(β) A _propositional function_ is necessary when all its values are true. This sense is not relevant to our present discussion. (γ) A _proposition_ is necessary with respect to a given constituent when it is the value, with that constituent as argument, of a necessary propositional function, in other words, when it remains true however that constituent may be varied. In this sense, in a deterministic system, the connection of a volition with its determinants is necessary, if the time at which the determinants occur be taken as the constituent to be varied, the time-interval between the determinants and the volition being kept constant. But this sense of necessity is purely logical, and has no emotional importance. We may now sum up our discussion of causality. We found first that the law of causality, as usually stated by philosophers, is false, and is not employed in science. We then considered the nature of scientific laws, and found that, instead of stating that one event A is always followed by another event B, they stated functional relations between certain events at certain times, which we called determinants, and other events at earlier or later times or at the same time. We were unable to find any _a priori_ category involved: the existence of scientific laws appeared as a purely empirical fact, not necessarily universal, except in a trivial and scientifically useless form. We found that a system with one set of determinants may very likely have other sets of a quite different kind, that, for example, a mechanically determined system may also be teleologically or volitionally determined. Finally we considered the problem of free will: here we found that the reasons for supposing volitions to be determined are strong but not conclusive, and we decided that even if volitions are mechanically determined, that is no reason for denying freedom in the sense revealed by introspection, or for supposing that mechanical events are not determined by volitions. The problem of free will _versus_ determinism is therefore, if we were right, mainly illusory, but in part not yet capable of being decisively solved. FOOTNOTES: [35] A propositional function is an expression containing a variable, or undetermined constituent, and becoming a proposition as soon as a definite value is assigned to the variable. Examples are: "A is A," "_x_ is a number." The variable is called the _argument_ of the function. [36] _Logic_, Bk. III, Chap. V, § 2. [37] _Time and Free Will_, p. 199. [38] _Time and Free Will._ p. 202. [39] _Loc. cit._, § 6 X KNOWLEDGE BY ACQUAINTANCE AND KNOWLEDGE BY DESCRIPTION