The authors effect two notable achievements. First, they develop a modified system of standard logic which takes account of time-related terms. Then they apply this logic to reconstruct and analyze the argument between Aristotle and Diodorus on fatalism.

The 'Augmented Standard Logic' (ASL) that the authors develop permits temporal constants and variables. These may represent durationless instants, discrete periods such as days or years, or boundless times. ASL includes a wide range of application. In an Appendix, Barker and Paxson develop 53 theorems for their system, based on a couple dozen definitions, axioms. primitive rules and postulates.

A pertinent detail about ASL is that it does not contain what they call the Unrestricted Temporal Bivalence Thesis: that of necessity every proposition is either true during any given time or false during that time. Instead, it contains a Restricted Temporal Bivalence Thesis: that of necessity every proposition affirming the occurrence of an event is either true during any given time or false during that time provided that the time is simultaneous with or later than the event to which the proposition refers. This distinction becomes the central issue in the debate between Aristotle and Diodorus.

Turning to this debate, the authors note that other scholars have questioned whether such a debate ever occurred, and, if it did, exactly what each side was arguing. Some have seen the debate as muddled. or even absurd. But with the help of ASL, Barker and Paxson create a plausible reconstruction of both opposing views, and they evaluate the outcome. In doing so, they rely on the assumption that both Aristotle and Diodorus were keen logical analysts who would not adopt absurd or obviously foolish positions. Where there are gaps in the historical record, therefore, the authors supply the strongest available connections to complete the arguments.

Aristotle's position requires somewhat less reconstruction than Diodorus's. Aristotle denied that everything that happens had to have been just as it is (fatalism). Since some events do originate through deliberation and voluntary action, it is true that at least these events are contingent: they might have been otherwise. Before these events occur it is incorrect to say that a proposition about their occurrence is either true or false, for the events at that point are merely possible.

Barker and Paxson provide detailed analysis of both the logical structure of this argument using ASL, and the text of Chapter 9 of De interpretatione. They argue that their reconstruction is consistent with the text, and that it also forms a cogent position such as one would expect Aristotle to have adopted.

Likewise, they reconstruct Diodorus's 'Master Argument' against Aristotle in a manner that gives credit to him, too. But in determining the complex structure, the authors also uncover its flaw. Although the historical evidence hints that Diodorus won the debate. Barker and Paxson show that he actually begged the question. Hidden within his argument is the assumption that if a proposition is not true during a specific time then its negation is true during that time (the Unrestricted Temporal Bivalence Thesis). This assumption by itself rules out any possibility of contingency, and therefore assures Diodorus of being able to demonstrate the validity of fatalism. But without this assumption, Diodorus cannot make his case.

The Augmented Standard Logic which the authors developed could also be used, they believe, to make plain the structure of other philosophical arguments having temporal or modal dimensions.

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