What are quantifiers? Quantifiers have been thought of things that ‘range over’ a set of objects. For example, if I say
There are people with blue eyes
this statement can be represented as (with the domain restricted to people):
∃x(Bx).
This statement says that there is at least one person with property B, blue eyes. So the ‘Ex’ is doing the work of looking at the people in the domain (all people) and picking out one with blue eyes. Without this ‘∃x’ we would just have Bx, or x has blue eyes.
This concept of ‘ranging over’ and selecting an individual with a specific property out of the whole group works in the vast majority of applications. However, I’ve pointed out a few instances in which it makes no sense to think of the domain as a predetermined group of objects, such as in natural language and relativistic situations. In these cases the domain cannot be defined until something about the people involved are known, if at all; people may have a stock set of responses to questions but can also make new ones up.
So, since the problem resides with a static domain being linked to specific people, I suggest that we find a way to link quantifiers to those people. This means that if two people are playing a logic game, each person will have their own quantifiers linked to their own domain. The domains will be associated with the knowledge (or other relevant property) of the people playing the game.
We could index individual quantifiers to show which domain they belong to, but game theory has a mechanism for showing which player is making a move by using negation. When a negation is reached in a logic game, it signals that it is the other player’s turn to make a move. I suggest negation should also signal a change in domains, as to mirror the other player’s knowledge.
Using negation to switch the domain that the quantifiers reference is more realistic/ natural treatment of logic: when two people are playing a game, one may know certain things to exist that the other does not. So using one domain is an unrealistic view of the world because it is only in special instances that two people believe the exact same objects to exist in the world. Of course there needs to be much overlap for two people to be playing the same game, but having individual domains to represent individual intelligences makes for a more realistic model of reality.
Now that each player in a game has his or her own domain, what is the activity of the quantifier? It still seems to be ranging over a domain, even if the domain is separate, so the problem raised above has not yet been dealt with.
Besides knowing different things, people think differently too. The different ways people deal with situations can be described as unique strategies. Between the strategies people have and their knowledge we have an approximate representation of a person playing a logic game.
If we now consider how quantifiers are used in logic games, whenever we encounter one we have to choose an element of the domain according to a strategy. This strategy is a set of instructions that will yield a specified result and are separate from the domain. So quantifiers are calls to use a strategy as informed by your domain, your knowledge. They do not ‘range over’ the domain; it is the strategies a person uses that take the domain and game (perhaps “game-state” is more accurate at this point) as inputs and returns an individual.
The main problem mentioned above can now be addressed: Instead of predetermining sets objects in domains, what we need to predetermine are the players in the game. The players may be defined by a domain of objects and strategies that will be used to play the game, but this only becomes relevant when a quantifier is reached in the game. Specifying the players is sufficient because each brings his or her own domain and strategies to the game, so nothing is lost, and the domain and strategies do no have to be predefined because they are initially called upon within the game, not before.
I don’t expect this discussion to cause major revisions to the way people go about practicing logic, but I do hope that it provides a more natural way to think about what is going on when dealing with quantifiers and domains, especially when dealing with relativistic or natural language situations.
