Reflections on methodology belongs to a serious of propaedeutics in the context of the research dissertation Principles of Liberty by Tabea Hirzel (2015). Reflections on methodology V is concerned about questions of actuality, infinity and certainty, and their relation to ethical questions in European traditions of logic.

1. Refelctions on Methodology V
Actuality, Infinity, Certainty
Doctoral Candidate: Tabea Hirzel
Program: Doctorate of Diplomacy/ Political Economy
University: SMC University, Zug, Switzerland
Date: 12.30.2011
© Tabea Hirzel 1

2. Aristotle & Infinity
Belief in the existence of the infinite comes mainly from five considerations:
1. From the nature of time – for it is infinite.
2. From the division of magnitudes – for the mathematicians also use the notion of the infinite.
3. If coming to be and passing away do not give out, it is only because that from which things come
to be is infinite. (linear thought)
4. Because the limited always finds its limit in something, so that there must be no limit, if everything
is always limited by something different from itself.
5. Most of all, a reason which is peculiarly appropriate and presents the difficulty that is felt by
everybody – not only number but also mathematical magnitudes and what is outside the heaven
are supposed to be infinite because they never give out in our thought. (Aristotle [1])
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3. Cantor’s three realms of infinity
(1) the infinity of God (which he called the "absolutum"),
(2) the infinity of reality (which he called "nature")
(3) the transfinite numbers and sets of mathematics.
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4. Pythagorean and Dharmic logic
• Must the believe in an actual infinity
necessarily imply «transition of souls»?
• Is their a relation between actual infinity
and the transition in logic searched by
Kant? (Pythagoreans  Galileo  modern
physics)
• Influence on scientific revolution
(Leonardo Da Vinci, Kepler, Copernicus)
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5. Plotinian syllogism
First principle*
(⌐p ∧ p/ ⊥/0)
Infinity (-)
(⌐p1)
Potentiality
(Power)
(⌐p2)
Conceilment
(⌐p3) Self-
Identity (Unity)
Actuality (+)
(p1) Act
(p2)
Manifestation
(p3) Rational
logic
* Plotinian descriptions of this first principle were:
Action
Liberty
God
Etc. Personal identity
(actual infinity) (∃⌐p
∧ p)
⌐p1 p1
⌐p2 p2
⌐p3 p3
Objective being
(∃⌐(⌐p) ⇔p)
⌐p1 p1
⌐p2 p2
⌐p3 p3
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6. Plotinian categories of being
First
principle* (0)
Actual
infinity
Subjectivity
Concrete
actuality
Space
Time
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7. ∃
∃
First principle
Actual infinity
Objective world
objects
infinity
actuality
subjects
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8. Threefold infinity Threefold actuality
Manifestation
(substance/
matter)
Act (process)
Logic
(causation)
objects
Potentiality
Conceilment
Self-identity
subjects
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9. ∃
∃
First principle
Actual infinity
Objective world
objects
infinity
actuality
subjects
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10. ∃
∃
First principle
Actual infinity
Objective world
Potentiality
Conceilment
Self-identity
Manifestation
(substance/
matter)
Act (process)
Logic
(causation)
objects
infinity
actuality
subjects
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11. ∃
∃
First principle
Actual infinity
Objective world
Potentiality
God (Plotiniant)
God (Thomist)
Person
(plotinian)
Conceilment
Self-identity
Manifestation
(substance/
matter)
Act (process)
Logic
(causation)
objects
infinity
actuality
subjects
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12. Hoppean ethics
The statement (p) that the other (⌐x) has no property right (?) is a contradiction
(⊥) to the act (?) of argumentation (?).
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13. Bivalent logic (Aristotelian)
Principle (or law) of bivalence:
∀P ∀x (x ∈ P ∨ ∉ P)
For all P that are all x, x is an element of set
P or is not an element of set P.
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14. THREE TRADITIONAL LAWS
Aristotelian Logic
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15. 1. Law of identity (LOI)
«Each thing is the same with itself and different from another»
(Aristotle. Metaphysics, Book VI, Part 4 (c) )
Fallacies:
Informal logical fallacies: equivocation
Formal representation
For any proposition A:A=A
∀ A: A=A
(A)A
For Locke not a priori/ innate (Essay Concerning Human Understanding)
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16. 2. Law of (non)contradiction
(LOC)
Alfred North Whitehead, Bertrand Russell (1910). Principia
Mathematica. Cambridge, p. 105 (*2.24)
«Nothing can both be and not be»
(Russell 1912:72,1997 edition)
Aristotle. Metaphysics, Book IV, Part 4
Formal representation
NOT(A = NOT-A)
﹁(A = ﹁ A)
⊢. ﹁ (p.﹁ p)
For Locke not a priori/ innate (Essay Concerning Human Understanding)
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17. 3. Law of excluded middle
(LOM)
Alfred North Whitehead, Bertrand Russell (1910). Principia
Mathematica. Cambridge, p. 105 (*2.11)
Aristotle, Metaphysics, Book IV, Part 7
Formal representation
For all A: A or ~A
∀ A: A∨﹁A
⊢.p ∨ ﹁ p
«Everything must either be or not be.» (Russel 1912)
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18. TWO ADDITIONAL LAWS
Leibnitz
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19. Principle of suffiecient reason
The principle has a variety of expressions, all of which are perhaps best summarized by the following:
For every entity X, if X exists, then there is a sufficient explanation for why X exists.
For every event E, if E occurs, then there is a sufficient explanation for why E occurs.
For every proposition P, if P is true, then there is a sufficient explanation for why P is true.
A sufficient explanation may be understood either in terms of reasons or causes, for like many
philosophers of the period, Leibniz did not carefully distinguish between the two.
Deductive reasoning
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20. Identity of indiscernibles
Entities x and y are identical if every predicate possessed by x is also possessed by y and vice versa.
A logical or an empirical principal?
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21. Law of continuity
Whatever succeeds for the finite, also succeeds for the infinite.
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22. Transcendental law of
homogeneity (TLH)
If a is finite and dx is infinitesimal, then one sets:
a+dx=a
Similar:
u dv + v du + du dv = u dv + v du
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23. FOUR LAWS
Schopenhauer
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24. Fourfold Root of the Principle of
Sufficient Reason
1. A subject is equal to the sum of its predicates, or a = a.
2. No predicate can be simultaneously attributed and denied to a subject, or a ≠ ~a.
3. Of every two contradictorily opposite predicates one must belong to every subject.
4. Truth is the reference of a judgment to something outside it as its sufficient reason or ground.
The laws of thought can be most intelligibly expressed thus:
1. Everything that is, exists.
2. Nothing can simultaneously be and not be.
3. Each and every thing either is or is not.
4. Of everything that is, it can be found why it is.
There would then have to be added only the fact that once for all in logic the question is about what is
thought and hence about concepts and not about real things.
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25. GENERAL PROBLEMS OF
BEING
Kant
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26. Problems
• The question of actual infinity
• The question of transition
• Synthetic a priorism
• Set of all sets
• Perceptible infinity
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27. Special predicate of being
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28. REASON
Kleene, Hamilton
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29. Domain (Kleene)
Domain (D) is not trivial (Kleene 1967:84)
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30. Hamilton on logic
“Logic is the science of the necessary forms of thought” (Hamilton 1860:17)
(1) “determined or necessitated by the nature of the thinking subject itself . . . it is subjectively, not
objectively, determined;
(2) “original and not acquired;
(3) “universal; that is, it cannot be that it necessitates on some occasions, and does not necessitate on
others.
(4) "it must be a law; for a law is that which applies to all cases without exception, and from which a
deviation is ever, and everywhere, impossible, or, at least, unallowed. . . . This last condition, likewise,
enables us to give the most explicit enunciation of the object-matter of Logic, in saying that Logic is the
science of the Laws of Thought as Thought, or the science of the Formal Laws of Thought, or the
science of the Laws of the Form of Thought; for all these are merely various expressions of the same
thing."
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31. Necessary vs. Contingen
(Hamilton)
The logical significance of this law: The logical significance of the law of Reason and
Consequent lies in this, - That in virtue of it, thought is constituted into a series of acts all
indissolubly connected; each necessarily inferring the other. Thus it is that the distinction and
opposition of possible, actual and necessary matter, which has been introduced into Logic, is a
doctrine wholly extraneous to this science.
Par. XVII. Law of Sufficient Reason, or of Reason and Consequent
The thinking of an object, as actually characterized by positive or by negative attributes,
is not left to the caprice of Understanding – the faculty of thought; but that faculty must
be necessitated to this or that determinate act of thinking by a knowledge of something
different from, and independent of; the process of thinking itself.
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32. PROBLEMS
Russel
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33. Induction principle
He makes an argument that this induction principle can neither be disproved or proved by
experience,[18] the failure of disproof occurring because the law deals withprobability of success rather
than certainty; the failure of proof occurring because of unexamined cases that are yet to be
experienced, i.e. they will occur (or not) in the future. "Thus we must either accept the inductive
principle on the ground of its intrinsic evidence, or forgo all justification of our expectations about the
future".
Russell offers other principles that have this similar property: "which cannot be proved or disproved by
experience, but are used in arguments which start from what is experienced." He asserts that
these "have even greater evidence than the principle of induction . . . the knowledge of them has the
same degree of certainty as the knowledge of the existence of sense-data. They constitute the means
of drawing inferences from what is given in sensation“ (Russell 1912:70, 1997 edition)  see Hoppe
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34. Inference principle
“anything implied by a true proposition is true” (Russell 1912:71, 1997 edition)
 modus ponens
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35. modus ponens
modus ponens vs. law
For example, Alfred Tarski (Tarski 1946:47) distinguishes modus ponens as one of three "rules of
inference" or "rules of proof", and he asserts that these "must not be mistaken for logical laws". The
two other such "rules" are that of "definition" and "substitution"; see the entry under Tarski.
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36. Tarsky’s rules of inference
Modus ponens (demonstration
Definition
Substitution
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37. Intuitionistic logic
or construtive logic
Replaces the traditional concept of truth with the concept of constructive provability.
Constructive proof: In mathematics, a constructive proof is a method of proof that demonstrates the
existence of a mathematical object by creating or providing a method for creating the object. This is in
contrast to a non-constructive proof (also known as an existence proof or pure existence theorem) which
proves the existence of a particular kind of object without providing an example.
Metatheorems:
1. The deduction theorem for first-order logic says that a sentence of the form φ→ψ is provable from
a set of axioms A if and only if the sentence ψ is provable from the system whose axioms consist of φ
and all the axioms of A.
2. Consistency proofs of systems such as Peano arithmetic
Existence proof (pure existence theorem):
Includes:
Axiom of infinity
Axiom of choice
Law of excluded middle
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38. SET THEORY
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39. Zermelo–Fraenkel set theory
(ZFC)
1. Axiom of extensionality
2. Axiom of regularity (also called the Axiom of foundation)
3. Axiom schema of specification (also called the axiom schema of separation
or of restricted comprehension)
4. Axiom of pairing
5. Axiom of union
6. Axiom schema of replacement
7. Axiom of infinity
8. Axiom of power set
9. Well-ordering theorem
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40. General set theory (GST)
(Boolos)
1. Axiom of Extensionality: The sets x and y are the same set if they have the
same members.
3. Axiom Schema of Specification (or Separation or Restricted
Comprehension): If z is a set and ϕ is any property which may be satisfied by
all, some, or no elements of z, then there exists a subset y of z containing just
those elements x in z which satisfy the property ϕ. The restriction to z is
necessary to avoid Russell's paradox and its variants.
5. Axiom of Adjunction: If x and y are sets, then there exists a set w, the
adjunction of x and y, whose members are just y and the members of x.
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41. Urelements
is an object (concrete or abstract) that is not
a set, but that may be an element of a set.
Urelements are sometimes called "atoms" or
"individuals."
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42. Kripke–Platek axioms of set
theory (KP)
1. Axiom of extensionality
2. Axiom of regularity (also called the Axiom of foundation)
4. Axiom of pairing
5. Axiom of union
7. Axiom of infinity
9. Well-ordering theorem
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43. Axiom of Infininty
In the formal language of the Zermelo–Fraenkel axioms, the axiom reads:
∃I=(∅ ∈ I ∧ ∀x ∈ I ((x ∪ {x}) ∈ I))
In words, there is a set I (the set which is postulated to be infinite), such that the empty set is in I and
such that whenever any x is a member of I, the set formed by taking the union of x with its singleton {x}
is also a member of I. Such a set is sometimes called an inductive set.
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44. Universal set (problem)
TWO PROBLEMS
Russell’s paradox
Zermelo–Fraenkel set theory and related set theories, which are based on the idea of the cumulative
hierarchy, do not allow for the existence of a universal set. Its existence would cause paradoxes which
would make the theory inconsistent.
Cantor’s theorem
A second difficulty with the idea of a universal set concerns the power set of the set of all sets.
Because this power set is a set of sets, it would automatically be a subset of the set of all sets,
provided that both exist. However, this conflicts with Cantor's theorem that the power set of any set
(whether infinite or not) always has strictly higher cardinality than the set itself.
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45. Universal set (solution)
Restricted comprehension
There are set theories known to be consistent (if the usual set theory is consistent) in which the
universal set V does exist (and V ∈ V is true). In these theories, Zermelo's axiom of comprehension
does not hold in general, and the axiom of comprehension of naive set theory is restricted in a different
way. A set theory containing a universal set is necessarily a non-well-founded set theory.
Foundation axiom
States that every non-empty set A contains an element that is disjoint from A
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46. Subjective being beyong logic!
• This is what intentionality means
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47. Existentional problem
All argumentation partners are persons
(have/own/need a body)
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48. Proof of contradiction
• reductio ad impossibilem
• a particular kind of the more general form
of argument known as reductio ad
absurdum
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49. ARGUMENT OF
ARGUMENTATION
Hoppe
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50. Induction
While the conclusion of a deductive argument is supposed to be certain, the truth of the conclusion of
an inductive argument is supposed to be probable, based upon the evidence given.
(Copi, I. M., Cohen, C., & Flage, D. E. (2007). Essentials of logic (2nd ed.). Upper Saddle River, NJ:
Pearson Education, Inc.)
Many dictionaries define inductive reasoning as reasoning that derives general principles from specific
observations, though some sources disagree with this usage.
Deductive and Inductive Arguments, Internet Encyclopedia of Philosophy, "Some dictionaries define
"deduction" as reasoning from the general to specific and "induction" as reasoning from the
specific to the general. While this usage is still sometimes found even in philosophical and
mathematical contexts, for the most part, it is outdated.“
One can understand abductive reasoning as "inference to the best explanation"
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51. Analogy
Argument from analogy is a special type of inductive argument, whereby perceived similarities are
used as a basis to infer some further similarity that has yet to be observed.
• a cognitive process of transferring information or meaning from a particular subject (the analogue
or source) to another particular subject (the target),
• or a linguistic expression corresponding to such a process.
In a narrower sense, analogy is
• an inference or an argument from one particular to another particular,
• as opposed to deduction, induction, and abduction,
• where at least one of the premises or the conclusion is general.
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52. argumentum e contrario
• Opposite to analogy
Strength of an analogy
Several factors affect the strength of the argument from analogy:
• The relevance (positive or negative) of the known similarities to the similarity inferred in the
conclusion.
• The degree of relevant similarity (or dissimilarity) between the two objects.
• The amount and variety of instances that form the basis of the analogy.
Counterarguments
• Disanalogy
• Counteranalogy
• Unintended consequences
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53. Teleological argument
• Existence of God
• Teleological argument vs. Intelligent design
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54. Ontological argument
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55. Problem of induction
C. D. Broad said that "induction is the glory of science and the scandal of philosophy.“
Pyrrhonism, Indian Carvaka school
The problem: In inductive reasoning, one makes a series of observations and infers a new claim
based on them.
1. If the past cannot predict the future  it is not certain, regardless of the number of observations
2. Problem of circularity: The observations themselves do not establish the validity of inductive
reasoning, except inductively  but this holds equally true for deductive reasoning!
W.V.O. Quine offers a practicable solution to this problem[18] by making the metaphysical claim that
only predicates that identify a "natural kind" (i.e. a real property of real things) can be legitimately used
in a scientific hypothesis.
What is «real»?
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56. Induction and ethics
According to Popper, the problem of induction as usually conceived is asking the wrong question: it is
asking how to justify theories given they cannot be justified by induction. Popper argued that
justification is not needed at all, and seeking justification "begs for an authoritarian answer".
Existential questions and Ethics require an «authoritarian answer»!
Karl Popper (1963). Conjectures and Refutations. p. 25. ISBN 0-06-131376-9.
"I propose to replace ... the question of the sources of our knowledge by the
entirely different question: 'How can we hope to detect and eliminate error?'"
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57. How to deal with uncertainity
• We do not exist because we are certain of
our existense
• We exist because we will exist
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58. p1: Infinit actuality (material self)
p∧﹁p≡T
P1: infinit actuality exists
∃ p∧﹁p
p2: Existens of infinit actuality is unique (personal identity)
∃! p∧﹁p
p∧﹁p → ﹁(p∧﹁p) ≡ ﹁ p∧﹁(﹁ p)
∃ x ∧ ⌐x
∃ x= self
∃ ⌐x= other
p2: There is argumentation (contradiction)
∃ x ∧ ⌐x → ⊥
p3: The existence of self implies otherness
∃ x → ⌐x
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59. Time (McTagger)
• A-series (tensed): series of temporal
positions as being in continual
transformation (timeline) (ordered by way
of the non-relational singular predicates)
• B-series (tenseless): "comes before" (or
precedes) and "comes after" (or follows)
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60. • Schutz: presentism (only the present
exists, past and future are mental
constructs)
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61. LOGIC IN ETHICS
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62. Why certainity?
• The existential question, noetic tension
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63. Way to certainity
1. Define thought, existence, and certainty
1. Is Thought an action or an act? (time problem)
2. Is existence a predicate?
3. Is certainty knowledge of truth?
2. Cogito, ergo sum?
1. Does it imply that what does not think does not exist?
2. Does «God» think?
3. Does «God» exist?
3. xx
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