Monday, December 11, 2006

Tense Operators Versus Quantifiers in Modal Logic

Professor Cannon suggested earlier this semester that the English tenses, past and present, behave surprisingly like pronouns, and this suggests encouragement of theories of natural language in which tenses get represented by the use of quantifiers and variables that range over times. He also suggested that these tenses cannot be represented by operators, as is customary in many versions of tense logic. But this is not obvious. Tense operators are in fact capable of manifesting at least some pronoun-like behavior.

Take "variable-binding" for example. Operators do something like this by means of scope. To illustrate with a non-tense example, we can write either:

1) daimond A


2) For some possible world w, A is true in w.

The appearance of 'A' within the scope of the operator 'daimond' serves the same purpose as the binding of w in 'A is true in w' by the quantifier of 2). Similarly with tenses, compare:

2) It was the case that A.


4) For some past time t, A is true at t.

The main advantage of the explicit quantification of times over the use of tense operators is that the former is a more powerful notation--it can be used to express things that cannot be expressed in the tense-operator notation. But if the expressive resources of English tenses do not exploit this extra expressive power, i.e. if the expressive resources of English tenses do not exceed the expressive power of tense operators, then this is an insight about English that is worth noting.

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