In (1964), W.P. Alston's paper "Linguistic Acts" appeared. There, he describes the connection between what he calls "linguistic acts" and certain rules which may perhaps be described as primary stage linguistic rules, or primary stage meaning rules. Although Alston does not employ the terms "illocutionary act" and "speech act" in this study, he clearly means to refer to those acts which Austin aimed at in How to Do Things With Words (cf. 1964, 138n1). (Incidentally, Alston's notion of a "linguistic act" seems to be inspired by Grice's notion of "meaning something by an utterance"; for details click here.)
Alston’s "linguistic act" is closely associated to an act 'of saying'; it involves one's subjection of an utterance to certain rules, which in turn are apparently conceived of as linguistic, or proto-linguistic, rules. Indeed, in his discussion of the nature of these acts Alston explicitly refers to Wittgenstein's experiment of saying "It's cold here" and meaning 'It's warm here' (1964, 142). The performance of the act involves the subjection of one's utterance (or, more neutrally, behaviour) to those (linguistic or proto-linguistic) rules.
In his later book on Illocutionary Acts and Sentence Meaning (2000), Alston changes over to the term "illocutionary act". Apparently, he is now extremely uncertain how to characterise the notion (yet he does not seem to notice the Ambiguity Problem). Alston characterises the “illocutionary act” as the act of "uttering a sentence (or sentence surrogate) with a certain content" (2000, 2), and he sets as his "guiding star" "the idea that an illocutionary act report is an oratio obliqua report" (2000, 14).
Alston clearly notices that many, if not the majority of the acts which Austin had referred to by the term "illocutionary act", do not fall under the definitions he suggests, and he clearly notices, too, that this is a dilemma (of either facing important counter-evidence or abandoning his definitions). (cf., e.g., 2000, 81-2). Thus, for example, promising involves placing oneself under an obligation to do the promised thing, and adjourning a meeting involves bringing it about that the meeting is no longer in session. Involving the production of these conventional effects, however, those acts are not (mere) acts with a content. In the end, Alston decides to maintain his definitions (2000, 91).
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Alston, W.P. (1964), "Linguistic Acts", American Philosophical Quarterly 1/2, 138-146.
Alston, W.P. (2000), Illocutionary Acts and Sentence Meaning, New York: Cornell University Press.