From Counterfactual Conditionals to Temporal Conditionals

Abstract

Although it receives less attention, (Lewis in Noûs 13:455–476, 1979. https://doi.org/10.2307/2215339) admitted that the branching-time(-like) model fits a wide range of counterfactuals, including (Nix) ‘If Nixon had pressed the button, there would have been a nuclear war’, which was raised by (Fine in Mind 84:451–458, 1975). However, Lewis then claimed that similarity analysis is more general than temporality analysis. In this paper, we do not scrutinise his claim. Instead, we re-analyse (Nix) not only model-theoretically but also proof-theoretically from the ‘meaning-as-use’ and ‘inferentialist’ points of view. Then, we re-formalise (Nix) in a natural extension of hybrid tense logic, which we refer to as hybrid tense logic for temporal conditionals (HTL$_{TC}$). Consequently, we find that not only among counterfactuals, but also among indicatives, there is a wide range of conditionals whose formalisation in HTL$_{TC}$ is appropriate. We refer to these conditionals as temporal conditionals. This suggests a new logical generality that temporality analysis has but similarity analysis does not, from which emerges a new logical perspective on conditionals in general: temporal ones and others.

1 Introduction

1.1 Fine Problem Revisited

Lewis (1973) symbolised counterfactual conditionals in the form ‘If it were the case that $\varphi$, then it would be the case that $\psi$’ as . He then presented the truth condition of , which roughly states:

(SA)¹ In all possible worlds that are most similar to the actual world except that $\varphi$ holds, $\psi$ also holds.

Thus, the meaning of was explained in terms of ‘similarity’ among possible worlds. Since then, Lewis’s analysis of counterfactuals has been influential and standard, particularly in analytic philosophy and linguistics.

However, Fine (1975) pointed out that (SA) had a serious problem, which we refer to as the ‘Fine Problem’. The problem is shown perspicuously by the following sentence:

(Nix) If Nixon had pressed the button, there would have been a nuclear war.

Although this is a counterfactual conditional that is surely true, it may be thought to be false according to (SA). If we take the meaning of ‘being similar’ literally, the situations in possible worlds in which a nuclear war has occurred because of Nixon’s pressing the button should be entirely different from the situation in our actual world in which no nuclear war has occurred yet. Instead, the situations in the possible worlds in which Nixon did press the button but miraculously no nuclear war has occurred may well remain largely unchanged compared to the actual situation. If so, we would have to say that the following sentence is true:

($\overline{\text {Nix}}$) Even if Nixon had pressed the button, there would not have been a nuclear war.

To avoid this absurd consequence, Lewis made the following reply (1979), which was so natural that anyone who wanted to maintain (SA) would adopt it: in the case of (Nix), such a miracle that Nixon pressed the button but no nuclear war occurred quite contradicts important laws, such as physical, social, political, and military laws, of the actual world. The deviations from such laws are much heavier than the disagreements with particular facts (the relatively peaceful course of events) in the actual world. Accordingly, there is a smaller difference from reality in the succedent of (Nix), which is only against particular facts, than in the succedent of ($\overline{\text {Nix}}$), which is against actual important laws. Therefore, (Nix) is true, and ($\overline{\text {Nix}}$) is false, as expected.

This is a fairly famous story in the literature. However, there is another story that seems to have received less attention. Before replying to the Fine Problem in this manner, Lewis temporarily took into consideration branching-time-like model of counterfactuals,,² which he referred to as ‘Analysis 1’, and admitted that Analysis 1 seems to fit a wide range of counterfactuals ((Lewis, 1979), p. 463). We quote his formulation of it here³:

ANALYSIS 1. Consider a counterfactual “If it were that A, then it would be that C” where A is entirely about affairs in a stretch of time $t_{\text {A}}$. Consider all those possible worlds w such that:

• (1) A is true at w;

• (2) w is exactly like our actual world at all times before a transition period beginning shortly before $t_{\text {A}}$;

• (3) w conforms to the actual laws of nature at all times after $t_{\text {A}}$; and

• (4) during $t_{\text {A}}$ and the preceding transition period, w differs no more from our actual world than it must to permit A to hold.

The counterfactual is true if and only if C holds at every such world w.

By contrast, his similarity analysis, the (SA) above, was referred to as ‘Analysis 2’, and restated as follows⁴:

ANALYSIS 2. A counterfactual “If it were that A, then it would be that C” is (non-vacuously) true if and only if some (accessible) world where both A and C are true is more similar to our actual world, overall, than is any world where A is true but C is false.

He then claimed that Analysis 2 is more general than Analysis 1 because there are many counterfactual suppositions that seem to have no connection with particular times, such as ‘If kangaroos had no tails...’ and ‘If gravity went by the inverse cube of distance...’.

1.2 Our Course of Logical Analysis of Counterfactuals

In this paper, we do not scrutinise Lewis’s above claim. Instead, we begin with a re-analysis of (Nix), following a paradigm of logical analysis of natural language in analytic philosophy: Tarski’s theory of truth (Tarski, 1944).

Our re-analysis of (Nix) proceeds as follows. In Sect. 2, we review Tarski’s theory of truth (Tarski, 1944) as a paradigm we follow for logical analysis of natural language, including counterfactuals. On a closer look, the paradigm turns out to be along the lines of the ‘meaning-as-use’ approach (Wittgenstein, 2001) and what later has been called ‘inferentialism’ (Brandom, 1994). In Sect. 3, following the procedure of the paradigm, we observe that there are at least three types of use of counterfactuals: historical/real, rhetorical, and theoretical. In Sect. 4, we select historical/real counterfactuals, of which (Nix) is typical, and specify an inference schema that constrains their use, i.e. counterfactual transitive inference (CT), which was dismissed as the fallacy of transitivity in one manner of formalisation by Lewis (1973) but can be validated in another manner of formalisation in his own system. In Sect. 5, we present another formalisation and deduction of (CT) in a natural extension of hybrid tense logic, i.e. a version of hybrid logic improved proof-theoretically by Braüner (2011), which we refer to as hybrid tense logic for temporal conditionals (HTL$_{TC}$).

This procedure may give one the impression that the result shall only have much lower generality than that obtained using the Stalnaker–Lewis approach, which is based on Stalnaker–Lewis reasoning: there is a uniformity of grammatical form that is classified as counterfactual in natural language; therefore, there should be a uniformity of logical form that is shared amongst almost all sentences in the grammatical form.⁵ (We shall briefly examine Stalnaker–Lewis reasoning in Sect. 3.) However, this impression turns out to be wrong. Instead, we shall see that the procedure brings a new generality, thereby suggesting the possibility of a logical reclassification of conditionals in general that might deconstruct such a grammatical dichotomy as indicative vs. subjunctive. In Sects. 6 and 7, we provide evidence for this claim by presenting a formalisation of an indicative version of (CT) in HTL$_{TC}$. In Sect. 8, we list some of many ramifications that our results would have for related works in philosophical logic and linguistics. Finally, in Sect. 9, we conclude this paper with a few more remarks about a crucial difference between our new analysis and Lewis’s classical analysis.

The “Appendix” presents a natural deduction system for HTL$_{TC}$ as a special case of Braüner’s systems (Braüner, 2011).

2 Our Paradigm of Logical Analysis of Natural Language: Tarski’s Theory of Truth

We can say that what Tarski’s historic work (Tarski, 1944) attained is a logical refinement of our informal conception of the word ‘true’ in natural language, which involves two points of view along the lines of the ‘meaning-as-use’ approach (Wittgenstein, 2001) and what later has been called ‘inferentialism’ (Brandom, 1994).⁶⁷

First, Tarski (1944) notes that there are different uses of the word ‘true’ and, accordingly, different conceptions of it. Further, he admits that he does not understand such a question as ‘What is the right conception of truth?’ (Tarski, 1944, p. 355). Then, from the various uses he selects a use that he thinks is intelligible and specifies what constrains the selected use, i.e. what he refers to as T-schema:

$$\begin{aligned} \hbox {(T)}\, X\,{ is}\,{ true}\,{ if},\,{ and}\,{ only}\,{ if},\, p, \end{aligned}$$

where ‘p’ stands for an arbitrary sentence and ‘X’ is the name of the sentence that ‘p’ stands for. Here, we can detect a ‘meaning-as-use’ point of view: to specify the meaning (conception) of a word or a sentence, we should specify the use of the word or the sentence.

Second, we can say that by (T), he also specifies a simple inference involving ‘true’: (i) we can infer the proposition that ‘p’ is true from the proposition that p; conversely, (ii) we can infer the proposition that p from the proposition that ‘p’ is true, where ‘p’ is substituted for X in (T). We can then regard the ‘if’ direction (i) as the introduction rule of ‘true’ and the ‘only if’ direction (ii) as the elimination rule of it. Here, we can detect an ‘inferentialist’ point of view: to specify the logical use of a word or a sentence, we should specify the inference (i.e. the logical context) in which the word or the sentence is used.

Thus, he first specified the requirement that a logical analysis of ‘true’ should satisfy⁸ the result should imply all the sentences of the form (T). He then proceeded to (show the outline of a way to) construct a formal language in which this requirement is met⁹ one in which the selected use of ‘true’ can be reconstructed formally (i.e. recursively defined in this case) so that all of the sentences of the form (T) can be asserted consistently.¹⁰¹¹

3 Three Types of Use of Counterfactuals: Historical/Real, Rhetorical and Theoretical

Our re-analysis of (Nix) follows the procedure of Tarski’s theory of truth (Tarski, 1944) involving the ‘meaning-as-use’ and ‘inferentialist’ points of view.

From the ‘meaning-as-use’ point of view in Tarski (1944), we suspend the reasoning that we referred to as Stalnaker–Lewis reasoning in Sect. 1.2: there is a uniformity of grammatical form that is classified as counterfactual in natural language; therefore, there should be a uniformity of logical form that is shared amongst almost all sentences in the grammatical form.¹²

In fact, it is clear that Stalnaker–Lewis reasoning is not valid for cases other than counterfactuals. We present a traditional counterexample: the indefinite article. Consider ‘Jemima is waiting for a mouse who lives in that hole’ $\cdots$(a).¹³ The indefinite article ‘a’ in (a) can be construed as an existential quantifier as well as a universal quantifier. Notoriously, the grammatical form of a sentence containing indefinite articles is far from a decisive evidence for determining its logical form.¹⁴ After all, we cannot rely on uniformity of the grammatical form of sentences or words, even though it is a non-negligible hint as to their logical form in many cases.

Curiously enough, at a level of conditionals in general, people seem to be fully conscious of this lesson. Today, most logicians, linguists, and analytic philosophers would never reason, ‘Conditionals have the uniform grammatical form “if..., then...”, so there should be a uniform logical form of conditionals in general’.¹⁵ On the contrary, there have been considerable efforts devoted to distinguishing conditionals as material, strict, intuitionistic, relevance, and, for that matter, indicative and counterfactual.

Why, then, do not we suspect that there might be logical differences among counterfactuals even though they have past-tense verbs in their antecedents and succedents in many languages quite uniformly? Why do not we apply the lesson about conditionals in general to counterfactual conditionals in particular recursively even if there should be family resemblances?

In fact, Placek and Müller (2007), Xu (1997), and Lewis himself (Lewis, 1996) detect a remarkable subclass of counterfactuals, which are referred to as ‘historical counterfactuals’ by Placek and Müller (2007) and ‘real counterfactuals’ by Xu (1997) and Lewis (1996). They are counterfactuals concerning ‘historical possibility’ (Placek & Müller, 2007) or ‘real possibility’ (Xu, 1997; Lewis, 1996). Although the characterisation of historical/real possibility is not strict but intuitive, it will suffice to borrow the characterisation stated by Lewis (1996): it is the possibility that conforms to actual history up to some past moment and the actual laws of nature as ever (Lewis, 1996, p. 552).

Then, (Nix), as repeated below, is thought to be a typical case of historical/real counterfactuals:

(Nix) If Nixon had pressed the button, there would have been a nuclear war.

In fact, (Nix) supposes historical due courses that coincide with actual history up to some past moment and subsequently proceed otherwise while conforming to the actual laws of nature.¹⁶ Another example ‘If Oswald had not killed Kennedy, then someone else would have’$\cdots$ (Osw), which is referred to in Lewis (1973), is also typical of this class; however, the sentence may very well be an example of false historical/real counterfactuals according to Lewis’s opinion (Lewis, 1973, p. 3).

By contrast, examples of counterfactuals that are obviously not historical/real include ‘If I were a bird, I could fly there!’$\cdots$ (b) and ‘If I were you, I would retort!’$\cdots$ (y). Example (b) may well be a rhetorical expression close to a sigh showing both the fact that the speaker wants to go there but cannot and his or her regret in that regard. As for (y), it may well be more realistic to consider it a euphemistic expression for both a criticism of the other’s failure to retort and advice for him or her to retort even after that time than to assign any philosophical interpretation to the phrase ‘I were you’. We loosely refer to such a class of counterfactuals as ‘rhetorical counterfactuals’.

A boundary case between historical/real and rhetorical counterfactuals is Lewis’s familiar one, i.e. ‘If kangaroos had no tails, they would topple over’$\cdots$ (k).¹⁷ As pointed out by Placek and Müller (2007) exactly, it may be possible to understand the sentence by seriously considering a possibility of evolution of kangaroos without tails, but it seems again more realistic to consider that it highlights the importance of their tails for their balance (Placek & Müller, 2007, p. 177).

Now, what becomes of another of Lewis’s examples, i.e. ‘If gravity went by the inverse cube of distance,...’ $\cdots$ (g)?¹⁸ This can represent a theoretical thought experiment (in physics, in this case) that might be comparable to, say, ‘If this axiom in this system were replaced by another,...’$\cdots$ (ax) or ‘If we modified the value of the parameter in this system,...’$\cdots$ (para), as when a logician, mathematician, or computer scientist is investigating the significance of an axiom or a parameter in a given system or developing variations of the system. If so, we may loosely refer to such counterfactuals as ‘theoretical counterfactuals’.

4 Counterfactual Transitive Inference

From the (at least) three uses of counterfactuals listed in Sect. 3, we select a historical/real use, of which (Nix) is to be typical.¹⁹ Then, from the ‘inferentialist’ point of view in Tarski (1944),²⁰ we ask the question: what inference can constrain the historical/real use of counterfactuals? We think that counterfactual transitive inference, i.e. transitive inference comprising counterfactual conditionals, can²¹:

The conceptual reason for our selection of this inference patten is as follows. We find that historical/real counterfactuals should be essentially transitive in a two-fold sense. First, their temporality should be certainly transitive in the sense that if the time of $\varphi$ is as early as the time of $\psi$ and the time of $\psi$ is as early as the time of $\chi$, then the time of $\varphi$ is as early as the time of $\chi$.²²Second, the influentialness of the state of affairs in their antecedent should be probably transitive in the sense that if the state $\varphi$ influences the state $\psi$ and the state $\psi$ influences the state $\chi$, then the state $\varphi$ influences the state $\chi$.²³

However, some readers familiar with Lewis’s work (Lewis, 1973) might suspect that (CT) is just what he referred to as the fallacy of transitivity and in fact proved to be invalid, as his example intuitively shows²⁴:

The background to (OAW) is that Otto and Waldo are rivals for Anna’s affections. Anna likes Otto; hence, the first premise is true. Waldo usually follows Anna around; hence, the second one is also true. However, Waldo never runs the risk of meeting Otto; hence, the conclusion is false.

However, this way of speaking is quite misleading. A little reflection will make us realise this simple fact: it is true that there are cases in which (CT) does not hold, whereas there are also cases in which it does hold. For example, normally, we would not bother to contrive counterexamples to the following instance of (CT), whose first premise is (Nix), although it is possible to do so if one wishes to²⁵:

Then, we claim that what is more important both logically and philosophically is to explain why (CT) holds when it does hold rather than why it does not hold when it does not.²⁶ In any case, the fact is that Lewis’s system can symbolise (CT) such that it does hold and such that it does not hold. Here are two ways.

As is well known, Lewis symbolised ‘if it were that $\varphi$, then it would be that $\psi$’ as ‘’. Accordingly, the simplest way to symbolise (CT) as a whole is as follows²⁷:

Thus, it is not (CT) itself but (FCT), a symbolisation of (CT) in Lewis’s system, that is rightly referred to as the fallacy of transitivity, as (FCT) is in fact invalid in Lewis’s semantics.²⁸ Then, (OAW) should be an instance of (FCT).

Meanwhile, there is another symbolisation of (CT) in Lewis’s system under which the resulting schema becomes valid in Lewis’s semantics. To see this, we interpret (CT) as an abbreviation for the following²⁹

The quite natural idea behind $(CT')$ is that in the succedent of the first premise, the condition $\psi$ is accumulated on the condition $\varphi$; hence, the antecedent of the second premise must take over the accumulated conditions $\varphi$ and $\psi$. Accordingly, $(CT')$ is straightforwardly symbolised in Lewis’s system as follows³⁰:

Then, (VCT) is perfectly valid in Lewis’s semantics,³¹ and can be deduced in the most basic version V of Lewis’s axiomatic system.³² Thus, (NND) should be an instance of (VCT) in Lewis’s system.

5 Deduction of Transitivity of Counterfactuals in Hybrid Tense Logic

By accepting (VCT) as a correct formalisation of (CT), we can say that Lewis’s analysis certainly fulfils a logical refinement of our informal use and conception of historical/real counterfactuals. However, we claim that there is another logical refinement of historical/real counterfactuals that is much more fine-grained than Lewis’s version and has another logical generality. The main purpose of this paper is, among other things, to demonstrate this claim.

5.1 Making Explicit Reference Times in Historical/Real Counterfactuals

In Fig. 1, we present our informal branching-time model for (Nix), which is a representative example of historical/real counterfactuals, using only classical temporal concepts. Although our model is based on the idea in Tsai (2014), which is relatively recent research on Japanese counterfactual conditionals, it is quite similar in particular to Thomason and Gupta’s (Thomason & Gupta, 1980). Hence, for comparison, we also display another informal branching-time model for (Nix) in Thomason and Gupta’s manner in Fig. 2 just beneath ours in Fig. 1. After explaining our informal model we shall briefly look at correspondences and differences between ours (Fig. 1) and theirs (Fig. 2) at this informal level.

Fig. 1

Informal branching-time model of (Nix) based on the idea in Tsai (2014)

Fig. 2

Informal branching-time model of (Nix) in the manner of Thomason and Gupta (1980)

In Fig. 1, the middle arrow represents the actual time series, on which the point Val represents ‘the valuation time’ at which we make a valuation of whether the sentence (Nix) is true or false.³³ The arrows branching (up and down, respectively) from the middle represent possible time series starting to follow different courses from the actual one at some past moment.³⁴ E$_{1}$ and E$_{2}$ on these arrows represent ‘the event times’ at which Nixon presses the button and a nuclear war occurs respectively.³⁵

The hypothesis proposed by Tsai (2014) is that, in addition to these event times, there exists ‘a reference time’, which is a classical concept from Reichenbach (1947), in each of the antecedent and the succedent in a Japanese counterfactual conditional. The significance of this hypothesis is that, by considering the reference time in the succedent in particular, we can explain variations in tense and aspect of the succedent of Japanese counterfactual conditionals.³⁶

In this paper, we presuppose that this hypothesis holds completely for English counterfactual conditionals as well. In the present case (Nix), for example, the perfective aspect of ‘have been’ in ‘there would have been a nuclear war’ can be explained by the configuration shown in Fig. 1 in which E$_{2}$ lies to the left of R$_{2}$. In addition, our hypothesis is that there exists another implicit reference time in (Nix), in addition to R$_{1}$ and R$_{2}$: a diverging time point R$_{0}$.

However, Fig. 1 involves technical improvements to the original version of model representation in Tsai (2014). Although Tsai (2014) represents the two reference times R$_{1}$ (in the antecedent) and R$_{2}$ (in the succedent) as points straightforwardly, we represent them as relations, which we refer to as ‘time reference relations’. This is because, while the original version of model representation in Tsai (2014) depicts only a single possible time series other than the actual one, the branching-time model qua the strict mathematical model can have multiple possible divergences from the actual time series. In such a case, a reference time may be located on each of the multiple possible time series. More specifically, in the present case, each R$_{1}$ and R$_{2}$ as respective points lies on each of the possible time series; hence, there are multiple R$_{1}$s and R$_{2}$s as points in the above-mentioned model as a whole. The valuator of (Nix) then refers to all multiple R$_{1}$s and R$_{2}$s located on different time series at the valuation time. To this end, in this paper we refine reference time points R$_{1}$, R$_{2}$ into time reference relations R$_{1}$, R$_{2}$ from the valuation time, which represent the distribution of multiple R$_{1}$s, R$_{2}$s as respective points. Similarly, we technically interpret the diverging time point R$_{0}$ as the diverging time reference relation R$_{0}$ from the valuation time Val.

Thus, we can see that (Nix) can be interpreted intuitively as ‘in any possible time series branching off from the actual one at some past moment referred to by R$_{0}$, if a moment is referred to by R$_{1}$ and E$_{1}$ occurs at that moment, then, thereafter, if a moment is referred to by R$_{2}$, then E$_{2}$ has occurred by that moment’, as shown in Fig. 1.

Now, let us compare our model (Fig. 1) with one represented in Thomason and Gupta’s manner (Fig. 2) at the informal level.

In Fig. 2, (Nix) is evaluated at moment i in history h at which no nuclear war has occurred. Moment $i_{1}$ on h is a past moment of i at which Nixon had a chance to press the button but did not do so. The counterfactual histories $h'$ and $h''$ are those in which Nixon pressed the button at moments $i_{2}$ and $i_{3}$, respectively, which are the ‘alternative presents’ to $i_{1}$ and said to be ‘copresent’ with $i_{1}$. This ‘copresence’ relation is indicated by $\simeq$. The solid line indicating $h'$ through $i_{2}$ and the dotted line indicating $h''$ through $i_{3}$ show that the pair $<i_{2}, h'>$ is selected to be the closest pair to $<i_{1}, h>$ at which ‘Nixon presses the button’ is true, and the pair $<i_{3}, h''>$ is not. (“The closest pair to $<i_{1}, h>$ at which ‘Nixon presses the button’ is true” is determined by the “selection function”.) Then, by the existence of $i_{4}$ on $h'$, at which there is a nuclear war, ‘There will have been a nuclear war’ is true at the closest pair $<i_{2}, h'>$ to $<i_{1}, h>$. Because such a moment $i_{1}$ is found to be in the past of i on h, (Nix) is considered to be true at i on h in Fig. 2.

We can then list some correspondences and differences between Figs. 1 and 2 that would already be clear. Correspondences: Val corresponds to i, E$_{1}$ corresponds to $i_{2}$ and $i_{3}$, E$_{2}$ corresponds to $i_{4}$ and $i_{5}$, and R$_{1}$ is comparable to $\simeq$ in the sense that both relate the valuation time (Val, i) to the counterfactual event time of Nixon’s pressing the button (E$_{1}$, $i_{2}$/$i_{3}$), where $\simeq$ does so via some past moment $i_{1}$ of i. Differences: R$_{0}$ and R$_{2}$ have no counterparts in Fig. 2; conversely, $i_{1}$ has no counterpart in Fig. 1; furthermore, while ‘the closest’ counterfactual history (time series) to the actual history is selected in Fig. 2, it is not selected in Fig. 1.

After presenting our formal language and model in Sects. 5.2 and 5.3, we shall make a more formal comparison between our formalism and Thomason and Gupta’s in Sect. 5.4.

5.2 Symbolising Historical Counterfactuals by Time Reference Operators

Note that Fig. 1, regarded as a branching-time model, comprises (1) the temporal order relation among moments, and (2) the time reference relations from the valuation time to the reference times. We then find that classical tense logic from Prior (1955), as it is, applies to (1). For (2), we note that tense logic can be regarded as a type of multi-modal logic that sets the label of the future relation as F and that of the past relation as P and, accordingly, introduces the indexed modal operators $\langle F \rangle$ (‘at some time in the future’) and $\langle P \rangle$ (‘at some time in the past’)—the abbreviations of which are F and P, respectively, as modal operators—into language. In accordance with this precedent, we can also convert (2) into symbols in a straightforward way; we prepare the labels $\text {R}_{n}$ ($n \in \{ 0,1,2,\ldots \}$) of the time reference relations and then introduce the indexed modal operators $\langle \text {R}_{n} \rangle$ ($n \in \{ 0,1,2,\ldots \}$), which we refer to as ‘time reference operators’, into language.

However, there are some complications to this approach. As seen from Fig. 1, a time reference relation is a relation that starts from the current valuation time Val and crosses time series; hence, it is connected independently of the temporal order. Although we omit the explanation of the difficulties and the details of the trial-and-error approach while designing the syntax,³⁷ we mention that it is difficult to express the behaviour of a time reference relation sufficiently by means of a simple form of classical tense logic.

To describe its behaviour adequately, we use (i) the downarrow binder $\downarrow$, an early version of which was introduced by Valentin Goranko (1994,1996), and (ii) the inverse operator $\langle \text {R}_{n}^{-} \rangle$ of the time reference operator $\langle \text {R}_{n} \rangle$. By means of the former, we can construct a sentence of the form ‘$\downarrow a. \varphi$’ with which we can express ‘Name the current state ‘a’ and $\varphi$ holds at a’, where $\varphi$ is an arbitrary formula that is possibly open; hence, a can occur in $\varphi$ such that we can make a reflexive reference to the current state a. For example, ‘$\downarrow a. GPa$’ is a (valid closed) formula that reads as ‘Name the current state ‘a’ and it henceforth holds that a held once’.

The latter, $\langle \text {R}_{n}^{-} \rangle$, corresponds to the inverse relation $\text {R}_{n}^{-}$ of the time reference relation $\text {R}_{n}$, where $\text {R}_{n}^{-}$ expresses the passive reference relation ‘being referred to by $\text {R}_{n}$’ and $\text {R}_{n}$ expresses the active reference relation ‘referring to by $\text {R}_{n}$’ (for the strict satisfaction relation, see Sect. 5.3 below). Note that this extension is also only an imitation of the precedent set by Prior (1955, 1967) in the sense that the pair of F and P is a typical instance of pairs of modal operators that are the inverse operators of each other.

Through the extension of classical tense logic by the two items (i) and (ii) above, for instance, ‘$\downarrow a. P F \langle \text {R}_{n}^{-} \rangle a$’ can express ‘Name the current state ‘a’ and at some moment in the past there exists some moment in the future that is being referred to by $\text {R}_{n}$ from a’, i.e. ‘In some possible future course starting from some moment in the past of the current state a, there exists a reference time $\text {R}_{n}$ from a’.

Using this expressive power, we consider a symbolisation of (Nix) that reflects Fig. 1. The most straightforward way to describe it is as follows, where p:= ‘Nixon presses the button’q:= ‘There is a nuclear war’, and G is the future-necessity tense operator meaning ‘At any moment in the future,’.

$\downarrow a. P (\langle \text {R}_{0}^{-} \rangle a ~\wedge ~$

           (a)  $F (\langle \text {R}_{1}^{-} \rangle a ~\wedge ~ p ~\wedge ~ F\langle \text {R}_{2}^{-} \rangle a) ~\wedge ~$

           (b)  $G (\langle \text {R}_{1}^{-} \rangle a \rightarrow (p \rightarrow G(\langle \text {R}_{2}^{-} \rangle a \rightarrow P q))))$                $\cdots$(Nix-S)

(Nix-S) says that there is a past moment $\text {R}_{0}$ such that (a) in a future course proceeding from it, p holds at $\text {R}_{1}$ and, thereafter, $\text {R}_{2}$ will come, and (b) in any future course proceeding from it, if p holds when $\text {R}_{1}$ comes, then q will have occurred when $\text {R}_{2}$ comes.

(Nix-S) is surely the strongest version of symbolisation of (Nix) (where ‘S’ denotes ‘strong’). The speaker of (Nix) may not assume the existence of all or some of the reference time points $\text {R}_{0}$, $\text {R}_{1}$, and $\text {R}_{2}$. For instance, the weakest version is surely the following, where H is the past-necessity tense operator meaning ‘At any moment in the past,’:

$\downarrow a. H (\langle \text {R}_{0}^{-} \rangle a \rightarrow G (\langle \text {R}_{1}^{-} \rangle a \rightarrow (p \rightarrow G(\langle \text {R}_{2}^{-} \rangle a \rightarrow P q))))$            $\cdots$(Nix-W)

(Nix-W) says that if there is a diverging time point $\text {R}_{0}$ in the past, then, if $\text {R}_{1}$ comes in the future after $\text {R}_{0}$ and p holds at $\text {R}_{1}$, then, if $\text {R}_{2}$ comes in the future after $\text {R}_{1}$, then q will have occurred at $\text {R}_{2}$. Thus, we can find that there are various options for the symbolisation of (Nix) depending on which of $\text {R}_{0}$, $\text {R}_{1}$, and $\text {R}_{2}$ is assumed by the speaker to exist in his/her model.

In the next section, we specify our language and its model formally.

5.3 HTL$_{TC}$ and Its Branching-Time Model

Let $\textsf {Prop}=\{ p, q, r,\ldots \}$ and $\textsf {Nom}=\{ a, b, c,\ldots \}$ be countably infinite sets of propositional variables and nominals (which we may regard as names of moments), respectively, which are mutually disjoint. Further, let $\{ \text {R}_{n} \}_{n \in \mathbb {N}}$ be a countably infinite set of relation symbols.

We then define the language to symbolise (Nix-S, W) using the syntax below, where $N \in \{G, H, [\text {R}_{n}], [\text {R}_{n}^{-}] \}$:

$$\begin{aligned} \varphi {:}{:}{=} p ~\vert ~ a ~\vert ~ \varphi ~\wedge ~ \psi ~\vert ~ \varphi \rightarrow \psi ~\vert ~ \bot ~\vert ~ N \varphi ~\vert ~ @_{a} \varphi ~\vert ~ \downarrow a. \varphi \end{aligned}$$

For other propositional connectives, we define $\lnot \varphi \equiv \varphi \rightarrow \bot$, $\top \equiv \lnot \bot$, $\varphi \vee \psi \equiv \lnot (\lnot \varphi ~\wedge ~ \lnot \psi )$, and $\varphi \leftrightarrow \psi \equiv (\varphi \rightarrow \psi ) ~\wedge ~ (\psi \rightarrow \varphi )$. For the future-possibility operator F and the past-possibility operator P, we define $F \varphi \equiv \lnot G \lnot \varphi$ and $P \varphi \equiv \lnot H \lnot \varphi$ derivatively from G and H, respectively. Similarly, we set $\langle \text {R}_{n} \rangle \varphi \equiv \lnot [\text {R}_{n}] \lnot \varphi$, $\langle \text {R}_{n}^{-} \rangle \varphi \equiv \lnot [\text {R}_{n}^{-}] \lnot \varphi$.

This language is only a small extension of hybrid tense logic $\mathcal {H}(\downarrow )$ given by Braüner (2011), which can be obtained by adding time reference operators and their inverses ($[\text {R}_{n}]$, $[\text {R}_{n}^{-}]$), as well as classical tense operators (G, H), to it. We refer to this language as hybrid tense logic for temporal conditionals and denote it by HTL$_{TC}$.

We next present a standard Kripke model for HTL$_{TC}$. First, we prepare a Kripke frame $\mathcal {T}=(T, \{ < \} \cup \{ \text {R}_{n} \}_{n \in \mathbb {N}})$ comprising a set T of moments and relations $<, \text {R}_{n}$ ($n \in \mathbb {N}$) among these moments. Given a valuation V for propositional variables, the pair $\mathcal {M}=(\mathcal {T}, V)$ is a Kripke model for HTL$_{TC}$.

In the above, < represents the strict temporal order on T, which is assumed to be transitive.³⁸

It is also assumed to be past-ward linear, i.e. it satisfies the following condition:

$$\begin{aligned}{} & {} { Past}\hbox {-}{} { ward}\,{ Linearity}\\{} & {} \quad \forall t, t', t'' \in T~ ((t'< t ~\text {and}~ t''< t) \Rightarrow (t'< t'' ~\text {or}~ t' = t'' ~\text {or}~ t'' < t')) \end{aligned}$$

This expresses a general constraint on the branching-time model in that it can branch forward but not backward, reflecting the idea that the past is determined and therefore linear, whereas the future is undetermined and branches into many possible time series.

Furthermore, we assume the following condition for the interaction between < and $\text {R}_{n}$³⁹

$$\begin{aligned}{} & {} { Intermediateness}\,{ of}\,{ Time}\, { Reference}\\{} & {} \quad (\text {R}_{n}< ^{\exists }\text {R}_{n+1}< \text {R}_{n+2})\\{} & {} \quad \forall t, t', t'' \in T ~ (t \text {R}_{n} t' ~\text {and}~ t'< t'' ~\text {and}~ t \text {R}_{n+2} t'' \\{} & {} \quad \Rightarrow \exists t''' \in T (t'< t''' ~\text {and}~ t \text {R}_{n+1} t''' ~\text {and}~ t''' < t'')), \\ \end{aligned}$$

which is equivalent to the $\text {R}_{n}$-inverse version:

$$\begin{aligned}{} & {} (\text {R}_{n}^{-}< ^{\exists }\text {R}_{n+1}^{-}< \text {R}_{n+2}^{-})\\{} & {} \quad \forall t, t', t'' \in T ~ (t' \text {R}_{n}^{-} t ~\text {and}~ t'< t'' ~\text {and}~ t'' \text {R}_{3}^{-} t \\{} & {} \quad \Rightarrow \exists t''' \in T (t''< t''' ~\text {and}~ t''' \text {R}_{2}^{-} t ~\text {and}~ t''' < t'')). \end{aligned}$$

An instance of this condition is

$$\begin{aligned}{} & {} (\text {R}_{1}^{-}< ^{\exists }\text {R}_{2}^{-}< \text {R}_{3}^{-})\\{} & {} \quad \forall t, t', t'' \in T ~ (t' \text {R}_{1}^{-} t ~\text {and}~ t'< t'' ~\text {and}~ t'' \text {R}_{3} t'\\{} & {} \quad \Rightarrow \exists t''' \in T (t''< t''' ~\text {and}~ t''' \text {R}_{2}^{-} t ~\text {and}~ t''' < t'')), \end{aligned}$$

whose corresponding inference rule is $(R_{1}^{-}< ^{\exists }R_{2}^{-} < R_{3}^{-})$ shown in Table 1 in Sect. 5.5, which, in turn, plays a crucial role in our formal deduction of counterfactual transitivity.

With the language and its model above, we state the satisfaction conditions only of the downarrow binder $\downarrow$, the time reference operator $\langle \text {R}_{n} \rangle$, and its inverse $\langle \text {R}_{n}^{-} \rangle$. The remaining items are defined as usual.

For $\downarrow$, we introduce a function g from nominals to moments and relativise the evaluation of formulas to g as follows:

$$\begin{aligned} \begin{array}{lll} \mathcal {M}, g, t \models \downarrow a. \varphi&\text {iff}&\mathcal {M}, g[a \mapsto t], t \models \varphi , \end{array} \end{aligned}$$

where $g[a \mapsto t]$ assigns the same moments to the nominals as g except that $g[a \mapsto t]$ assigns t to a.

The satisfaction conditions for $[\text {R}_{n}]$ and $[\text {R}_{n}^{-}]$ are given as follows:

$$\begin{aligned} \begin{array}{lll} \mathcal {M}, g, t \models [\text {R}_{n}] \varphi &{}\text {iff} &{} \forall t' \in T~ (t \text {R}_{n} t' \Rightarrow \mathcal {M}, t' \models \varphi ), \\ \mathcal {M}, g, t \models [\text {R}_{n}^{-}] \varphi &{}\text {iff} &{} \forall t' \in T~ (t' \text {R}_{n} t \Rightarrow \mathcal {M}, t' \models \varphi ). \end{array} \end{aligned}$$

A natural deduction system for HTL$_{TC}$, which is, again, only a special case of Braüner’s system (Braüner, 2011), is presented in “Appendix A”. In particular, note that all rules corresponding to conditions on the accessibility relation, listed in Table 5 in “Appendix A”, are instances of the form for geometric theories,⁴⁰

5.4 Comparison with Thomason and Gupta’s Theory

Our resulting formalism is similar to that of Thomason and Gupta (1980), which analyses counterfactual conditionals (and conditionals in general) using branching-time structures. We list three similarities between our approach and theirs that illuminate the respective formalisms:

First, we share with Thomason and Gupta (1980) the idea of using basic formalism of classical tense logic originating from Prior (1955, 1967) to analyse counterfactual conditionals (and conditionals in general).

Second, their symbolisation of a supposedly typical example of historical/real counterfactuals shows signs of our Reichenbachian idea based on Tsai (2014): a historical/real counterfactual conditional generally has an implicit reference time in each of the antecedent and the succedent.⁴¹

Third, there is a technical correspondence. In the semantics of Thomason and Gupta (1980), the ‘copresence’ relation $\simeq$ is used to refer to the ‘alternative presents’ at which the antecedent of a conditional holds.⁴² We can think that this ‘copresence’ relation $\simeq$ is the counterpart of our ‘time reference’ relation R$_{n}$.

However, there are crucial differences between our formalism and that of 1980, of which we list three that also illuminate both formalisms:

First, the semantics of Thomason and Gupta (1980) involves quantification over histories (or scenarios), i.e. sets of moments, which ours does not.⁴³ This implies that their semantics departs from the standard Kripke semantics, while ours does not.

Second, following Stalnaker (1968), Thomason and Gupta (1980) used the special single connective > for the conditional, and present complex semantics with it.⁴⁴ (In short, the syntax is simple but the semantics is complex.) By contrast, we dispense with such a special connective. Instead, we use material implication $\rightarrow$ combined with the downarrow binder, the classical tense operators, and their natural (Reichenbachian) extensions (namely the time reference operators and their inverses) and present straightforward Kripke semantics with them.⁴⁵ (In short, the syntax is complex but the semantics is straightforward.)

Third, our formalism has a specific purpose not shared by that of Thomason and Gupta (1980)⁴⁶: to explain why counterfactual transitive inference (CT) holds when it does hold.⁴⁷ We pursued this proof-theoretically as well as model-theoretically, as discussed in the next section.

5.5 Deduction of Counterfactual Transitivity in HTL$_{TC}$

Now, we are prepared for a formalisation of (CT) in a manner different from Lewis’s approach. We symbolise three sentence-schemas constituting (CT) in HTL$_{TC}$ in the same way as that for (Nix) into (Nix-W), which is much simpler than (Nix-S). The resulting schema is as follows⁴⁸:

We note two points. First, all three sentence-schemas of (SCT) share the same diverging time point $\text {R}_{0}$. This implies that the speaker of (CT) keeps the context unchanged throughout his/her inference. Second, (SCT) involves three reference times — $\text {R}_{1}$, $\text {R}_{2}$ and $\text {R}_{3}$ — in addition to $\text {R}_{0}$, where $\text {R}_{2}$ is both the reference time in the succedent of the first premise and the one in the antecedent of the second. This reflects an obvious idea on temporal relation in the whole inference structure: $\text {R}_{2}$ should be located between $\text {R}_{1}$ and $\text {R}_{3}$.

We can then construct the formal proof of (SCT) in the natural deduction system for HTL$_{TC}$, which is a special case of Braüner’s system (Braüner, 2011). In Fig. 3, $\Phi$, $\Psi$, and X are subformulas of the three formulas of (SCT), which exclude $\downarrow a.$, and $\Phi [i/a]$ is the result of replacing all the occurrences of a in $\Phi$ by i. Figure 4 elucidates the meaning of each step of the proof.

Fig. 3

Natural deduction derivation of (SCT)

Fig. 4

Intuitive understanding of natural deduction derivation of (SCT)

Let us observe the deduction in Fig. 3. Every inference rule used in each step has been made explicit. Thus, we find that, although the deduction is lengthy, nearly all of the steps are routine practices of standard natural deduction and the rules on the accessibility relation are used in only two steps. One is common in modal logic, i.e. (Trans-F), corresponding to transitivity of future relation (see Table 5 in “Appendix A”). The other is unique to the present context, which requires that (CT) and, accordingly, (SCT) should hold. It is as follows:

This rule (Table 1) is also an instance of the form for geometric theories,⁴⁹ The meaning is obvious: if reference times $\text {R}_{1}$ and $\text {R}_{3}$ are on the same timeline, then there is a reference time $\text {R}_{2}$ between them.

We note two points. (1) The routine part of the deduction comprises eliminations and introductions of $\rightarrow$ and the classical tense operators H, G in addition to two eliminations and one introduction of the $\downarrow$ binder. This means that, essentially, (SCT) is an inference involving nested-tense operators interposed among implications. (2) The ‘nested-tense’ inference presupposes a special accessibility condition, i.e. intermediateness of $\text {R}_{2}$ between $\text {R}_{1}$ and $\text {R}_{3}$, as well as a usual condition, i.e. transitivity of the temporal order relation <.

This is our answer to the question why (CT) holds when it does hold. In summary, it is by virtue of (1) the elimination/introduction rules of the $\downarrow$ binder and implication and classical tense operators and (2) transitivity of the temporal order relation and intermediateness of the second reference time between the first and third reference times.

Table 1 Special accessibility rule: $(R_{1}^{-}< ^{\exists }R_{2}^{-} < R_{3}^{-})$

As an immediate consequence, we can translate (NND), an instance of (CT), into $(NND')$ below, which is, in turn, an instance of (SCT). In the following, $\varphi {:}{=}p$, $\psi {:}{=}P q$, and $\chi {:}{=}P r$, where p:= ‘Nixon presses the button’, q:= ‘There is a nuclear war’, r:= ‘There is a depopulation’, and P is the past-possibility tense operator meaning ‘At some moment in the past,’.

Incidentally, when $\text {R}_{1}$=$\text {R}_{2}$=$\text {R}_{3}$, i.e. when the reasoner of (CT) assumes that the reference times in the antecedent and succedent remain the same throughout his/her inference, the result becomes simpler. In such a case, (CT) is symbolised as

and can be deduced as in Fig. 5 below.

Fig. 5

Natural deduction derivation of $(SCT [R_{1}=R_{2}=R_{3}])$

As seen in Fig. 5, (Trans-F) and ($R_{1}^{-}< ^{\exists }R_{2}^{-} < R_{3}^{-}$) are not used anymore, and therefore our answer can be simplified as follows: when the distinction between the reference times in the antecedent and succedent collapses, it is by virtue of the elimination/introduction rules of the $\downarrow$ binder and implication and the classical tense operators that (CT) holds when it does hold. In other words, in such a case we can say that (CT) is almost purely a ‘nested-tense’ inference with no additional accessibility conditions required.

In addition, by the soundness result in Braüner (2011), $(SCT {[R_{1}=R_{2}=R_{3}]})$ is valid in the class of all frames of < and $\text {R}_{n}$, as is (SCT) in the class of frames satisfying transitivity of < (corresponding to the rule (Trans-F)) and intermediateness of $\text {R}_{2}$ between $\text {R}_{1}$ and $\text {R}_{3}$ (corresponding to the rule ($R_{1}^{-}< ^{\exists }R_{2}^{-} < R_{3}^{-}$)).⁵⁰

6 Deduction of Indicative Transitivity in HTL$_{TC}$

We now consider the following syllogism, assuming that it was spoken in 1962:

As is seen, this is an indicative version of (NND). We can symbolise this in HTL$_{TC}$ as follows, where again p:= ‘Nixon presses the button’ q:= ‘There is a nuclear war’, and r:= ‘There is a depopulation’:

and can be regarded as instances of (IT) and (SIT), respectively⁵¹⁵²:

(SIT), in turn, can be presented with a formal deduction in HTL$_{TC}$, which we omit, as it can be executed in a manner similar to and simpler than (SCT).

7 A New Generality: Emergence of ‘Temporal’ Conditionals

Compare (SCT) with (SIT):

Obviously, (SIT) is a special case of (SCT), where ‘the head’ of the three formulas constituting the latter, i.e. ‘$H (\langle \text {R}_{0}^{-} \rangle a \rightarrow$’, is removed.⁵³ This implies that there are many uses of conditionals in which the indicative transitivity

is just a special case of the counterfactual transitivity

This, in turn, implies that there are many uses of conditionals in which the indicative conditional

$$\begin{aligned} { `If}\, { it}\,{ is}\,{ the}\,{ case}\,{ that}\, \varphi ,\,{ then}\,{ it}\,{ will}\,{ be}\,{ the}\,{ case}\,{ that}\, \psi ' \end{aligned}$$

is just a special case of the counterfactual conditional

$$\begin{aligned} { `If}\,{ it}\,{ were}\,{ the}\,{ case}\,{ that}\, \varphi ,\, { then}\,{ it}\,{ would}\,{ be}\,{ the}\,{ case}\,{ that}\, \psi '. \end{aligned}$$

Thus, we discover a new class of conditionals in which an indicative conditional is a special case of a counterfactual conditional in its logical form. We refer to such a class as ‘temporal’ conditionals. A distinctive characteristic of the class is given roughly as follows:

(TC)⁵⁴ For any member of the class of ‘temporal’ conditionals, if it is grammatically counterfactual, then we can construct its intelligible indicative counterpart by cutting ‘the head’ $H (\langle \text {R}_{0}^{-} \rangle a \rightarrow$ of (the weak version of) its translation in HTL$_{TC}$, and conversely, if it is grammatically indicative, then we can construct its intelligible counterfactual counterpart by adding ‘the head’ $H (\langle \text {R}_{0}^{-} \rangle a \rightarrow$ to (the weak version of) its translation in HTL$_{TC}$.

(Nix) is a paradigm of this class. We can construct its indicative counterpart ‘If Nixon presses the button, there will be a nuclear war’ in the specified way. Similarly, this can be done for (Osw) in Sect. 3 and the three constituents of (OAW) in Sect. 4, although there are cases in which (OAW) does not hold as a whole.⁵⁵

Meanwhile, consider (b), (y), (k), and (g) in Sect. 3. The indicative counterparts of (b) and (y) are complete nonsense. For (k), its reasonable counterpart seems to be ‘If kangaroos lose tails, they will topple over’; however, this is at least quite unnatural. For (g), its counterpart would be ‘If gravity goes by the inverse cube of distance,...’; however, this is again almost nonsense.

Interestingly, for this criterion, what we want to refer to as a historical counterfactual may nevertheless turn out to be not ‘temporal’. Consider ‘No Hitler, no A-bomb’$\cdots$ (Hit).⁵⁶ Its indicative counterpart would be something such as ‘If Hitler does not appear, A-bombs will not be developed’; however, it has unnaturalness similar to that of ‘If kangaroos lose tails,...’. This suggests that the former is also close to a rhetoric that highlights the historical importance of Hitler for the actual development of atomic bombs.

In Table 2, we display the result of applying (TC) to the samples of counterfactuals in this paper:

Table 2 Classification of counterfactual samples in this study

Certainly, there will be considerable scope to scrutinise (TC) above.⁵⁷ However, I would like the reader to at least accept the following fact: there are many conditionals for which our symbolisation in HTL$_{TC}$ is appropriate. If this fact is accepted, there seems to emerge a new logical perspective on conditionals in general: temporal ones and others.

More specifically, the above results show that, among both indicative and counterfactual conditionals, there are those that can be symbolised appropriately in a natural extension of classical tense logic, and there are those that cannot. In this case, we find out that the former become much clearer from a logical point of view, whereas the latter are less clear and miscellaneous; hence, further logical investigation is required, which, in turn, perhaps requires different formal systems in which they can be reconstructed otherwise logically.

8 Related Works and Discussion

Our results would have many ramifications for related works in philosophical logic and linguistics. We list some in this section.

Two previous works, Thomason and Gupta (1980) and Placek and Müller (2007), analysed counterfactuals in terms of historical/real possibility modelled by branching-time structures. One of the differences in semantics between these approaches and ours is that while Thomason and Gupta (1980) and Placek and Müller (2007) involve quantification over histories, ours does not. Among the differences in syntax, these approaches use a special single connective (Stalnaker-style > for Thomason and Gupta (1980), Lewis-style for Placek and Müller (2007)), whereas ours does not. (For these differences, see Sect. 5.4.)

As a formal system, our approach, i.e. HTL$_{TC}$, is close to M. J. Cresswell’s system (Cresswell, 2010), which enables temporal references corresponding to ‘now’ and ‘then’ in natural language. However, there are two main differences between his system and ours. First, and less important, Cresswell’s logic is a type of predicate tense logic, whereas ours is a type of hybrid tense logic.⁵⁸ Second, and more importantly, Cresswell’s logic is a linear-time tense logic, whereas ours is a (possibly) branching-time tense logic, which enables us to express many possible timelines upward and also make temporal references in them.⁵⁹

Among recent literature, W. B. Starr’s approach (Starr, 2014) seems antithetical to ours, as it proposes ‘a uniform theory of conditionals’ (its title), which is based on an extension of Stalnaker’s system and, more fundamentally, what we referr to as Stalnaker reasoning: there is a uniformity of grammatical form among conditionals; therefore, there should be a uniformity of logical form among them.⁶⁰ However, this certainly does not mean that our results conflict with Starr’s theory; instead, it only suggests that the targets of Starr’s theory include temporal and other conditionals. Therefore, it would be expected that Starr’s theory would turn out to be more suitable for the latter (other) class than for the former (temporal) class.

Meanwhile, Rothschild (2015) maintains propositionalism about conditionals, which is the conservative view that conditionals, as well as non-conditional declarative sentences, have propositional content identified with their truth conditions that can adequately account for their meaning; however, as Rothschild (2015) reports, there are more serious challenges to this view concerning conditionals than there are concerning non-conditionals. Our analysis could be a partial but strong support for the propositionalism about conditionals: it could be a partial support, as it is specific to temporal conditionals; it could, on the other hand, be a strong support because of its specificity, as it has presented their fine-grained syntactic structures and correspondingly their fine-grained truth conditions. Thus, it would be worthwhile to scrutinise how far the specified truth conditions of temporal conditionals can vindicate the propositionalism about temporal conditionals.

Finally, there seems to be an overall characteristic of our approach that is distinct from any other in the literature on counterfactuals. As we mentioned in Sect. 4, since Lewis (1973), it has been agreed that counterfactual transitive inference (CT) does not necessarily hold. However, this is just another way of saying that there are cases in which (CT) does not hold as well as cases in which it does hold. Previous studies appear to have focussed mainly on the former fallacious cases; however, this paper focusses on the latter successful cases and asked the question why does (CT) hold when it does hold?

9 Concluding Remarks

In conclusion, we present a few more remarks about a crucial difference between our new analysis and Lewis’s classical analysis, in the light of (CT). In this paper, we faithfully follow the ‘meaning-as-use’ and ‘inferentialist’ approach involved in the procedure of Tarski’s theory of truth,⁶¹ whereas Lewis appears to have had no intention of doing so. We specify what constrains the use of counterfactuals in inference, i.e. (successful) counterfactual transitive inference (CT), and, as a result, there emerges a set of logical constraints on the notion of ‘temporality’; Lewis, on the other hand, does not seem to have adopted such a procedure concerning the notion of ‘similarity’.

This difference is reflected clearly in respective formal deductions with regard to (CT). In our deduction of (SCT), there appear inference rules that are certain to constrain the notion of ‘temporality’, namely the elimination/introduction rules of tense operators, (Trans-F), and ($R_{1}^{-}< ^{\exists }R_{2}^{-} < R_{3}^{-}$). Meanwhile, for Lewis’s deduction of (VCT),⁶²no such rules that constrain ‘similarity’ seem to be required. Other than rules and axioms of propositional logic, the only rule and axiom required there are (DWC) and (VA(5)):

Deduction within ConditionalsFor any $n \ge 1$,

Axiom (5)

However, these do not seem to have any direct connection with ‘similarity’. Roughly speaking, (DWC) is a type of necessitation of actual implication,⁶³ and (VA(5)) is a weakened analogy of the theorem $((\varphi \wedge \psi ) \rightarrow \chi ) \leftrightarrow (\varphi \rightarrow (\psi \rightarrow \chi ))$ in propositional logic. After all, at least proof-theoretically, it seems that the notion of ‘similarity’ is not required for deduction of (VCT).⁶⁴

Thus, the alleged success of our new analysis can be attributed to the two classical points of view, namely the ‘meaning-as-use’ and ‘inferentialist’ points of view, detected in Tarski’s theory of truth in Sect. 2. We believe that it is also a demonstration of their significance for logical analysis of natural language.⁶⁵

Appendix A: A Natural Deduction System for HTL\(_{TC}\)

A natural deduction system for HTL$_{TC}$, which is only a special case of Braüner’s system (Braüner, 2011), is presented below (Tables 3, 4, 5).

Table 3 Natural deduction rules for connectives

Table 4 Natural deduction rules for nominals

Table 5 Rules corresponding to conditions on the accessibility relation