true if and only if it is not the case that 'p' is false in at least one possible world.
Rule P: ◊p ↔ ¬□¬p. That is to say, 'p' is possibly true if and only if it is not the case that 'p' is false in all possible worlds.
Modal logic is essentially propositional logic combined with the modal operators □ and ◊ as defined above. At this point we have laid the foundation necessary to understand Middle Knowledge (MK) within the framework of possible world semantics. My next post will begin my analysis.
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