Steve Watson

As an example of the kind of definition that is proposed to cover the latter conception they offer the following: “A has power over B when A can change B’s actions, or sanction B, and so forth.” This definition is only offered as an indicator by those authors, and it is, of course, unsatisfactory as it stands. For example, it does not distinguish between power and mere influence. To distinguish ‘power’ in the sense that it is usually used – and in which sense it has a special significance in informal sociological explanations – from ‘power’ as the mere existence of the effect of one agent on another, we need to concentrate on the implied coercive capabilities of A over B. Thus we would need to include the fact that B changes his behaviours because he believes that A has the ability to punish him if he fails to comply. Moreover, he believes that A actually will punish him if he fails to comply.

We can attempt a definition using the established formalism as follows:

x has power over y iff (∃b_response ∈ E(y))[ B(x) ] which satisfies the conditions

1. E(y)[ C(b_response, x) ] = c_response
2. E(y)[ TS(c_response, I(y), y) ] << E(y)[ TS(c_now, I(y), y) ]
3. (∃b ∈ E(y)[ B(y) ]) [(E(y)[ A(y)=b → A(x)=b_response ] & (E(y)[ (A(y)≠b → A(x)≠b_response) ]]

• Whenever x and y are in a relationship such as this we say that x Dominates y and write x/y
• As presented the relationship of dominance appears to have no essential structure. It is not necessarily transitive, and it is not necessarily non-symmetric.
• Degrees of power can be traced back to the degrees by which the total satisfaction functions are altered in 2.
• Note that E(y)[ statement ] in condition 3 is to be read as the subjective estimation of the fuzzy truth value of ‘statement,’ or the subjective degree of belief of y in ‘statement.’
• Complexity can be added by noting that in most cases b_response is merely one of a class of behaviours that are available to x which can punish y in various degrees, that b in condition 3 is merely one of a class of behaviours that are available to y which can prompt the response from x.
• c_now in condition 2 is the context at the time of consideration, but it should really refer to the context that exists when x does not respond punitively to the b in condition 3.
• Note that this is power through punishment, but there is a similar definition available for power through reward: simply replace condition 2 with the condition
  

  E(y)[ TS(c_response, I(y), y) ] >> E(y)[ TS(c_now, I(y), y) ]

In the light of the notes above, refine the original definition thus:

x dominates y iff
(∃B_response(x) ∈ E(y)[ B(x) ])
(∃B_neutral(x) ∈ E(y)[ B(x) ]-B_response(x))
(∃B_stimulus(y) ∈ E(y)[ B(y) ])
the following conditions are satisfied:

1. (∀b_stimulus ∈ B_stimulus(y)) (∀b_response ∈ B_response(x))
   E(y)[ P(B_neutral, x) | A(y)=b_stimulus & A(x)≠b_response ] ~ 1
2. ∀b_stimulus ∈ B_stimulus(y)) (∃b_response ∈ B_response(x))
   [(E(y)[ A(y)=b_stimulus → A(x)=b_response ]
3. ∀b_response ∈ B_response(x)) (∃b_stimulus ∈ B_stimulus(y))
   [(E(y)[ A(x)=b_response → A(y)=b_stimulus ]
4. ∀c_response ∈ E(y)[ C(B_response, x) ]) (∀c_neutral ∈ E(y)[ C(B_neutral, x) ])
   [E(y)[ TS(c_response, I(y), y) ] << E(y)[ TS(c_neutral, I(y), y) ]