The Inherent Structure of Experiments
as a Constraint to Spatial Analysis and Modeling

We use words for kinds of things (predicators) and individual things of some kind (nominators). In addition to predicators for space and time which range over individual locations and moments in time (in spatial and temporal reference systems), we use the possibility of forming amounts of space and time [25], such as regions and time intervals. The former can be used to talk about the amount of space occupied by certain things. Similar predicators we use for amounts of stuff or objects [23]. Furthermore, we call all these predicators for space, objects, stuff, and their amounts endurances, meaning that they play a particular role in describing situations: they can change in time, whereas occurrences are the things that are going on in time, reflecting a common distinction in information ontology.

Nominators allow us to refer to particular things, either by introducing names, or by using (in a common situation of speech) indicators (“this”, “that”) together with predicators:

In the following we use various nominators for each predicator above, including names for persons, objects etc.

Other predicators stand for different occurrences, to say what “goes on” with things. We distinguish dynamic from static occurrences using process predicators (involving some change of a situation that happens at a moment in time) and state predicators (involving some situation is static at a moment in time). Furthermore, we use a special class of predicators for talking about what can be done (do-predicator):

Do-predicators are distinct from other occurrences, since they stand for kinds of actions that can be attributed to the persons performing them, including their purposes [9, 12]:

The copula $\kappa$ is used to form situations with occurrences, to say that some occurrence has happened, and $\pi$ to form situations with do-predicators, to denote action performances:

Appredicators are expressions that further specify the occurrence, which may use prepositions together with nominators. A happening uses a temporal nominator and the copula $\kappa$ with some predicator for occurrences. For example:

“at that time is raining this amount of water”
“now is growing”

In a similar way, we use the copula $\pi$ for reporting on action performances:

“at that time does stay at this house”
“now does run home”

Note that we can always interpret an action performance as if it was a process, i.e., a behavior [28], since do-predicators are occurrences. Situations are either happenings or actions that are controlled by endurance nominators, referring to those things to which this happens/who control the action. In particular, we require a person in control of actions:

For example:

“this person at this time does stay at this house”
“here at this time is raining this amount of water”
“this tree now is growing”

The distinctive role of situations, which are sorts of time-dependent propositions, has been recognized early on in artificial intelligence, where they are called fluents [19].

Propositions are used to make defensible claims. From a pragmatic perspective, the latter are speech acts, actions that can be performed by persons in a dialogue. To be able to express such acts, we introduce a way of saying that someone makes a claim using any proposition formed from the grammar above.

For example, Nora now makes the claim that it will be raining tomorrow:

“Nora now $\pi$(here tomorrow is raining)”

Goals are propositions intended by persons. They can be wished without ever pursuing an action (wishful thinking), but in the more practically relevant cases, we talk about goals that actually can be pursued via actions. We form goals from propositions using a conjunction “such that” or $⊩$. For example, if I am traveling, I might wish to be at home at a certain time:

“such that I then do stay at home”

We can distinguish goals based on what kind of proposition is used. Whenever we are using a situation as a goal, we are wishing that the latter may come about:

An example for a modificative goal is my wish to be at home (above), meaning a modification of the place at which I am staying. Imperatives are speech acts that prompt some action from a person. This can be expressed either by indicating the action directly, or by requesting a goal and leaving the action that implements the goal open to the person addressed. In order to express imperatives, we use the copula $!$:

For example, a mother may request from her daughter Nora to be at home in time for dinner:

“Nora ! $⊩$ at this time are having dinner”
“Peter ! at this time do cycle home”

The first imperative is a request to bring about some situation using some modificative goal³³3“Aufforderung zur Herbeifuehrung eines Sachverhaltes”, see [18, p. 44]. This leaves it open to Nora how and when she takes action to meet the goal. The second imperative, in contrast, requests an action explicitly. Following Lorenzen [18, p.45], we call the first case final imperatives, and the second a-final imperatives. Finally, we allow for a corresponding speech act, a request, which expresses that someone is performing a request using an imperative.

For example, Nora’s mother Ellie requests Nora to run some errands later:

“Ellie now $\pi$(Nora ! today do run this errand)”,

stands for the corresponding request. If we leave away the person nominators in such acts, we mean that the person who utters the request is requesting something from herself, meaning the person sets herself a goal. For example, I might now set myself the goal of running errands later today:

“now $\pi$(! today do run this errand)”

In pragmatic philosophy, knowledge is understood as a form of know-how, meaning it must be actionable: knowledge enables action, encompassing the skills necessary to achieve goals, articulate and pursue interests, and ultimately navigate life within a heterogeneous society [17, 12]. What distinguishes knowledge from mere opinion is the notion of validity: a valid claim is a proposition that is successfully justifiable, which in turn requires the success of the actions underlying its defense, including the successful execution of experiments.

To be valid, claims must be generalizable across multiple examples. To express such generalizable claims, we employ standard logical connectives: disjunction ($\vee$) for “or,” conjunction ($\wedge$) for “and,” and negation ($\neg$) for “it is not the case that.” These can be combined to form complex propositions. Additionally, we use the implication operator ($\to$) to denote conditional statements: “if the first proposition is true, then the second must also be true.” For example, the logical formula $A\vee (\neg B\wedge (\neg (C\to D)))$ expresses a structured claim where $A,B,C,$ and $D$ are arbitrary propositions.

Quantifiers extend conjunction and disjunction over arbitrarily many propositions by introducing variables. Variables are placeholders for elements within a specified domain – a collection of nominators that share a common predicate. To denote domains, we use upper-case symbols corresponding to predicators in Sect. 2.1. For example, the domain Person consists of nominators referring to individuals. Variables such as $x,y,z$ can be substituted by any element from their respective domains. The universal quantifier ($\bigwedge$) generalizes conjunction across all elements of a domain, asserting that a proposition holds for every substitution:

${\bigwedge }_{x\in \text{Space}}x$ now is raining $\wedge$ $x$ now is wet

This states that it is raining and wet everywhere in space. Conversely, the existential quantifier ($\bigvee$) generalizes disjunction, asserting that a proposition holds for at least one substitution:

${\bigwedge }_{x\in \text{Space}}{\bigvee }_{y\in \text{Time}}x$ $y$ is raining $\wedge$ $x$ $y$ is wet

This expresses that at every location in space, there exists some point in time where it is raining and wet. We refer to such quantified logical expressions as formulas. Formulas can be used to describe complex situations involving actions or processes.

A crucial aspect of pragmatic knowledge is understanding how actions lead to consequences. We distinguish between conditions, which must hold at the time an action is performed, and consequences, which describe the expected results. An action is deemed unsuccessful with respect to a goal if its consequences do not fulfill that goal. The reason for failure can often be traced back to unmet conditions. This leads to the notion of knowledge about the consequences of actions⁴⁴4“Handlungsfolgenwissen” [12, 9]. Such knowledge is formalized using consequential rules, which capture the expected outcomes of actions under specific conditions:

Here, $R(x,\mathrm{\dots },y)$ denotes a formula capturing requirements (conditions necessary for the action), and $EC(x,\mathrm{\dots },y)$ denotes a formula capturing the expected consequences. For example:

${\bigwedge }_{x\in \mathrm{𝐶𝑎𝑛𝑑𝑙𝑒}}{\bigwedge }_{y\in \mathrm{𝑀𝑎𝑡𝑐ℎ𝑒𝑠}}$ Nora now uses $y$ on $x$ $\to$ $x$ then is burning.

This rule asserts that lighting a candle with a match under Nora’s agency results in the candle burning – though this claim is context-dependent. It holds for an adult on Earth but fails for a child or in a zero-oxygen environment. In pragmatics, this only demonstrates the need to refine requirements for assuring validity. Progression rules describe changes in state over time due to processes rather than actions:

The temporal ordering implicitly assumes that the antecedent conditions occur before the consequent state. For example:

${\bigwedge }_{x\in \mathrm{𝐿𝑎𝑘𝑒}}x$ now contains this amount of water $\wedge$ here now raining that amount of water) $\to x$ then contains (this + that) amount of water.

Progression rules need to be justified by experiments (see below) or derived from other knowledge. A set of such rules forms a rule base: $\mathrm{𝐶𝑅𝐵}$ for consequential rules and $\mathrm{𝑃𝑅𝐵}$ for progression rules. Together with a set of formulas describing the current situation $S(t)$, we obtain a knowledge base: ${\mathrm{𝐶𝐾𝐵}}_{S(t)}=\mathrm{𝐶𝑅𝐵}\cup S(t)$ or ${\mathrm{𝑃𝐾𝐵}}_{S(t)}=\mathrm{𝑃𝑅𝐵}\cup S(t)$. If we can infer a formula $F$ from such a knowledge base using logical inference, we write $\mathrm{𝐾𝐵}\prec F$.

In addition to knowledge about consequences of actions and progressions, we also require knowledge about people’s behavior in terms of speech acts. These are actions like claims and requests in which some explicit knowledge base is required. Correspondingly, we introduce rules of inference (for actions that derive claims from other claims) as well as decision rules (for deriving goals from other claims or other goals):

A particularly relevant example of a decision is to plan. Pragmatically, plans are understood as artifacts that are a result of a process of planning [8]. However, they are more than that: Plans are also symbolic manifestations of imperatives (formalized by using a request $\pi (!)$). For one, we plan according to a planning goal, which can be understood as a final imperative specifying an intended situation that should be realized by a plan. The plan itself manifests likewise a final or an a-final imperative, consisting of a series of actions to be performed or of subgoals to be pursued in order to reach this goal. A successful plan, thus, satisfies a conditioned imperative: it needs to successfully realize the goal whenever we follow it in an experiment. We can express this kind of knowledge also in terms of rules.

Based on such knowledge bases, we can assess what can be done. Namely in the sense of knowing whether an expected consequence $A$ is achievable in a given situation. The latter can be defined based on whether A is logical implied by consequential rules in this situation:

Literally, ${△}_{{\mathrm{𝐶𝐾𝐵}}_{S(t)}}^{\pi }A$, or A is achievable means that some expected consequence described by the formula A can be justified by (repeatedly) applying consequential rules from the knowledge base to the situation $S(t)$. When it is clear which knowledge base is meant, we can also leave away the subscript: ${△}^{\pi }A$.

The power of this practical modal logic [17, 18] is to capture everyday notions of dispositions and action potentials relative to a situation. This becomes clear when we define the modal variants:

If a consequence is avoidable, this means its contrary can be achieved. If it is unachievable, we fail to justify it can be achieved. And if it is unavoidable, we fail to justify that it can be avoided. For example, in a situation where a state launches atomic missiles to attack another state, which also possesses atomic missiles, an atomic war is unavoidable. This is because, according to our knowledge of consequential rules of warfare and assuming a certain behavior, namely that the corresponding protocols are implemented by the group of people responsible for them, we fail to find a path of action that would not involve launching a counter-attack, and thus we may not find a way to prevent a war in this situation.

Controllable situations are both achievable and avoidable. Sometimes we can avoid a consequence only constructively, based on changing a situation described in a corresponding formula using another nominator, i.e., to switch nominators. This leads to a more specific case of value controllability:

The atomic counter-attack is a case in point, because there needs to be a switch for controlling the missile launch, and this switch is always in some position.

In an equivalent way, we can use modal logic to reason with knowledge of a situation and some progression model, which can be expressed as a collection of progression rules:

Literally, $A$ is a necessary consequence of a given situation $S(t)$ at time $t+\delta$, under the assumption that the progression rules and the situation descriptions are defendable, and if $A(t+\delta )$ is a logical implication. By abstracting from the particular base ${\mathrm{𝑃𝐾𝐵}}_{S(t)}$, we also write $▽A(t)$ for the situations that will happen as a consequence of this situation at some time $t$. For example, in case we have a progression model of rainfall covering the extent of a lake, we may be able to predict the amount of water of that lake at a time after the rainfall stopped, given that we know its water content in the current situation. The definitions of these so called mellontic modalities [17] are equivalent to the practical ones above, including possible ($△$), impossible ($\overline{△}$), and contingent (  ⋈ ). Contingent consequences are those that are possible yet we still fail to show that they are logically implied. That is, based on our progression model, we just don’t know.

The empirical (a-posteriori) knowledge [17] that we can obtain from an experiment can be written down in the form of experiential rules that are very similar to consequential rules introduced above, except that they involve the triggering of a process p (grammatic rule 4):

Literally, if we do something to start process $p$ under the condition $S_c$, then we can expect situation $S_m$ to occur [16]. Knowledge obtained from a given experiment can include many such rules. Experiential rules constitute both constructive building blocks and tests for empirical theories. In the latter case, by using a theory to infer an experiential rule that is compared with the result of a corresponding experiment, in the former case, by directly generalizing from experiential rules.

Yet, like all actions, experiments can fail, and in consequence, rules become invalid. How exactly can experiments fail? This depends on their purpose [10, 13]. The purpose of an experiment [16] derives from the trans-subjectivity of empirical knowledge: it is to reproduce the process p such that it leads to similar situations (consequences) under the same conditions, regardless of who is triggering the process and with which instruments (under which further circumstances). Conditions can be either fixed (not changed in the experiment) or controlled (changed in the experiment). This means that all conditions must be achievable via actions (definition 3), while controls need to be, in addition, controllable (can be switched on or off) (definition 7). In addition, we often need to leave some other situations contingent (“free”, or not pre-determined) (section 3.2, last paragraph). Conditions and contingent situations are required to prevent the experiment from being disturbed. The situations (grammatic rule 8) that are the consequences (schema 1) of the experiment can be represented by measures. Altogether, we call this the experimental reproducibility norm, and it has the following general form:

The nominators $f_1,\mathrm{\dots },f_n$ (fixes, taken from domains $F_i$), $c_1,\mathrm{\dots },c_n$ (controls, taken from domains $C_i$), $u_1,\mathrm{\dots },u_m$ (contingents from domains $U_i$), $m_1,\mathrm{\dots },m_u$ (measures, taken from domains $M_i$) thereby serve to identify and reproduce the respective situations.

For example, an experimental norm for a simple spatio-temporal experiment about growing crops in a geographic region could look like this:

This norm defines an experiment ($\mathrm{𝑅𝑒𝑔𝑖𝑜𝑛},\mathrm{𝑔𝑟𝑜𝑤}\mathrm{𝑏𝑒𝑎𝑛𝑠},\mathrm{𝐴𝑚𝑜𝑢𝑛𝑡𝑜𝑓𝐵𝑒𝑎𝑛𝑠}$) to determine how many beans can be produced in a region $r$, independent of who performs it ($o$) or when ($t$). The experiment requires sowing beans in $r$ at $t$ (controllable situation $S_c$) and ensuring that later sales ($t+\delta$) do not interfere, avoiding market disturbances. If beans are sown and properly cultivated ($p=\mathrm{𝑔𝑟𝑜𝑤}$), the norm expects that by ($t+\delta$), an approximate amount $m$ of beans will be produced. This norm is a priori: it does not specify the exact yield but requires that outcomes be reproducible up to equivalence. Experiments implementing this norm either fix or control or leave contingent conditions when triggering the process. If reproducibility fails – e.g., due to lack of seeds, planting restrictions, or market constraints – the experiment fails.

In case of failure, we can adjust an experimental norm to ensure valid experiential rules. Lange [16] suggested the following principle ways to deal with such disturbances:

1. 1.

   Isolating disturbances through shielding (possible in labs or simulations).

2. 2.

   Cleaning up disturbances by controlling, fixing, or rendering them contingent (e.g., via randomization).

3. 3.

   Incorporating disturbances as errors, increasing the tolerance of equivalences.

These adjustments constitute what Lange calls fault avoidance knowledge (referred to as exhaustion in [16]). For example, if bean growth depends on weather conditions or market quotas, fixing the yearly weather conditions and removing quota constraints could make the experiment reproducible. Note that inferential statistics, at its core, is a method for incorporating the disturbances of repeatable experiments using stochastic models (i.e., random generators) [18]. Methodologically, it comes after the introduction of experiments, not before.

Causal experiments play an exceptional role for science, since they allow us to determine causes. Yet, distinguishing causes from other experimental relations likewise requires pragmatic knowledge, an insight gained early by Georg Hendrik von Wright [28] in terms of his interventionist causality norm, and much later picked up in contemporary causal inference theory [20]. The corresponding experimental norm for causal experiments is more strict as it requires in addition a particular counterfactual situation, i.e., considering a consequential situation that occurs if we had not taken an action [21]. The norm requires that if some controls are not achieved, then the corresponding measures need to be different:

If an experiment satisfies such a norm, there is a one-to-one correspondence between possible control situations and measure situations. This is the case, e.g., when we run randomized control trials, where a control group lacks the condition, and the experiment is successful in case that group also lacks the expected consequence [21]. We can then call the control domain a cause of the measure domain. In case of failure to satisfy such a norm, we can clean up disturbances, i.e., by incorporating conditions, or by adding contingencies into the norm. The corresponding strategies are well known from the causal reasoning literature [20], including fixing confounders (common causes of conditions and consequences), and leaving contingent intermediators (effects of controls that are causes of consequences) and colliders (common effects of controls and consequences) [21].

Data record experiential rules in terms of the underlying nominators (in our bean growing example $(r_1,m_1),\mathrm{\dots },(r_k,m_k)$). Yet, such data records leave away many details needed to understand the underlying experiment. This includes not only the irrelevant further circumstances (here: time and person), but in particular, the fact that fixes and controls uniquely determine (are keys for) measures, and the question what kind of situations are controlled, fixed, or measured. To keep some of this information in an abbreviated form, we use the following notation for the type of experiential knowledge base that corresponds to an experimental norm:

Thus, for experiments, we usually control (c), fix (f) or measure (m) some domains D. For experiments that include claims, we additionally control, fix or measure knowledge claims ($\pi (\mathrm{𝐾𝐵})$) (which of course may be justified by further experiments). And for experiments that include goals, decisions and plans, we control, fix or measure requests for bringing about a situation in which we can make knowledge claims ($\pi (!⊩\pi (\mathrm{𝐾𝐵}))$. For the fixed conditions, we also write down constants instead of the domain from which they stem.