Point-accessibility operators for temporal logic

Point-accessibility operators (PA-operators) are temporal operators, also known as modal operators, for branching and linear Temporal logic. PA-operators are a generalization of Georg Henrik von Wright's diachronic modalities. Priorian temporal operators can express propositions such as “It is possible from the aspect of the present time that such and such event will take place at some time in the future as a whole” and “It is necessary from the aspect of the present time that such and such event took place at some time in the past as a whole”, but they cannot express more specific propositions such as “It is possible from the aspect of the present time that it is sunny tomorrow in Helsinki” or that “It is necessary from the aspect of the present time that it was sunny last year in Helsinki.” PA-operators are sufficiently fine-grained and expressive for formalizing such propositions. In general, PA-operators capture the meaning of temporal propositions that are stated at a specific point of time and targeted at a specific point of time (or at one or more points or intervals of time).

One may proceed directly into forward-directed and backward-directed and synchronic operators below. But in order to thoroughly grasp the PA-operators, it is important to see that they are instances of the point-accessibility operator schema. So, we have to first look at point-accessibility schema, then point-accessibility operator schema, and only then individual PA-operators.

• Definition. Point-Accessibility Schema: $t⊲t^{\prime }$, where $t$ is the aspect time, $⊲$ is the point-accessibility relation, and $t^{\prime }$ is the target time. In a temporal proposition, $t$ and $t^{\prime }$ are assigned and $t⊲t^{\prime }$ is replaced by a set of one or more temporal states of the Universe (${𝒯}$s) that are, were or will be accessible at $t^{\prime }$ from the aspect of time $t$. (Note: modalities are later defined as quantifiers over $t⊲t^{\prime }$.) In the following, the elements of $t⊲t^{\prime }$ are explained with all combinations of $t,t^{\prime },{𝒫}$, where $t\le {𝒫}$, and where ${𝒫}$ denotes the present time.
  • ${𝒫}$-forward: ${𝒫}⊲t^{\prime }$, where $t^{\prime }>{𝒫}$. In a linear system ${𝒫}⊲t^{\prime }$ is a set of exactly one causal successor of ${𝒫}$, which is realizable at $t^{\prime }$ from the aspect of ${𝒫}$. In a branching system ${𝒫}⊲t^{\prime }$ is a set of several mutually disconnected causal successors of ${𝒫}$ that are realizable at $t^{\prime }$ from the aspect of ${𝒫}$
  • ${𝒫}$-backward: ${𝒫}⊲t^{\prime }$, where $t^{\prime }<{𝒫}$. The only element of ${𝒫}⊲t^{\prime }$ is a single causal predecessor of ${𝒫}$ that was realized at $t^{\prime }$ from the aspect of ${𝒫}$
  • ${𝒫}$-synchronic: ${𝒫}⊲t^{\prime }$, where $t^{\prime }={𝒫}$. The single element of ${𝒫}⊲{𝒫}$ is ${𝒫}$, which is realized from the aspect of ${𝒫}$.
  • $t<{𝒫}<t^{\prime }:t⊲t^{\prime }$ is a set of one ${𝒯}$ that was realizable (in a linear system) or several ${𝒯}$s that were realizable (in a branching system) at $t^{\prime }$ from the aspect of $t$. From the aspect of ${𝒫}$, the same ${𝒯}$ is realizable at $t^{\prime }$ (in a linear system), or the elements of ${𝒫}⊲t^{\prime }\subset t⊲t^{\prime }$ are realizable at $t^{\prime }$ (in a branching system).
  • $t<t^{\prime }={𝒫}:t⊲t^{\prime }$ is a set of one ${𝒯}$ that was realizable (in a linear system) or several ${𝒯}$s which were realizable (in a branching system) at $t^{\prime }$ from the aspect of $t$. Exactly one of them is realized at $t^{\prime }={𝒫}$.
  • $t<t^{\prime }<{𝒫}:t⊲t^{\prime }$ is a set of one ${𝒯}$ that was realizable (in a linear system) or several ${𝒯}$s which were realizable (in a branching system) at $t^{\prime }$ from the aspect of $t$. From the aspect of ${𝒫}$, exactly one of them was realized at $t^{\prime }$.

• Definition. Point-accessibility operator schema. Length 1 point-accessibility (PA-1) operator schema ${{ℳ}}_t{\varphi }_{t^{\prime }}$ applies a PA-1 accessibility chain $t⊲t^{\prime }$. ${{ℳ}}_t{\varphi }_{t^{\prime }}$ is read as: It (is, was, will be) ${ℳ}$ from the aspect of $t$ that $\varphi$ (is, was, will be) realized at $t^{\prime }$, where ${ℳ}$ is a modality, $t$ is the aspect time, $t^{\prime }$ is the target time, and $\varphi$ is a property or a disjunction of properties, or in von Wright's words "a grammatically complete sentence which, however, does not express a true or false proposition unless it is qualified with respect to time." An example of such sentence is "It rains in Helsinki."
• Definition. PA-1 operator is an instance of a PA-1 operator schema where ${ℳ}$ is assigned a specific modality: possible (pos), contingent (con), necessary (nec), impossible (imp), or neutral (neu). (The modalities are written in natural language instead of Priorian one-character symbols (F, G, P, H) to improve readability. In addition, there is no standard symbol for neutrality.)
• Definition. PA-1 proposition is an instance of a PA-1 operator schema where $t$, $t^{\prime }$, $\varphi$ and ${ℳ}$ are assigned. In a linear system a PA-1 proposition is either true or false, with any $t,t^{\prime }$ combination. In a branching system a PA-1 proposition is either true, false or indeterminate with any $t,t^{\prime }$ combination where $t\le {𝒫}$. Formally, a PA-1 proposition states that a specific number of elements of the set $t⊲t^{\prime }$ conform to $\varphi$, i.e., modalities are quantifiers over $t⊲t^{\prime }$. Consider some examples.
  • pos${}_t{\varphi }_{t^{\prime }}$ is true iff at least one element of $t⊲t^{\prime }$ conforms to $\varphi$.
  • nec${}_t{\varphi }_{t^{\prime }}$ is true iff every element of $t⊲t^{\prime }$ conforms to $\varphi$.
  • con${}_t{\varphi }_{t^{\prime }}$ is true iff at least one but not every element of $t⊲t^{\prime }$ conforms to $\varphi$.
  • imp${}_t{\varphi }_{t^{\prime }}$ is true iff no element of $t⊲t^{\prime }$ conforms to $\varphi$.
  • neu${}_t{\varphi }_{t^{\prime }}$ is true iff nec${}_t{\varphi }_{t^{\prime }}$ is true; false iff imp${}_t{\varphi }_{t^{\prime }}$ is true; indeterminate iff con${}_t{\varphi }_{t^{\prime }}$ is true.

The following rules hold for PA-1 propositions:

• ${{ℳ}}_t{\varphi }_{t^{\prime }}\wedge {{ℳ}}_t{\varphi }_{t^{″}}\equiv {{ℳ}}_t{\varphi }_{t^{\prime }\wedge t^{″}}.$
• ${{ℳ}}_t{\varphi }_{t^{\prime }}\vee {{ℳ}}_t{\varphi }_{t^{″}}\equiv {{ℳ}}_t{\varphi }_{t^{\prime }\vee t^{″}}.$
• ${{ℳ}}_t{\varphi }_{t^{\prime }}\vee {{{ℳ}}^{\prime }}_t{\varphi }_{t^{\prime }}\equiv {ℳ}\vee {{{ℳ}}^{\prime }}_t{\varphi }_{t^{\prime }}.$
• Definition. Length $n$ PA (PA-$n$) operator schema ${{ℳ}}_{t_0}\dots {{ℳ}}_{t_{n-1}}^{\text{'}}{\varphi }_{t_n}$ applies a PA-$n$ chain, which is read as: It (is, was, will be) ${ℳ}$ from the aspect of $t_0$ that $\dots$ it (is, was, will be) ${{ℳ}}^{\text{'}}$ from the aspect of $t_{n-1}$ that $\varphi$ (is, was, will be) realized at $t_n$.
• Definition. PA-n proposition is an instance of a PA-n operator schema where $\varphi$, ${ℳ},\dots ,{{ℳ}}^{\text{'}}$ and $t_0,\dots ,t_n$ are assigned. In a linear system a PA-n proposition is either true or false. In a branching system a PA-n proposition where $t_0\le {𝒫}$ is either true, false or indeterminate.
• Definition. A PA proposition of the type ${{ℳ}}_{t_0}\dots {{{ℳ}}^{\prime }}_{t_{n-1}}{\varphi }_{t_n}$ is a single-chain proposition, as it applies a single chain of accessibility $t_0⊲\dots ⊲t_{n-1}⊲t_n$. Exactly one ${𝒯}$ suffices in making a single-chain proposition either true, false or indeterminate. Such ${𝒯}$ is the earliest of $t_0,\dots ,t_n$. As $t_0\le {𝒫}$, the earliest of $t_0,\dots ,t_n$ is earlier than or equal to ${𝒫}$. This is intelligible in a system where the past and ${𝒫}$ are fixed, and the earliest of $t_0,\dots ,t_n$ causes all possibilities that are relevant to the focal proposition. A multi-chain proposition has two or more chains of accessibility. Accordingly, more than one ${𝒯}$ may be required in making a multi-chain proposition true, false or indeterminate.
• Definition. Forward-directed PA-1 propositions. The direction of a PA-1 proposition is the direction of its accessibility relation $t⊲t^{\prime }$. In forward-directed PA-1 propositions $t<t^{\prime }$, i.e., the aspect time is earlier than the target time. Forward-directed PA-1 propositions of the type ${{ℳ}}_{{𝒫}}{\varphi }_{t>{𝒫}}$ are present propositions about the future, where ${𝒫}⊲t$ is a set of one ${𝒯}$ that is (in a linear system) or several ${𝒯}$s that are (in a branching system) from the aspect of ${𝒫}$ realizable at $t$. foggy is an abbreviation of "It is foggy in Helsinki."
  • A: possible${}_{{𝒫}}$foggy${}_{t^{\prime }>{𝒫}}\equiv$ "It is possible from the aspect of ${𝒫}$ that it will be foggy in Helsinki at $t^{\prime }$." True iff it is foggy in Helsinki in at least one element of ${𝒫}⊲t^{\prime }$. Otherwise false.
  • B: necessary${}_{{𝒫}}$foggy${}_{t^{\prime }>{𝒫}}\equiv$ "It is necessary from the aspect of ${𝒫}$ that it will be foggy in Helsinki at $t^{\prime }$." True iff it is foggy in Helsinki in every element of ${𝒫}⊲t^{\prime }$. Otherwise false.
  • C: impossible${}_{{𝒫}}$foggy${}_{t^{\prime }>{𝒫}}\equiv$ "It is impossible from the aspect of ${𝒫}$ that it will be foggy in Helsinki at $t^{\prime }$." True iff it is foggy in Helsinki in no element of ${𝒫}⊲t^{\prime }$. Otherwise false.
  • D: contingent${}_{{𝒫}}$foggy${}_{t^{\prime }>{𝒫}}\equiv$ "It is contingent from the aspect of ${𝒫}$ that it will be foggy in Helsinki at $t^{\prime }$." True iff it is foggy in Helsinki in at least one but not in every element of ${𝒫}⊲t^{\prime }$. Otherwise false. In other words, true iff imp${}_{{𝒫}}$foggy${}_{t^{\prime }}$ and nec${}_{{𝒫}}$foggy${}_{t^{\prime }}$ are false, and false iff imp${}_{{𝒫}}$foggy${}_{t^{\prime }}$ or nec${}_{{𝒫}}$foggy${}_{t^{\prime }}$ is true.
  • E: neutral${}_{{𝒫}}$foggy${}_{t^{\prime }>{𝒫}}\equiv$ "It is neutral from the aspect of ${𝒫}$ that it will be foggy in Helsinki at $t^{\prime }$" $\equiv$ "It will be foggy in Helsinki at $t^{\prime }$." True iff it is foggy in Helsinki in every element of ${𝒫}⊲t^{\prime }$, i.e., iff nec${}_{{𝒫}}$foggy${}_{t^{\prime }}$ is true. False iff it is foggy in no element of ${𝒫}⊲t^{\prime }$, i.e., iff imp${}_{{𝒫}}$foggy${}_{t^{\prime }}$ is true. Indeterminate iff it is foggy in Helsinki in at least one but not in every element of ${𝒫}⊲t^{\prime }$, i.e., iff con${}_{{𝒫}}$foggy${}_{t^{\prime }}$ is true. E is thereby a future contingent when D is true.

• Definition. Backward-directed and synchronic PA-1 propositions. In backward-directed PA-1 propositions $t^{\prime }<t.$ In synchronic PA-1 propositions $t=t^{\prime }.$ Backward-directed PA-1 propositions of the type ${{ℳ}}_{{𝒫}}{\varphi }_{t^{\prime }<{𝒫}}$ are present propositions about the past. Synchronic PA-1 propositions of the type ${{ℳ}}_{{𝒫}}{\varphi }_{{𝒫}}$ are present propositions about the present. In backward-directed and synchronic PA-1 propositions possibility, necessity and neutrality are equivalent, and contingency statements are false. Von Wright calls this "a modal logic of a universe of propositions which has no room for contingent propositions but in which every truth is a necessity and every falsehood is an impossibility." For, the past and ${𝒫}$ are unchanging from the aspect of ${𝒫}$, even if they could have been realized differently. Therefore, backward-directed and synchronic propositions of the types pos${}_{{𝒫}}{\varphi }_{t^{\prime }\le {𝒫}}$, nec${}_{{𝒫}}{\varphi }_{t^{\prime }\le {𝒫}}$, neu${}_{{𝒫}}{\varphi }_{t^{\prime }\le {𝒫}}$ can be written as ${\varphi }_{t^{\prime }}$, and read as $\varphi$ was/is the case at t'. Backward-directed and synchronic propositions of the type imp${}_{{𝒫}}{\varphi }_{t^{\prime }\le {𝒫}}$ can be written as $\mathrm{\neg }{\varphi }_t$, and read as $\varphi$ was/is not the case at t'.
  • A: possible${}_{{𝒫}}$foggy${}_{t^{\prime }\le {𝒫}}\equiv {\varphi }_{t^{\prime }\le {𝒫}}$. True iff it is foggy in Helsinki in at least one element of ${𝒫}⊲t^{\prime }$, i.e., in its only element.
  • B: necessary${}_{{𝒫}}$foggy${}_{t^{\prime }\le {𝒫}}\equiv {\varphi }_{t^{\prime }\le {𝒫}}$. True iff it is foggy in Helsinki in every element of ${𝒫}⊲t^{\prime }$, i.e., in its only element.
  • C: impossible${}_{{𝒫}}$foggy${}_{t^{\prime }\le {𝒫}}\equiv \mathrm{\neg }{\varphi }_{t^{\prime }\le {𝒫}}$. True iff it is foggy in Helsinki in no element of ${𝒫}⊲t^{\prime }$, i.e., not in its only element.
  • D: contingent${}_{{𝒫}}$foggy${}_{t^{\prime }\le {𝒫}}$. True iff it is foggy in Helsinki in at least one but not in every element of ${𝒫}⊲t^{\prime }$, i.e., never true.
  • E: neutral${}_{{𝒫}}$foggy${}_{t^{\prime }\le {𝒫}}\equiv {\varphi }_{t^{\prime }<{𝒫}}$. True iff it is foggy in Helsinki in every element of ${𝒫}⊲t^{\prime }$. False iff it is foggy in no element of ${𝒫}⊲t^{\prime }$. Indeterminate iff it is foggy in Helsinki in at least one but not in every element of ${𝒫}⊲t^{\prime }$, i.e., never indeterminate.

References

As of this edit, this article uses content from "Tense Logic and Ontology of Time", authored by Avril Styrman, which is licensed in a way that permits reuse under the Creative Commons Attribution-ShareAlike 3.0 Unported License, but not under the GFDL. All relevant terms must be followed.

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