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Formal properties of ``now''

H. Kamp. Formal properties of ``now''. Theoria, 37:227–273, 1971.

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Abstract:

This paper is a development of an earlier paper by A. N. Prior [Nous 2 (1968), 101--119]. A semantical framework is described in which tense logics can be extended to include a unary propositional connective $<b>N</b>$ corresponding to the function of "now" in ordinary discourse. A number of results are established, the most important of which are of the following form: Let $\scr L(N)$ be a tense logic with the "now"-operator and let $\scr L$ be the calculus obtained by omitting $N$ from $\scr L(N)$; if an axiom system $\scr A$ for $\scr L$ is semantically complete, then so is an axiom system $\scr A'$ for $\scr L(N)$, where $\scr A'$ is closely related to $\scr A$. The paper is in two parts, the first restricted to propositional calculi, where it is shown that the "now"-operator is always eliminable, while the second is devoted to (first order) predicate calculi (with identity), where it is shown that the "now"-operator is not always eliminable. If $\scr T=\langle T,&lt;\rangle$ represents the structure of time (which for the most part the author assumes to be a partial ordering) and $\germ M$ is an interpretation relative to $\scr T$, the obvious characterization of the truth-value of a formula $\phi$ in $\germ M$ at a moment $t\in T$ is shown to require an additional parameter in order to be extended to an appropriate semantical clause for the "now"-operator. This is essentially because the word "now" when used in ordinary discourse refers back to the moment of utterance, in which case the concept to be characterized is that of the truth-value of $\phi$ in $\germ M$ at $t$ when part of an expression uttered at $t'$. A formula is then to be regarded as valid (relative to $\scr T$) if in every interpretation it is true when uttered. The author also defines and uses alternative but equivalent notions in which an interpretation is a pair $\langle\germ M,t\sb 0\rangle$, where $\germ M$ is as before and $t\sb 0$ represents the present moment. In addition, instead of restricting questions of completeness to a single temporal structure $\scr T$, the author considers non-empty classes of partial orderings and characterizes notions of strong and weak $K$-completeness. He also shows that some axiom systems are weakly but not strongly $<b>K</b>$-complete, for appropriate classes $K$. Finally, it should also be noted that the results described in this paper apply to a rather general notion of a tense operator. Unlike truth-functions, the relevant arguments and values of a tense operator are not truth-values but "courses of truth-values through time", i.e., functions from moments to truth-values.

BibTeX: (download)

@ARTICLE{kamp71:_formal,
  author = {H. Kamp},
  title = {Formal properties of ``now''},
  journal = {Theoria},
  year = {1971},
  volume = {37},
  pages = {227--273},
  abstract = {
	This paper is a development of an earlier paper by A. N. Prior [Nous
	2 (1968), 101--119]. A semantical framework is described in which
	tense logics can be extended to include a unary propositional connective
	${\bf N}$ corresponding to the function of "now" in ordinary discourse.
	A number of results are established, the most important of which
	are of the following form: Let $\scr L({N})$ be a tense logic with
	the "now"-operator and let $\scr L$ be the calculus obtained by omitting
	${N}$ from $\scr L({N})$; if an axiom system $\scr A$ for $\scr L$
	is semantically complete, then so is an axiom system $\scr A'$ for
	$\scr L({N})$, where $\scr A'$ is closely related to $\scr A$.
	The paper is in two parts, the first restricted to propositional calculi,
	where it is shown that the "now"-operator is always eliminable, while
	the second is devoted to (first order) predicate calculi (with identity),
	where it is shown that the "now"-operator is not always eliminable.
	If $\scr T=\langle T,&lt;\rangle$ represents the structure of time
	(which for the most part the author assumes to be a partial ordering)
	and $\germ M$ is an interpretation relative to $\scr T$, the obvious
	characterization of the truth-value of a formula $\phi$ in $\germ
	M$ at a moment $t\in T$ is shown to require an additional parameter
	in order to be extended to an appropriate semantical clause for the
	"now"-operator. This is essentially because the word "now" when used
	in ordinary discourse refers back to the moment of utterance, in
	which case the concept to be characterized is that of the truth-value
	of $\phi$ in $\germ M$ at $t$ when part of an expression uttered
	at $t'$. A formula is then to be regarded as valid (relative to $\scr
	T$) if in every interpretation it is true when uttered. The author
	also defines and uses alternative but equivalent notions in which
	an interpretation is a pair $\langle\germ M,t\sb 0\rangle$, where
	$\germ M$ is as before and $t\sb 0$ represents the present moment.
	In addition, instead of restricting questions of completeness to
	a single temporal structure $\scr T$, the author considers non-empty
	classes of partial orderings and characterizes notions of strong
	and weak ${K}$-completeness. He also shows that some axiom systems
	are weakly but not strongly ${\bf K}$-complete, for appropriate classes
	${K}$.
	Finally, it should also be noted that the results described in this
	paper apply to a rather general notion of a tense operator. Unlike
	truth-functions, the relevant arguments and values of a tense operator
	are not truth-values but "courses of truth-values through time",
	i.e., functions from moments to truth-values. }
}

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