Philosophy of Science: Explanation Models

Wetenschapsfilosofie II: Scientific Explanation

• Focus on Hempel and Kitcher.
• Theories of explanations and explanatory models from Hempel and Kitcher, as well as providing a toolbox to describe and evaluate explanatory practices will be examined. A pragmatic approach to scientific explanation will also be presented.

Carl Hempel and Philosophy of Explanation

• Carl Hempel is considered a key figure in the philosophy of explanation.
• Hempel articulated precisely an idea which had received a hazy formulation from traditional empiricists such as Hume and Mill.

Key Concepts

• "Explain" appears as a binary relation as, the 'explanans' explains the 'explanandum'.
• Explanandum: The phenomenon one seeks to explain.
• Explanans: The statements that provide the explanation for the explanandum.

Hempel's Deductive-Nomological (DN) Model

• To successfully explain an explanandum, certain conditions must be met.
• The explanandum must be a logical consequence of the explanans - deductive.
• The sentences forming the explanans must be true.
• The explanans must contain at least one natural law, which must form a necessary premise in the logical derivation - nomological.
• According to Hempel, explanation begins with derivation, either deductively or inductively, of a sentence that describes the phenomenon being explained (explanandum) from a set of sentences (explanans) containing at least one general law.
• The covering law model has fallen on hard times.
• The ability to derive a description of a phenomenon from premises containing a law seems quite tangential to our understanding of the phenomenon.

Hempel's DN Model / Covering Law Model

• An ordered pair (L, C) constitutes a potential explanation for a singular sentence E, if and only if:
• L is a purely descriptive sentence.
• C is a singular sentence.
• E is deductively derivable from the conjunction of L and C.
• E is not deductively derivable from C alone.
• Difficulties stem from the fact that when viewed as providing a set of necessary and sufficient conditions for explanation, it is far too liberal.
• The model is quite powerless to adjudicate the more subtle considerations about explanatory adequacy which are the focus of scientific debate.

The Simplest Illustration

• The simplest object is a mirror, and the law for a mirror is that when the light hits the mirror, it does not continue in a straight line, but bounces off the mirror into a new straight line. The light striking a mirror travels in such a way that the two angles, between each beam and the mirror, are equal.

DN-Model example

• Explanans is that the reflected light beam 'a' has an angle of 45 degrees relative to the mirror from where the light is reflected.
• The condition is that the angle of incidence of 'a' relative to the mirror is 45 degrees.
• The law states for all light beams reflecting off a mirror, if the angle of incidence of 'a' relative to the mirror is 45 degrees, then the reflected beam also has an angle of 45 degrees relative to the mirror.

Symbolic Representation of DN Model

• C represents the condition as in the angle of incidence of a relative to the mirror is 45 degrees, which is symbolically represented as Pa
• L represents the Law as in 'for all light beams that reflects off a mirror: if the angle of incidence to the mirror is 45, then the reflected beam also has an angle of 45' which is expressed as (√x)(Px → Qx)
• E represents the explanandum as in reflected light 'a' has an angle of 45 to the mirror of the light where it is reflected which is expressed as Qa

True vs Well-Confirmed Explanans

• (L,C) is a true explanans for E if and only if (L,C) is a potential explanans for E and both L and C are true.
• (L,C) is a well-confirmed explanans for E if and only if (L,C) is a potential explanans for E and both L and C are well-confirmed.

Aluminum Rod Expansion Example

• Utilizes DN model with the law that dL = 0.0000222 × Lo × dt, where dL is the expansion in meters, Lo is the initial length in meters, and dt is the temperature difference in Celsius.
• Initial conditions specify an aluminum rod with an original length of 1 meter at 50°C, which is then heated to 250°C, resulting in a final length of 1.0044 meters.

Aluminium Rod Expansion Example with symbolic expression

• C1: This aluminum rod was heated from 50 to 250 °C is expressed as P₁α
• C2: This aluminum rod has an initial length (at 50 °C) of 1 m is expressed as P2α
• L: All aluminum rods are heated from 50 to 250 °C and their initial length is 1 m, then they are 4.44 mm longer at 250 °C is expressed as (√x)(P₁x (√x)(P₁x P2x^･･ Pnx) → Qx)
• E: The aluminium rod is 4,44 mm longer than it was before it was heated is expressed as Da

Important point

• (L, C) constitutes a potential explanation for a singular sentence E, if and only if L is a purely descriptive sentence, C is a singular sentence, E is deductively derivable from the conjunction of L and C, and E is not deductively derivable from C alone.

Circular Reasoning Avoidance

• The DN model excludes circular reasoning.
• Circular reasoning example: the substance has a pH value >7, therefore substance has a pH >7.

Aim of DN Explanation

• A D-N explanation answers the question of why the explanandum-phenomenon occurred.
• It demonstrates that the phenomenon resulted from certain particular circumstances, specified in C1, C2,...,Ck, in accordance with the laws L1, L2,...,Lr
• Given particular circumstances and the laws in question, the occurrence of the phenomenon was to be expected, it enables understanding of why the phenomenon occurred

Explanation - Understanding

• A D-N explanation offers a formal way to show that a phenomenon logically comes from general laws and specific circumstances.
• A D-N explanation describes not only what happened, but shows that it was inevitable, given the circumstances and laws.
• Answering "why?" involves demonstrating that the phenomenon necessarily follows from laws and circumstances, and is thus to be expected.

Hempel's Inductive-Statistical (IS) Model

• DN explanation uses deterministic natural laws, IS explanation uses statistical laws.
• DN explanation is deductive, IS explanation is inductive.
• DN explanation results in necessary expectation, IS explanation results in degrees of expectation.

IS-Model

• Explains why patient P recovers from disease Z after taking medicine M
• The DN model entails that by taking medicine, one always recovers
• The IS model suggests medicine only increases the likelihood of recovery; for example, there is an 80% chance of recovery, but it does not guarantee it

IS-model Example

• L: Prob (G|F) = r: The probability of someone recovering, given they take the medicine, is 80%.
• C: Fp: Patient p has taken the medicine.
• E: Gp: Patient p is recovered.
• r = degree of inductive support that the premises offer.

Circular reasoning Exclusions in the IS Model

• L: Prob (G|G) = 1
• C: Gb
• E: Gb

Additional points on the is Model

• HPR: High probability requirement, where r > ε and ε ≥ 0.5
• Example: 2 % of inhabits of Sweden are Roman Catholic, Petersen is a Swede, and Petersen is Roman Catholic

Ambiguity in IS Model

• L1: The chance that someone over 65 dies after a heart attack is 80%.
• L2: The chance that someone in excellent physical condition survives a heart attack is 90%.
• Problem: Person b is 65+ and in excellent physical condition; which law should be applied, given contradicting survival probabilities (20% vs. 90%) based on L1 and L2, respectively.
• This ambiguity is an IS explanation.

Resolution: Requirement of Maximal Specificity (RMS)

• If K is the set of all statements accepted at the given time, let k be a sentence that is logically equivalent to K.
• To be rationally acceptable in the knowledge situation represented by K, the proposed explanation must meet the following condition:
• If k implies that b belongs to a class F1, and that F1 is a subclass of F, then k must also imply a statement specifying the statistical probability of G in F1, say Prob(G| F₁) = r1.
• Here, r₁ must equal r unless the probability statement just cited is simply a theorem of mathematical probability theory.

Breakdown of RMS

• K is a set of beliefs that are generally accepted on the given time.
• Situation k is rationally acceptable when explaining using the IS model
• Situation k implies the object being explained to a class F1 that's the subclass of F
• Requirement over waarschijnlijkheid explains the likelihood is in class G.
• Prob (G|F₁) = r₁

Further characteristics

• If r₁≠ r implies specification is too vague
• If r₁ = r implies no extra additional specificity.
• r₁ = r = 1 implies mathematical certainty, in that there is no contribution to explanation that offers understanding

IS Model example

• F: Patients with some symptoms F , with Prob (G|F) = 0,7 of recovery
• F1: Patients with some symptoms F and a genetic marker M, with Prob (G|F₁) = 0.9 of recovery
• Then, Wiskundige stelling implies we can set prob G and G or (Prob (G|F&G) = 1
• The above shows it best to go with r=0.9 gebruiken and Wiskundige stelling uitsluiten

Problems with Hempel's Model

• Harry is bald, because all members of the Greenbury School Board of 1964 are bald and Harry is a member of the school board in 1964
• Aluminium expands when heated, because all metals expand when they are heated and aluminum is a metal.

Accidentele generalisatie or Accidental generalization

• Greenhury school board is based on specific observations of specific observations but there are no natural laws, plus generalization is limited in time and context and doesn't explain why Harry specifically is bald
• On the other hand: Metais expand with heat based on physical properties of thermal expansion, is universal for all metals and there are underlying natural laws. The generalization is common where it explains why that aluminium is expanding under these conditions.

Irrelevant premises

• In deductive logic, irrelevant premises don't affect the validity of an argument and in explanation premesis has to be relevant and information can affect the clarity and reduction of effectivity of explantaion.

Asymmetry

• Logical deduction is symmetrical
• There is clarity in intuition about explanations (asymmetry)

Daniel Hausman

• Build on DN model but adds Independent Alterability to address asymmetry and irrelevant problems

Independent Alterability

• Causal Derivations builds in D-N Explanation, where independent alterability addresses the issue of asymmetry and irrelevant premises.
• Hausman: For every pair of variables, X and Y, whose values are specified in a derivation, if the value of X were changed by intervention, then the value of Y would be unchanged

More about causality

• In Hempel's model, explanations are purely arguments.
• In Hausman model causal links are considered
• Hausman: explanations are arguments, but only arguments that go from cause to effect are explanations.

Philip Kitcher

• Is a proponent of the Unification account on explanation
• The unification view can overcome some of the most serious difficulties besetting the covering law model.
• Kitcher's (1976) analysis of unification is a defense of the problems for Friedman's approach.
• It is possible to defend the thesis that historical appeals to the explanatory power of theories involve recognition of a virtue over and beyond considerations of simplicity and predictive power.
• The task is to articulate how E(K) unifies.

Philip Kitcher’s View on Scientific Explanations

• Scientific explanations enhance forecastability/understandability and unificaition
• Unification involves explaining different phenomena in such a way to form a singular common framework to give basic principles.
• Hempel's problems can be solved via unification, rather than causality.
• Embraces an unofficial view of explanation as unification.
• Aims to show that this view can be developed to provide insight into major episodes in the history of science.
• The natural sciences do not merely pile up unrelated items of knowledge of more or less practical significance, but that they increase our understanding of the world, increasing understanding.

Unification

• Scientific explanations are sought through unification by the construction of arguments that derive a part of our knowledge from other parts.
• If K is a set of belief, then each is a set of agreements where the premise and conlusion belong to a systematization of K
• Systematization of K is done using a unification
• Has gradations in intensity
• It is about comprehending a maximum of facts and regularities in terms of a minimum of theoretical concepts and assumptions.
• Achieved by exhibiting phenomena as manifestations of common, underlying structures and processes that conform to specific, testable, basic principles
• Achieved by using similar arguments in the derivation of many accepted sentences.
• Two historical episodes illustrate the desire for unification that played a crucial role:
  • Prior to the articulation of a theory with high predictive power, certain proposals for theory construction are favored on grounds of their explanatory promise
  • The explanatory power of embryonic theories is explicitly tied to the notion of unification.
  • Particular featured of the theories are taken to support these claims to unification.
• Scientists are interested in stringent patterns of argument, patterns which contain some nonlogical expressions and which are fairly similar in terms of logical structure.
• Newton's work demonstrated that one pattern of argument could be used again and again in the derivation of a wide range of accepted sentences.
• The explanatory store over K is that systematization whose basis does best by the criteria of unifying power.
• Is achieved by generating a large number of accepted sentences as the conclusions of acceptable arguments which instantiate a few, stringent patterns.
• Newtonian theory's core pattern involves computing equations of motion, also showing how conclusions generated by instantiating the core pattern derive further conclusions, supplemented by a problem-reducing pattern.

Argument patterns are composed of

• Schematic sentences: non-logical terms replaced by dummy letters (cfr. Formalization in Logica)
• A set of sets of filling instructions: guidelines for replacing the dummy letters
• Classification: inferential characteristics of the series of schematic sentences (what is the premise, what is the conclusion, which logical rules)
• Requires an argument, must have features to serve as the basis of an act of explanation
• Explanations are ordered Paris of a proposition and an act type.
• Science doesn't supply individual arguments, but a reserve of explanatory arguments.
• The most obvious way to categorize explanation is to view it as an activity, answering actual or anticipated questions by presenting reasons.

Example of an argument pattern

• Schematic sentences include a is P, All P's are bold then a is kaal so is Harry.
• Harry. Instructions say A is the name of an individual and to replace all with same instructions
• An argument pattern is a triple consisting of a schematic argument, a set of sets of filling instructions, and a classification for the schematic argument.
• A set of sentences instantiates the general argument pattern just in case it meets the following conditions:
  • The sequence has the same number of terms as the schematic argument
  • Each sentence in the sequence is obtained from its corresponding schematic sentence in accordance with the appropriate set of filling instructions.
  • It is possible to construct a chain of reasoning assigning each sentence the status of its corresponding schematic sentence.
• Newtonian Example:
  • (1) The force on a is 3.
  • (2) The acceleration of a is y.
  • (3) Force = mass-acceleration.
  • (4) (Mass of a)-(y) = ,3
  • (5) 8 = 0
  • This argument allows mathematical assumptions without occurring as terms, and does not give a complete description of the route.
• Argument patterns of scientists results from a compromise in demanding similarity either in logical structure or nonlogical vocabulary.
• Is determined by two constraints: the conditions on the substitution of expressions for dummy letters; and the conditions on the logical structure imposed by the classification, to avoid degeneration.

Factors of Unified Power

• Four factors of unifying power
  • Number the amount of explained phenomena, Schaarste when not all phenomena can be explained using it, stringent when strict pattern is made and similarity when using simpler explanation
• Striktheid: Strengthen the effect or not use that explanation.
• Similarity use that argument if its a good analogy explanation
• The unifying power of a basis varies directly with the stringency of the patterns, and inversely with the number of members of the basis.

Classifications of sentences

• G and G' are offspring of a common ancestor Go as well as being from the same parent
• One can put new instructions as well as put the proper logic so everything makes sence.

Explanation versus non exemplary arguments

• Accidentel Genralisatie: Not a good explanation to base something of one argument that has limited reason why like "all men on Greenbury school council 1964 are blad"
• Irllevante Premissen: Cannot relate to what that reason is then is not a good argument "Every man has pill because they don't want to get pregnant"
• The unifying power of the basis will be less than that of the set that we normally accept as explanatory.

Asymmetry

• No real base explanation like "A mast is tall just cause the mast is 45 degree.
• Comes from deriving something from a specification from the period and same law, seems to get things backward.
• You can explain why a simple pendulum has it's period is does by deriving a specification of its period from a specification of the length and law
• The covering law model fails

Samenvatting and Important points

• Samenvatting 15 point:
• Good patterns good explanation and more force it make the points
• Explain in model has is that make them not be all bad

Barometer and coming storm

• An oefening: is that there always been coming storm where no matter is it reading the barometer
• You do not use arguments that can provide for the best unification of our beliefs

Newtonian Program

• Achieved in dynamics, astronomy, inspired some successor to call dynamic corpuscularianism.
• Principia had shown how to obtain the motions of bodies from a knowledge of forces and possibility of dealing gravity system in a unified way.
• The step was to isolate force laws, and apply these laws, so that all phenomena of nature could be derived.

Darwin's Evolutionary Theory

• Unifies a host of biological phenomena.
• Offers a pattern, not detailed explanations.

Stringency

• If both conditions are relaxed completely then the notion of pattern degenerates so as to admit any argument.
• A Stringent pattern uses theoretical concepts.

Spurious Unification

• If the filling instructions associated with a pattern P could be replaced by different filling instructions, allowing for the substitution of a class of expressions of the same syntactic category, to yield a pattern P’ and if P’ would allow the derivation of any sentence, then the unification achieved by P is spurious.

How scientific explanation yields understanding.

• Minimize the number of types of premises we must take as underived - reduce the number of brute facts.
• Scientists favors patterns, to understand and share.