I was thinking that upside down A and backwards E were feeling lonely. Yes, ∀ and ∃ love each other very much, but they could really use a new friend. Introducing Angle I:
Now, Angle I, , is just like her friends ∀ and ∃. She can be used in a formula such as ∀x∃yz(Pxyz).
But how should we understand what is going on with the failure of the quantified tertium non datur? With that advent of a third quantifier, what’s to stop us from having a fourth, fifth or n quantifiers?
The Fregean tradition of quantifiers states that the upside down A means ‘for any” and the backwards E mean ‘there exists some’. So ‘∀x∃yPxy’ means ‘for any x, there exists some y, such that x and y are related by property P’. For instance we could say that for any rational number x there exists some other rational number y such that y=x/2.
If we, however, follow closer to the game-theoretic tradition of logic, then the quantifiers no longer need take on their traditional role. The two quantifiers act like players in a game, in which the object is to make the total statement true or false. In our above example, we would say that backwards E would win the game, because no matter what number upside down A picks, there is always some number that ∃ could find that is twice the number ∀ chose.
Under this view of quantifiers, quantifiers acting as players in a game, there is no reason why there can’t be any number of players. (Personally, I like the idea of continuing down the list of vowels: upside-down A, backwards E, angle I, then inverted O, O, maybe angle U? Go historical with Abelard, hEloise, and then Fulbert? Suggestions?)
Now, what is it good for? Let’s play a game of Agent Logic!
The purpose of a game of Agent Logic is to determine the loyalties of the agents in that game, i.e. discover any secret agents. A game consists of a particular logical situation, as given by formulae of independence friendly logic, with at least three different agents, each of which is represented by a quantifier: ∀, ∃, angle I, inverted O, etc. Each agent has an associated ‘domain’, and for the game to be non-trivial the intersection of the domains must have at least one element.
A game of Agent Logic is played by determining the information dependencies required to derive the target formulae from the premise formulae. Once the required information dependencies are known, then the strategies and loyalties of the agents have used may be inferred. The simplest solution to a game is one in which an information dependence indicates a loyalty: if an agent has access to certain information, then that agent must have a specific loyalty.
The person running the game is the Intelligence Director, given by the quantifier angle-I. This is you! All other agents are possible opposing Intelligence Directors or secret agents of the opposing Intelligence Directors. It is your job to figure out who has given who access to information and how that agent has acted upon it. Any information or strategy that is not derivable from the premises are considered acts of treason against you, the Intelligence Director. If the target premise (conclusion) is derivable from the premises alone, no determination of loyalty can be made.
The ‘domain’ of angle-I consists of what you depend upon, i.e. what you believe to exist and what you believe the other agent’s believe to exist. (Though it is a premise itself.) Recall that the backslash, , means ‘is dependent upon’ and the forward slash, /, means ‘is independent of’.
∀ (a, b, c),
∃ (a, b, c, d),
a, b, c, d
In this ‘domain’ of angle-I, the Intelligence Director is dependent upon ∀ depending upon the existence of a, b and c, and being independent of ∃, that ∃ depends on the existence of a, b, c and d, and the director herself depends upon the existence of a, b, c, and d.
Now, since angle-I depends upon ∀ not depending upon d, there is no way to derive the target from the premises. However, since ∃ does depend upon d, if ∀ depends upon ∃, then agent ∀ has access to d.
Therefore, given treason,
4. ∀ (∃(d)) [premise of treason – ∀ receives information from ∃, specifically d ]
5. Pd [instantiation from 2, 4]
This shows that the conclusion can be reached if ∀ is treasonous, a secret agent of ∃, i.e. ∀ is loyal to ∃ and not angle-I.