Category Archives: Popper

POPPER and EMPIRICAL THEORY. A conceptual Experiment


eJournal: uffmm.org
ISSN 2567-6458, 12.March 22 – 16.March 2022, 11:20 h
Email: info@uffmm.org
Author: Gerd Doeben-Henisch
Email: gerd@doeben-henisch.de

BLOG-CONTEXT

This post is part of the Philosophy of Science theme which is part of the uffmm blog.

PREFACE

In a preceding post I have outline the concept of an empirical theory based on a text from Popper 1971. In his article Popper points to a minimal structure of what he is calling an empirical theory. A closer investigation of his texts reveals many questions which should be clarified for a more concrete application of his concept of an empirical theory.

In this post it will be attempted to elaborate the concept of an empirical theory more concretely from a theoretical point of view as well as from an application point of view.

A Minimal Concept of an Empirical Theory

The figure shows the process of (i) observing phenomena, (ii) representing these in expressions of some language L, (iii) elaborating conjectures as hypothetical relations between different observations, (iv) using an inference concept to deduce some forecasts, and (v) compare these forecasts with those observations, which are possible in an assumed situation.

Empirical Basis

As starting point as well as a reference for testing does Popper assume an ’empirical basis’. The question arises what this means.

In the texts examined so far from Popper this is not well described. Thus in this text some ‘assumptions/ hypotheses’ will be formulated to describe some framework which should be able to ‘explain’ what an empirical basis is and how it works.

Experts

Those, who usually are building theories, are scientists, are experts. For a general concept of an ’empirical theory’ it is assumed here that every citizen is a ‘natural expert’.

Environment

Natural experts are living in ‘natural environments’ as part of the planet earth, as part of the solar system, as part of the whole universe.

Language

Experts ‘cooperate’ by using some ‘common language’. Here the ‘English language’ is used; many hundreds of other languages are possible.

Shared Goal (Changes, Time, Measuring, Successive States)

For cooperation it is necessary to have a ‘shared goal’. A ‘goal’ is an ‘idea’ about a possible state in the ‘future’ which is ‘somehow different’ to the given actual situation. Such a future state can be approached by some ‘process’, a series of possible ‘states’, which usually are characterized by ‘changes’ manifested by ‘differences’ between successive states. The concept of a ‘process’, a ‘sequence of states’, implies some concept of ‘time’. And time needs a concept of ‘measuring time’. ‘Measuring’ means basically to ‘compare something to be measured’ (the target) with ‘some given standard’ (the measuring unit). Thus to measure the height of a body one can compare it with some object called a ‘meter’ and then one states that the target (the height of the body) is 1,8 times as large as the given standard (the meter object). In case of time it was during many thousand years customary to use the ‘cycles of the sun’ to define the concept (‘unit’) of a ‘day’ and a ‘night’. Based on this one could ‘count’ the days as one day, two days, etc. and one could introduce further units like a ‘week’ by defining ‘One week compares to seven days’, or ‘one month compares to 30 days’, etc. This reveals that one needs some more concepts like ‘counting’, and associated with this implicitly then the concept of a ‘number’ (like ‘1’, ‘2’, …, ’12’, …) . Later the measuring of time has been delegated to ‘time machines’ (called ‘clocks’) producing mechanically ‘time units’ and then one could be ‘more precise’. But having more than one clock generates the need for ‘synchronizing’ different clocks at different locations. This challenge continues until today. Having a time machine called ‘clock’ one can define a ‘state’ only by relating the state to an ‘agreed time window’ = (t1,t2), which allows the description of states in a successive timely order: the state in the time-window (t1,t2) is ‘before’ the time-window (t2,t3). Then one can try to describe the properties of a given natural environment correlated with a certain time-window, e.g. saying that the ‘observed’ height of a body in time-window w1 was 1.8 m, in a later time window w6 the height was still 1.8 m. In this case no changes could be observed. If one would have observed at w6 1.9 m, then a difference is occurring by comparing two successive states.

Example: A County

Here we will assume as an example for a natural environment a ‘county’ in Germany called ‘Main-Kinzig Kreis’ (‘Kreis’ = ‘county’), abbreviated ‘MKK’. We are interested in the ‘number of citizens’ which are living in this county during a certain time-window, here the year 2018 = (1.January 2018, 31.December 2018). According to the statistical office of the state of Hessen, to which the MKK county belongs, the number of citizens in the MKK during 2018 was ‘418.950’.(cf. [2])

Observing the Number of Citizens

One can ask in which sense the number ‘418.950’ can be understood as an ‘observation statement’? If we understand ‘observation’ as the everyday expression for ‘measuring’, then we are looking for a ‘procedure’ which allows us to ‘produce’ this number ‘418.950’ associated with the unit ‘number of citizens during a year’. As everybody can immediately realize no single person can simply observe all citizens of that county. To ‘count’ all citizens in the county one had to ‘travel’ to all places in the county where citizens are living and count every person. Such a travelling would need some time. This can easily need more than 40 years working 24 hours a day. Thus, this procedure would not work. A different approach could be to find citizens in every of the 24 cities in the MKK [1] to help in this counting-procedure. To manage this and enable some ‘quality’ for the counting, this could perhaps work. An interesting experiment. Here we ‘believe’ in the number of citizens delivered by the statistical office of the state of Hessen [2], but keeping some reservation for the question how ‘good’ this number really is. Thus our ‘observation statement’ would be: “In the year 2018 418.950 citizens have been counted in the MKK (according to the information of the statistical office of the state of Hessen)” This observation statement lacks a complete account of the procedure, how this counting really happened.

Concrete and Abstract Words

There are interesting details in this observation statement. In this observation statement we notice words like ‘citizen’ and ‘MKK’. To talk about ‘citizens’ is not a talk about some objects in the direct environment. What we can directly observe are concrete bodies which we have learned to ‘classify’ as ‘humans’, enriched for example with ‘properties’ like ‘man’, ‘woman’, ‘child’, ‘elderly person’, neighbor’ and the like. Bu to classify someone as a ‘citizen’ deserves knowledge about some official procedure of ‘registering as a citizen’ at a municipal administration recorded in some certified document. Thus the word ‘citizen’ has a ‘meaning’ which needs some ‘concrete procedure to get the needed information’. Thus ‘citizen’ is not a ‘simple word’ but a ‘more abstract word’ with regard to the associated meaning. The same holds for the word ‘MKK’ short for ‘Main-Kinzig Kreis’. At a first glance ‘MKK’ appears as a ‘name’ for some entity. But this entity cannot directly be observed too. One component of the ‘meaning’ of the name ‘MKK’ is a ‘real geographical region’, whose exact geographic extensions have been ‘measured’ by official institutions marked in an ‘official map’ of the state of Hessen. This region is associated with an official document of the state of Hessen telling, that this geographical region has to be understood s a ‘county’ with the name MKK. There exist more official documents defining what is meant with the word ‘county’. Thus the word ‘MKK’ has a rather complex meaning which to understand and to check, whether everything is ‘true’, isn’t easy. The author of this post is living in the MKK and he would not be able to tell all the details of the complete meaning of the name ‘MKK’.

First Lessons Learned

Thus one can learn from these first considerations, that we as citizens are living in a natural environment where we are using observation statements which are using words with potentially rather complex meanings, which to ‘check’ deserves some serious amount of clarification.

Conjectures – Hypotheses

Changes

The above text shows that ‘observations as such’ show nothing of interest. Different numbers of citizens in different years have no ‘message’. But as soon as one arranges the years in a ‘time line’ according to some ‘time model’ the scene is changing: if the numbers of two consecutive years are ‘different’ then this ‘difference in numbers’ can be interpreted as a ‘change’ in the environment, but only if one ‘assumes’ that the observed phenomena (the number of counted citizens) are associated with some real entities (the citizens) whose ‘quantity’ is ‘represented’ in these numbers.[5]

And again, the ‘difference between consecutive numbers’ in a time line cannot be observed or measured directly. It is a ‘second order property’ derived from given measurements in time. Such a 2nd order property presupposes a relationship between different observations: they ‘show up’ in the expressions (here numbers), but they are connected back in the light of the agreed ‘meaning’ to some ‘real entities’ with the property ‘overall quantity’ which can change in the ‘real setting’ of these real entities called ‘citizens’.

In the example of the MKK the statistical office of the state of Hessen computed a difference between two consecutive years which has been represented as a ‘growth factor’ of 0,4%. This means that the number of citizens in the year 2018 will increase until the year 2019 as follows: number-citizens(2019) = number-citizens(2018) + (number of citizens(2018) * growth-factor). This means number-citizens(2019) =418.950 + (418.950 * 0.004) = 418.950 + 1.675,8 = 420.625,8

Applying change repeatedly

If one could assume that the ‘growth rate’ would stay constant through the time then one could apply the growth rate again and again onto the actual number of citizens in the MKK every year. This would yield the following simple table:

YearNumberGrowth Rate
2018418.950,00,0040
2019420.625,80
2020422.308,30
2021423.997,54
2022425.693,53
2023427.396,30
Table: Simplified description of the increase of the number of citizens in the Main-Kinzig county in Germany with an assumed growth rate of 0,4% per year.

As we know from reality an assumption of a fixed growth rate for complex dynamic systems is not very probable.

Theory

Continuing the previous considerations one has to ask the question, how the layout of a ‘complete empirical theory’ would look like?

As I commented in the preceding post about Popper’s 1971 article about ‘objective knowledge’ there exists today no one single accepted framework for a formalized empirical theory. Therefore I will stay here with a ‘bottom-up’ approach using elements taken from everyday reasoning.

What we have until now is the following:

  1. Before the beginning of a theory building process one needs a group of experts being part of a natural environment using the same language which share a common goal which they want to enable.
  2. The assumed natural environment is assumed from the experts as being a ‘process’ of consecutive states in time. The ‘granularity’ of the process depends from the used ‘time model’.
  3. As a starting point they collect a set of statements talking about those aspects of a ‘selected state’ at some time t which they are interested in.
  4. This set of statements describes a set of ‘observable properties’ of the selected state which is understood as a ‘subset’ of the properties of the natural environment.
  5. Every statement is understood by the experts as being ‘true’ in the sense, that the ‘known meaning’ of a statement has an ‘observable counterpart’ in the situation, which can be ‘confirmed’ by each expert.
  6. For each pair of consecutive states it holds that the set of statements of each state can be ‘equal’ or ‘can show ‘differences’.
  7. A ‘difference’ between sets of statements can be interpreted as pointing to a ‘change in the real environment’.[5]
  8. Observed differences can be described by special statements called ‘change statements’ or simply ‘rules’.
  9. A change statement has the format ‘IF a set of statements ST* is a subset of the statements ST of a given state S, THEN with probability p, a set of statements ST+ will be added to the actual state S and a set of statements ST- will be removed from the statements ST of a given state S. This will result in a new succeeding state S* with the representing statements ST – (ST-) + (ST+) depending from the assumed probability p.
  10. The list of change statements is an ‘open set’ according to the assumption, that an actual state is only a ‘subset’ of the real environment.
  11. Until now we have an assumed state S, an assumed goal V, and an open set of change statements X.
  12. Applying change statements to a given state S will generate a new state S*. Thus the application of a subset X’ of the open set of change statements X onto a given state S will here be called ‘generating a new state by a procedure’. Such a state-generating-procedure can be understood as an ‘inference’ (like in logic) oder as a ‘simulation’ (like in engineering).[6]
  13. To write this in a more condensed format we can introduce some signs —– S,V ⊩ ∑ X S‘ —– saying: If I have some state S and a goal V then the simulator will according to the change statements X generate a new state S’. In such a setting the newly generated state S’ can be understood as a ‘theorem’ which has been derived from the set of statements in the state S which are assumed to be ‘true’. And because the derived new state is assumed to happen in some ‘future’ ‘after’ the ‘actual state S’ this derived state can also be understood as a ‘forecast’.
  14. Because the experts can change all the time all parts ‘at will’ such a ‘natural empirical theory’ is an ‘open entity’ living in an ongoing ‘communication process’.
Second Lessons Learned

It is interestingly to know that from the set of statements in state S, which are assumed to be empirically true, together with some change statements X, whose proposed changes are also assumed to be ‘true’, and which have some probability P in the domain [0,1], one can forecast a set of statements in the state S* which shall be true, with a certainty being dependent from the preceding probability P and the overall uncertainty of the whole natural environment.

Confirmation – Non-Confirmation

A Theory with Forecasts

Having reached the formulation of an ordinary empirical theory T with the ingredients <S,V,X,⊩ > and the derivation concept S,V ⊩ ∑ X S‘ it is possible to generate theorems as forecasts. A forecast here is not a single statement st* but a whole state S* consisting of a finite set of statements ST* which ‘designate’ according to the ‘agreed meaning’ a set of ‘intended properties’ which need a set of ‘occurring empirical properties’ which can be observed by the experts. These observations are usually associated with ‘agreed procedures of measurement’, which generate as results ‘observation statements’/ ‘measurement statements’.

Within Time

Experts which are cooperating by ‘building’ an ordinary empirical theory are themselves part of a process in time. Thus making observations in the time-window (t1,t2) they have a state S describing some aspects of the world at ‘that time’ (t1,t2). When they then derive a forecast S* with their theory this forecast describes — with some probability P — a ‘possible state of the natural environment’ which is assumed to happen in the ‘future’. The precision of the predicted time when the forecasted statements in S* should happen depends from the assumptions in S.

To ‘check’ the ‘validity’ of such a forecast it is necessary that the overall natural process reaches a ‘point in time’ — or a time window — indicated by the used ‘time model’, where the ‘actual point in time’ is measured by an agreed time machine (mechanical clock). Because there is no observable time without a time machine the classification of a certain situation S* being ‘now’ at the predicted point of time depends completely from the used time machine.[7]

Given this the following can happen: According to the used theory a certain set of statements ST* is predicted to be ‘true’ — with some probability — either ‘at some time in the future’ or in the time-window (t1,t2) or at a certain point in time t*.

Validating Forecasts

If one of these cases would ‘happen’ then the experts would have the statements ST* of their forecast and a real situation in their natural environment which enables observations ‘Obs’ which are ‘translated’ into appropriate ‘observation statements’ STObs. The experts with their predicted statements ST* know a learned agreed meaning M* of their predicted statements ST* as intended-properties M* of ST*. The experts have also learned how they relate the intended meaning M* to the meaning MObs from the observation statements STobs. If the observed meaning MObs ‘agrees sufficiently well’ with the intended meaning M* then the experts would agree in a statement, that the intended meaning M* is ‘fulfilled’/ ‘satisfied’/ ‘confirmed’ by the observed meaning MObs. If not then it would stated that it is ‘not fulfilled’/ ‘not satisfied’/ ‘not confirmed’.

The ‘sufficient fulfillment’ of the intended meaning M* of a set of statements ST* is usually translated in a statement like “The statements ST* are ‘true'”. In the case of ‘no fulfillment’ it is unclear: this can be interpreted as ‘being false’ or as ‘being unclear’: No clear case of ‘being true’ and no clear case of ‘being false’.

Forecasting the Number of Citizens

In the used simple example we have the MKK county with an observed number of citizens in 2018 with 418950. The simple theory used a change statement with a growth factor of 0.4% per year. This resulted in the forecast with the number 420.625 citizens for the year 2019.

If the newly counting of the number of citizens in the years 2019 would yield 420.625, then there would be a perfect match, which could be interpreted as a ‘confirmation’ saying that the forecasted statement and the observed statement are ‘equal’ and therefore the theory seems to match the natural environment through the time. One could even say that the theory is ‘true for the observed time’. Nothing would follow from this for the unknown future. Thus the ‘truth’ of the theory is not an ‘absolute’ truth but a truth ‘within defined limits’.

We know from experience that in the case of forecasting numbers of citizens for some region — here a county — it is usually not so clear as it has been shown in this example.

This begins with the process of counting. Because it is very expensive to count the citizens of all cities of a county this happens only about every 20 years. In between the statistical office is applying the method of ‘forecasting projection’.[9] The state statistical office collects every year ‘electronically’ the numbers of ‘birth’, ‘death’, ‘outflow’, and ‘inflow’ from the individual cities and modifies with these numbers the last real census. In the case of the state of Hessen this was the year 2011. The next census in Germany will happen May 2022.[10] For such a census the data will be collected directly from the registration offices from the cities supported by a control survey of 10% of the population.

Because there are data from the statistical office of the state of Hessen for June 2021 [8:p.9] with saying that the MKK county had 421 936 citizens at 30. June 2021 we can compare this number with the theory forecast for the year 2021 with 423 997. This shows a difference in the numbers. The theory forecast is ‘higher’ than the observed forecast. What does this mean?

Purely arithmetically the forecast is ‘wrong’. The responsible growth factor is too large. If one would ‘adjust’ it in a simplified linear way to ‘0.24%’ then the theory could get a forecast for 2021 with 421 973 (observed: 421 936), but then the forecast for 2019 would be 419 955 (instead of 420 625).

This shows at least the following aspects:

  1. The empirical observations as such can vary ‘a little bit’. One had to clarify which degree of ‘variance’ is due to the method of measurement and therefore this variance should be taken into account for the evaluation of a theoretical forecast.
  2. As mentioned by the statistical office [9] there are four ‘factors’ which influence the final number of citizens in a region: ‘birth’, ‘death’, ‘outflow’, and ‘inflow’. These factors can change in time. Under ‘normal conditions’ the birth-rate and the death-rate are rather ‘stable’, but in case of an epidemic situation or even war this can change a lot. Outflow and inflow are very dynamic depending from many factors. Thus this can influence the growth factor a lot and these factors are difficult to forecast.
Third lessons Learned

Evaluating the ‘relatedness’ of some forecast F of an empirical theory T to the observations O in a given real natural environment is not a ‘clear-cut’ case. The ‘precision’ of such a relatedness depends from many factors where each of these factors has some ‘fuzziness’. Nevertheless as experience shows it can work in a limited way. And, this ‘limited way’ is the maximum we can get. The most helpful contribution of an ‘ordinary empirical theory’ seems to be the forecast of ‘What will happen if we have a certain set of assumptions’. Using such a forecast in the process of the experts this can help to improve to get some ‘informed guesses’ for planning.

Forecast

The next post will show, how this concept of an ordinary empirical theory can be used by applying the oksimo paradigm to a concrete case. See HERE.

Comments

[1] Cities of the MKK-county: 24, see: https://www.wegweiser-kommune.de/kommunen/main-kinzig-kreis-lk

[2] Forecast for development of the number of citizens in the MMK starting with 2018, See: the https://statistik.hessen.de/zahlen-fakten/bevoelkerung-gebiet-haushalte-familien/bevoelkerung/tabellen

[3] Karl Popper, „A World of Propensities“,(1988) and „Towards an Evolutionary Theory of Knowledge“, (1989) in: Karl Popper, „A World of Propensities“, Thoemmes Press, Bristol, (1990, repr. 1995)

[4] Karl Popper, „All Life is Problem Solving“, original a lecture 1991 in German, the first tome published (in German) „Alles Leben ist Problemlösen“ (1994), then in the book „All Life is Problem Solving“, 1999, Routledge, Taylor & Francis Group, London – New York

[5] This points to the concept of ‘propensity’ which the late Popper has discussed in the papers [3] and [4].

[6] This concept of a ‘generator’ or an ‘inference’ reminds to the general concept of Popper and the main stream philosophy of a logical derivation concept where a ‘set of logical rules’ defines a ‘derivation concept’ which allows the ‘derivation/ inference’ of a statement s* as a ‘theorem’ from an assumed set of statements S assumed to be true.

[7] The clock-based time is in the real world correlated with certain constellations of the real universe, but this — as a whole — is ‘changing’!

[8] Hessisches Statistisches Landesamt, “Die Bevölkerung der hessischen
Gemeinden am 30. Juni 2021. Fortschreibungsergebnisse Basis Zensus 09. Mai 2011″, Okt. 2021, Wiesbaden, URL: https://statistik.hessen.de/sites/statistik.hessen.de/files/AI2_AII_AIII_AV_21-1hj.pdf

[9] Method of the forward projection of the statistical office of the State of Hessen: “Bevölkerung: Die Bevölkerungszahlen sind Fortschreibungsergebnisse, die auf den bei der Zensuszählung 2011
ermittelten Bevölkerungszahlen basieren. Durch Auswertung von elektronisch übermittelten Daten für Geburten und Sterbefälle durch die Standesämter, sowie der Zu- und Fortzüge der Meldebehörden, werden diese nach einer bundeseinheitlichen Fortschreibungsmethode festgestellt. Die Zuordnung der Personen zur Bevölkerung einer Gemeinde erfolgt nach dem Hauptwohnungsprinzip (Bevölkerung am Ort der alleinigen oder der Hauptwohnung).”([8:p.2]

[10] Statistical Office state of Hessen, Next census 2022: https://statistik.hessen.de/zahlen-fakten/zensus/zensus-2022/zensus-2022-kurz-erklaert

POPPER – Objective Knowledge (1971). Summary, Comments, how to develope further


eJournal: uffmm.org
ISSN 2567-6458, 07.March 22 – 12.March 2022, 10:55h
Email: info@uffmm.org
Author: Gerd Doeben-Henisch
Email: gerd@doeben-henisch.de

BLOG-CONTEXT

This post is part of the Philosophy of Science theme which is part of the uffmm blog.

PREFACE

In this post a short summary of Poppers view of an empirical theory is outlined as he describes it in his article “Conjectural Knowledge: My Solution of the Problem of Induction” from 1971.[1] The view of Popper will be commented and the relationsship to the oksimo paradigm of the author will be outlined.

Empirical Theory according to Popper in a Nutshell

Figure: Popper’s concept from 1971 of an empirical theory, compressed in a nutshell. Graphic by Gerd Doeben-Henisch based on the article using Popper’s summarizing ideas on the pages 29-31

POPPER’S POSITION 1971

In this article from 1971 Popper discusses several positions. Finally he offers the following ‘demarcation’ between only two cases: ‘Pseudo Science’ and ‘Empirical Science’.(See p.29) In doing so this triggers the question how it is possible to declare something as an ‘objective empirical theory’ without claiming to have some ‘absolute truth’?

Although Popper denies to have some kind of absolute truth he will “not give up the search for truth”, which finally leads to a “true explanatory theory”.(cf. p.29) “Truth” plays the “role of a regulative idea”.(cf. p.30) Thus according to Popper one can “guess for truth” and some of the hypotheses “may well be true”.(cf.p.30)

In Popper’s view finally ‘observation’ shows up as that behaviour which enables the production of ‘statements’ as the ’empirical basis’ for all arguments.(cf.p.30) Empirical statements are a ‘function of the used language’.(cf. p.31)

This dimension of language leads Popper to the concept of ‘deductive logic’ which describes formal mechanisms to derive from a set of statements — which are assumed to be true — those statements, which are ‘true’ by logical deduction only. If statements are ‘logically false’ then this can be used to classify the set of assumed statements as ‘logically not consistent’. (cf. p.31)

comments on popper’s 1971-position 50 years later

The preceding outline of Popper’s position reveals a minimalist account of the ingredients of an ‘objective empirical theory’. But we as the readers of these ideas are living 50 years later. Our minds are shaped differently. The author of this text thinks that Popper is basically ‘true’, although there are some points in Popper’s argument, which deserve some comments.

Subjective – Absolute

Popper is moving between two boundaries: One boundary is the so called ‘subjective believe’ which can support any idea, and which thereby can include pure nonsense; the other boundary is ‘absolute truth’, which is requiring to hold all the time at all places although the ‘known world’ is evidently showing a steady change.

Empirical Basis

In searching for a possible position between these boundaries, which would allow a minimum of ‘rationality’, he is looking for an ’empirical Basis’ as a point of reference for a ‘rational theory’. He is locating such an empirical basis in ‘observation statements’ which can be used for ‘testing a theory’.

In his view a ‘rational empirical theory’ has to have a ‘set of statements’ (often called ‘assumptions’ of the theory or ‘axioms’) which are assumed to ‘describe the observable world’ in a way that these statements should be able to be ‘confirmed’ or be ‘falsified’.

Confirmation – Falsification

A ‘confirmation’ does not imply that the confirmed statement is ‘absolutely true’ (his basic conviction); but one can experience that a confirmed statement can function as a ‘hypothesis/ conjecture’ which ‘workes in the actual observation’. This does not exclude that it perhaps will not work in a future test. The pragmatical difference between ‘interesting conjectures’ and those which are of less interest is that a ‘repeated confirmation’ increases the ‘probability’, that such a confirmation can happen again. An ‘increasing probability’ can induce an ‘increased expectation’. Nevertheless, increased probabilities and associated increased expectations are no substitutes for ‘truth’.

A test which shows ‘no confirmation’ for a logically derived statement from the theory is difficult to interpret:

Case (i): A theory is claiming that a statement S refers to a proposition A to be ‘true in a certain experiment’, but in the real experiment the observation reveals a proposition B which translates to non-A which can interpreted as ‘the opposite to A is being the case’ (= being ‘true’). This outcome will be interpreted in the way that the proposition B interpreted as ‘non-A’ contradicts ‘A’ and this will be interpreted further in the way, that the statement S of the theory represents a partial contradiction to the observable world.

Case (ii): A theory is claiming that a statement S refers to a proposition A to be ‘true in a certain experiment’, but in the real experiment the observation reveals a proposition B ‘being the case’ (= being ‘true’) which shows a different proposition. And this outcome cannot be related to the proposition ‘A’ which is forecasted by the theory. If the statement ‘can not be interpreted sufficiently well’ then the situation is neither ‘true’ nor ‘false’; it is ‘undefined’.

Discussion: Case (ii) reveals that there exist an observable (empirical) fact which is not related to a certain ‘logically derived’ statement with proposition A. There can be many circumstances why the observation did not generate the ‘expected proposition A’. If one would assume that the observation is related to an ‘agreed process of generating an outcome M’, which can be ‘repeated at will’ from ‘everybody’, then the observed fact of a ‘proposition B distinguished from proposition A’ could be interpreted in the way, that the expectation of the theory cannot be reproduced with the agreed procedure M. This lets the question open, whether there could eventually exist another procedure M’ producing an outcome ‘A’. This case is for the actors which are running the procedure M with regard to the logically derived statement S talking about proposition A ‘unclear’, ‘not defined’, a ‘non-confirmation’. Otherwise it is at the same time no confirmation either.

Discussion: Case (i) seems — at a first glance — to be more ‘clear’ in its interpretation. Assuming here too that the observation is associated with an agreed procedure M producing the proposition B which can be interpreted as non-A (B = non-A). If everybody accepts this ‘classification’ of B as ‘non-A’, then by ‘purely logical reasons’ (depending from the assumed concept of logic !) ‘non-A’ contradicts ‘A’. But in the ‘real world’ with ‘real observations’ things are usually not as ‘clear-cut’ as a theory may assume. The observable outcome B of an agreed procedure M can show a broad spectrum of ‘similarities’ with proposition A varying between 100% and less. Even if one repeats the agreed procedure M several times it can show a ‘sequence of propositions <B1, B2, …, Bn>’ which all are not exactly 100% similar to proposition A. To speak in such a case (the normal case!), of a logical contradiction it is difficult if not impossible. The idea of Popper-1971 with a possible ‘falsification’ of a theory would then become difficult to interpret. A possible remedy for this situation could be to modify a theory in the way that a theory does forecast only statements with a proposition A which is represented as a ‘field of possible instances A = <a1, a2, …, am>’, where every ‘ai‘ represents some kind of a variation. In that modified case it would be ‘more probable’ to judge a non-confirmation between A as <a1, a2, …, am> and B as <B1, B2, …, Bn>, if one would take into account the ‘variability’ of a proposition.[3]

Having discussed the case of ‘non-confirmation’ in the described modified way this leads back again to the case of ‘confirmation’: The ‘fuzziness’ of observable facts even in the context of agreed procedures M of observation, which are repeatable by everyone (usually called measurement) requires for a broader concept of ‘similarity’ between ‘derived propositions’ and ‘observed propositions’. This is since long a hot debated point in the philosophy of science (see e.g. [4]). Until now does no general accepted solution exist for this problem.

Thus the clear idea of Popper to associate a theory candidate with a minimum of rationality by relating the theory in an agreed way to empirical observations becomes in the ‘dust of reality’ a difficult case. It is interesting that the ‘late Popper’ (1988-1991) has modified his view onto this subject a little bit more into the direction of the interpretation of observable events (cf. [5])

Logic as an Organon

In the discussion of the possible confirmation or falsification of a theory Popper uses two different perspectives: (i) in a more broader sense he is talking about the ‘process of justification’ of the theoretical statements with regard to an empirical basis relying on the ‘regulative idea of truth’, and (ii) in a more specialized sense he is talking about ‘deductive logic as an organon of criticism’. These two perspectives demand for more clarification.

While the meaning of the concept ‘theory’ is rather vague (statements, which have to be confirmed or falsified with respect to observational statements), the concept ‘deductive logic as an organon’ isn’t really clearer.

Until today we have two big paradigms of logic: (i) the ‘classical logic’ inspired by Aristotle (with many variants) and (ii) ‘modern formal logic’ (cf. [6]) in combination with modern mathematics (cf. [7],[8]). Both paradigms represent a whole universe of different variants, whose combinations into concrete formal empirical theories shows more than one paradigm.(cf. [4], [8], [10])

As outlined in the figure above the principal idea of logic in general follows the following schema: one has a set of expressions of some language L for which one assumes at least, that these expressions are classified as ‘true expressions’. According to an agreed procedure of ‘derivation’ one can derive (deduce, infer, …) other expressions of the language which are assumed to be classified as ‘true’ if the assumptions hold.[11]

The important point here is, that the modern concept of logic does not explain, what ‘true’ means nor exists there an explanation, how exactly a procedure looks like which enables the classification of an expression as ‘being true’. Logic works with the minimalist assumption that the ‘user of logic’ is using statements which he assumes to be ‘true’ independent of how this classification came into being. This frees the user of logic to deal with the cumbersome process of clarifying the meaning and the existence of something which makes a statement ‘true’, but on the other side the user of modern logic has no real control whether his ‘concept of derivation’ makes any sense in a real world, from which observation statements are generated claiming to be ’empirically true’, and that the relationships between these observational statements are appropriately ‘represented’ by the formal derivation concept. Until today there exists no ‘meta-theory’ which explains the relationship between the derivation concept of formal logic (there are many such concepts!) and the ‘dynamics of real events’.

Thus, if Popper mentions formal logic as a tool for the handling of assumed true statements of a theory, it is not really clear whether such a formal logical derivation really is appropriate to explain the ‘relationships between assumed true statements’ without knowing, which kind of reality is ‘designated’/ ‘referred to’ by such statements and their relationships between each other.

(Formalized) Theory and Logic

In his paper Popper does not explain too much what he is concretely mean with a (formalized) theory. Today there exist many different proposals of formalized theories for the usage as ’empirical theories’, but there is no commonly agreed final ‘template’ of a ‘formal empirical theory’.

Nevertheless we need some minimal conception to be able to discuss some of the properties of a theory more concretely. I will address this problem in another post accompanied with concrete applications.

COMMENTS

[1] Karl R.Popper, Conjectural Knowledge: My Solution of the Problem of Induction, in: [2], pp.1-31

[2] Karl R.Popper, Objective Knowledge. An Evolutionary Approach, Oxford University Press, London, 1972 (reprint with corrections 1973)

[3] In our everyday use of our ‘normal’ language it is the ‘normal’ case that a statement S like ‘There s a cup on the table’ can be interpreted in many different ways depending which concrete thing (= proposition B of the above examples) called a ‘cup’ or called ‘table’ can be observed.

[4] F. Suppe, Ed., The Structure of Scientific Theories, University of
Illinois Press, Urbana, 2nd edition, 1979.

[5] Gerd Doeben-Henisch, 2022,(SPÄTER) POPPER – WISSENSCHAFT – PHILOSOPHIE – OKSIMO-DISKURSRAUM, in: eJournal: Philosophie Jetzt – Menschenbild, ISSN 2365-5062, 22.-23.Februar 2022,
URL: https://www.cognitiveagent.org/2022/02/22/popper-wissenschaft-philosophie-oksimo-paradigma/

[6] William Kneale and Martha Kneale, The development of logic, Oxford University Press, Oxford, 1962 with several corrections and reprints 1986.

[7] Jean Dieudonnè, Geschichte der Mathematik 1700-1900, Friedrich Viehweg & Sohn, Braunschweig – Wiesbaden, 1985 (From the French edition “Abrégé d’histoire des mathématique 1700-1900, Hermann, Paris, 1978)

[8] Philip J.Davis & Reuben Hersh, The Mathematical Experience, Houghton Mifflin Company, Boston, 1981

[9] Nicolas Bourbaki, Elements of Mathematics. Theory of Sets, Springer-Verlag, Berlin, 1968

[10] Wolfgang Balzer, C.Ulises Moulines, Joseph D.Sneed, An Architectonic for Science. The Structuralist Program,D.Reidel Publ. Company, Dordrecht -Boston – Lancaster – Tokyo, 1987

[11] The usage of the terms ‘expression’, ‘proposition’, and ‘statement’ is in this text as follows: An ‘expression‘ is a string of signs from some alphabet A and which is accepted as ‘well formed expression’ of some language L. A ‘statement‘ is an utterance of some actor using expressions of the language L to talk ‘about’ some ‘experience’ — from the world of bodies or from his consciousness –, which is understood as the ‘meaning‘ of the statement. The relationship between the expressions of the statement and the meaning is located ‘in the actor’ and has been ‘learned’ by interactions with the world and himself. This hypothetical relationship is here called ‘meaning function  φ’. A ‘proposition‘ is (i) the inner construct of the meaning of a statement (here called ‘intended proposition’) and (ii) that part of the experience, which is correlated with the inner construct of the stated meaning (here called ‘occurring proposition’). The special relationship between the intended proposition and the occurring proposition is often expressed as ‘referring to’ or ‘designate’. A statement is called to ‘hold’/ to be ‘true’ or ‘being the case’ if there exists an occurring proposition which is ‘similar enough’ to the intended proposition of the statement. If such an occurring proposition is lacking then the designation of the statement is ‘undefined’ or ‘non confirming’ the expectation.

Follow-up Post

For a follow-up post see here.

OKSIMO MEETS POPPER. Popper’s Position

eJournal: uffmm.org
ISSN 2567-6458, 31.March – 31.March  2021
Email: info@uffmm.org
Author: Gerd Doeben-Henisch
Email: gerd@doeben-henisch.de

CONTEXT

This text is part of a philosophy of science  analysis of the case of the oksimo software (oksimo.com). A specification of the oksimo software from an engineering point of view can be found in four consecutive  posts dedicated to the HMI-Analysis for  this software.

POPPERs POSITION IN THE CHAPTERS 1-17

In my reading of the chapters 1-17 of Popper’s The Logic of Scientific Discovery [1] I see the following three main concepts which are interrelated: (i) the concept of a scientific theory, (ii) the point of view of a meta-theory about scientific theories, and (iii) possible empirical interpretations of scientific theories.

Scientific Theory

A scientific theory is according to Popper a collection of universal statements AX, accompanied by a concept of logical inference , which allows the deduction of a certain theorem t  if one makes  some additional concrete assumptions H.

Example: Theory T1 = <AX1,>

AX1= {Birds can fly}

H1= {Peter is  a bird}

: Peter can fly

Because  there exists a concrete object which is classified as a bird and this concrete bird with the name ‘Peter’ can  fly one can infer that the universal statement could be verified by this concrete bird. But the question remains open whether all observable concrete objects classifiable as birds can fly.

One could continue with observations of several hundreds of concrete birds but according to Popper this would not prove the theory T1 completely true. Such a procedure can only support a numerical universality understood as a conjunction of finitely many observations about concrete birds   like ‘Peter can fly’ & ‘Mary can fly’ & …. &’AH2 can fly’.(cf. p.62)

The only procedure which is applicable to a universal theory according to Popper is to falsify a theory by only one observation like ‘Doxy is a bird’ and ‘Doxy cannot fly’. Then one could construct the following inference:

AX1= {Birds can fly}

H2= {Doxy is  a bird, Doxy cannot fly}

: ‘Doxy can fly’ & ~’Doxy can fly’

If a statement A can be inferred and simultaneously the negation ~A then this is called a logical contradiction:

{AX1, H2}  ‘Doxy can fly’ & ~’Doxy can fly’

In this case the set {AX1, H2} is called inconsistent.

If a set of statements is classified as inconsistent then you can derive from this set everything. In this case you cannot any more distinguish between true or false statements.

Thus while the increase of the number of confirmed observations can only increase the trust in the axioms of a scientific theory T without enabling an absolute proof  a falsification of a theory T can destroy the ability  of this  theory to distinguish between true and false statements.

Another idea associated with this structure of a scientific theory is that the universal statements using universal concepts are strictly speaking speculative ideas which deserve some faith that these concepts will be provable every time one will try  it.(cf. p.33, 63)

Meta Theory, Logic of Scientific Discovery, Philosophy of Science

Talking about scientific theories has at least two aspects: scientific theories as objects and those who talk about these objects.

Those who talk about are usually Philosophers of Science which are only a special kind of Philosophers, e.g. a person  like Popper.

Reading the text of Popper one can identify the following elements which seem to be important to describe scientific theories in a more broader framework:

A scientific theory from a point of  view of Philosophy of Science represents a structure like the following one (minimal version):

MT=<S, A[μ], E, L, AX, , ET, E+, E-, true, false, contradiction, inconsistent>

In a shared empirical situation S there are some human actors A as experts producing expressions E of some language L.  Based on their built-in adaptive meaning function μ the human actors A can relate  properties of the situation S with expressions E of L.  Those expressions E which are considered to be observable and classified to be true are called true expressions E+, others are called false expressions  E-. Both sets of expressions are true subsets of E: E+ ⊂ E  and E- ⊂ E. Additionally the experts can define some special  set of expressions called axioms  AX which are universal statements which allow the logical derivation of expressions called theorems of the theory T  ET which are called logically true. If one combines the set of axioms AX with some set of empirically true expressions E+ as {AX, E+} then one can logically derive either  only expressions which are logically true and as well empirically true, or one can derive logically true expressions which are empirically true and empirically false at the same time, see the example from the paragraph before:

{AX1, H2}  ‘Doxy can fly’ & ~’Doxy can fly’

Such a case of a logically derived contradiction A and ~A tells about the set of axioms AX unified with the empirical true expressions  that this unified set  confronted with the known true empirical expressions is becoming inconsistent: the axioms AX unified with true empirical expressions  can not  distinguish between true and false expressions.

Popper gives some general requirements for the axioms of a theory (cf. p.71):

  1. Axioms must be free from contradiction.
  2. The axioms  must be independent , i.e . they must not contain any axiom deducible from the remaining axioms.
  3. The axioms should be sufficient for the deduction of all statements belonging to the theory which is to be axiomatized.

While the requirements (1) and (2) are purely logical and can be proved directly is the requirement (3) different: to know whether the theory covers all statements which are intended by the experts as the subject area is presupposing that all aspects of an empirical environment are already know. In the case of true empirical theories this seems not to be plausible. Rather we have to assume an open process which generates some hypothetical universal expressions which ideally will not be falsified but if so, then the theory has to be adapted to the new insights.

Empirical Interpretation(s)

Popper assumes that the universal statements  of scientific theories   are linguistic representations, and this means  they are systems of signs or symbols. (cf. p.60) Expressions as such have no meaning.  Meaning comes into play only if the human actors are using their built-in meaning function and set up a coordinated meaning function which allows all participating experts to map properties of the empirical situation S into the used expressions as E+ (expressions classified as being actually true),  or E- (expressions classified as being actually false) or AX (expressions having an abstract meaning space which can become true or false depending from the activated meaning function).

Examples:

  1. Two human actors in a situation S agree about the  fact, that there is ‘something’ which  they classify as a ‘bird’. Thus someone could say ‘There is something which is a bird’ or ‘There is  some bird’ or ‘There is a bird’. If there are two somethings which are ‘understood’ as being a bird then they could say ‘There are two birds’ or ‘There is a blue bird’ (If the one has the color ‘blue’) and ‘There is a red bird’ or ‘There are two birds. The one is blue and the other is red’. This shows that human actors can relate their ‘concrete perceptions’ with more abstract  concepts and can map these concepts into expressions. According to Popper in this way ‘bottom-up’ only numerical universal concepts can be constructed. But logically there are only two cases: concrete (one) or abstract (more than one).  To say that there is a ‘something’ or to say there is a ‘bird’ establishes a general concept which is independent from the number of its possible instances.
  2. These concrete somethings each classified as a ‘bird’ can ‘move’ from one position to another by ‘walking’ or by ‘flying’. While ‘walking’ they are changing the position connected to the ‘ground’ while during ‘flying’ they ‘go up in the air’.  If a human actor throws a stone up in the air the stone will come back to the ground. A bird which is going up in the air can stay there and move around in the air for a long while. Thus ‘flying’ is different to ‘throwing something’ up in the air.
  3. The  expression ‘A bird can fly’ understood as an expression which can be connected to the daily experience of bird-objects moving around in the air can be empirically interpreted, but only if there exists such a mapping called meaning function. Without a meaning function the expression ‘A bird can fly’ has no meaning as such.
  4. To use other expressions like ‘X can fly’ or ‘A bird can Y’ or ‘Y(X)’  they have the same fate: without a meaning function they have no meaning, but associated with a meaning function they can be interpreted. For instance saying the the form of the expression ‘Y(X)’ shall be interpreted as ‘Predicate(Object)’ and that a possible ‘instance’ for a predicate could be ‘Can Fly’ and for an object ‘a bird’ then we could get ‘Can Fly(a Bird)’ translated as ‘The object ‘a Bird’ has the property ‘can fly” or shortly ‘A Bird can fly’. This usually would be used as a possible candidate for the daily meaning function which relates this expression to those somethings which can move up in the air.
Axioms and Empirical Interpretations

The basic idea with a system of axioms AX is — according to Popper —  that the axioms as universal expressions represent  a system of equations where  the  general terms   should be able to be substituted by certain values. The set of admissible values is different from the set of  inadmissible values. The relation between those values which can be substituted for the terms  is called satisfaction: the values satisfy the terms with regard to the relations! And Popper introduces the term ‘model‘ for that set of admissible terms which can satisfy the equations.(cf. p.72f)

But Popper has difficulties with an axiomatic system interpreted as a system of equations  since it cannot be refuted by the falsification of its consequences ; for these too must be analytic.(cf. p.73) His main problem with axioms is,  that “the concepts which are to be used in the axiomatic system should be universal names, which cannot be defined by empirical indications, pointing, etc . They can be defined if at all only explicitly, with the help of other universal names; otherwise they can only be left undefined. That some universal names should remain undefined is therefore quite unavoidable; and herein lies the difficulty…” (p.74)

On the other hand Popper knows that “…it is usually possible for the primitive concepts of an axiomatic system such as geometry to be correlated with, or interpreted by, the concepts of another system , e.g . physics …. In such cases it may be possible to define the fundamental concepts of the new system with the help of concepts which were originally used in some of the old systems .”(p.75)

But the translation of the expressions of one system (geometry) in the expressions of another system (physics) does not necessarily solve his problem of the non-empirical character of universal terms. Especially physics is using also universal or abstract terms which as such have no meaning. To verify or falsify physical theories one has to show how the abstract terms of physics can be related to observable matters which can be decided to be true or not.

Thus the argument goes back to the primary problem of Popper that universal names cannot not be directly be interpreted in an empirically decidable way.

As the preceding examples (1) – (4) do show for human actors it is no principal problem to relate any kind of abstract expressions to some concrete real matters. The solution to the problem is given by the fact that expressions E  of some language L never will be used in isolation! The usage of expressions is always connected to human actors using expressions as part of a language L which consists  together with the set of possible expressions E also with the built-in meaning function μ which can map expressions into internal structures IS which are related to perceptions of the surrounding empirical situation S. Although these internal structures are processed internally in highly complex manners and  are — as we know today — no 1-to-1 mappings of the surrounding empirical situation S, they are related to S and therefore every kind of expressions — even those with so-called abstract or universal concepts — can be mapped into something real if the human actors agree about such mappings!

Example:

Lets us have a look to another  example.

If we take the system of axioms AX as the following schema:  AX= {a+b=c}. This schema as such has no clear meaning. But if the experts interpret it as an operation ‘+’ with some arguments as part of a math theory then one can construct a simple (partial) model m  as follows: m={<1,2,3>, <2,3,5>}. The values are again given as  a set of symbols which as such must not ave a meaning but in common usage they will be interpreted as sets of numbers   which can satisfy the general concept of the equation.  In this secondary interpretation m is becoming  a logically true (partial) model for the axiom Ax, whose empirical meaning is still unclear.

It is conceivable that one is using this formalism to describe empirical facts like the description of a group of humans collecting some objects. Different people are bringing  objects; the individual contributions will be  reported on a sheet of paper and at the same time they put their objects in some box. Sometimes someone is looking to the box and he will count the objects of the box. If it has been noted that A brought 1 egg and B brought 2 eggs then there should according to the theory be 3 eggs in the box. But perhaps only 2 could be found. Then there would be a difference between the logically derived forecast of the theory 1+2 = 3  and the empirically measured value 1+2 = 2. If one would  define all examples of measurement a+b=c’ as contradiction in that case where we assume a+b=c as theoretically given and c’ ≠ c, then we would have with  ‘1+2 = 3′ & ~’1+2 = 3’ a logically derived contradiction which leads to the inconsistency of the assumed system. But in reality the usual reaction of the counting person would not be to declare the system inconsistent but rather to suggest that some unknown actor has taken against the agreed rules one egg from the box. To prove his suggestion he had to find this unknown actor and to show that he has taken the egg … perhaps not a simple task … But what will the next authority do: will the authority belief  the suggestion of the counting person or will the authority blame the counter that eventually he himself has taken the missing egg? But would this make sense? Why should the counter write the notes how many eggs have been delivered to make a difference visible? …

Thus to interpret some abstract expression with regard to some observable reality is not a principal problem, but it can eventually be unsolvable by purely practical reasons, leaving questions of empirical soundness open.

SOURCES

[1] Karl Popper, The Logic of Scientific Discovery, First published 1935 in German as Logik der Forschung, then 1959 in English by  Basic Books, New York (more editions have been published  later; I am using the eBook version of Routledge (2002))