Title: Courses / History of Logic
Tags: Course.HistoryOfLogic Slide.02 Slide.07 Slide.03 Slide.04 Slide.08 Slide.05 Slide.06 Slide.01

Q: The history of western logic can be classified into five periods:
A: 1. ancient Greece (until the 6th century a.d.);
2. Middle Ages (from the 7th to the 11th century);
3. scholasticism (from the 11th to the 15th century);
4. modern “classical” logic (16th to 19th century);
5. mathematical logic (since the mid 19th century).

Q: Aristotle’s treatises on logic were grouped together by ancient
commentators under the title [...], comprising the
following parts:
1. [...]
A: Organon (“Instrument”); Categories: introduces a hierarchical classification based on ten basic categories; ambiguities of natural language; and different forms of opposites.
2. On Interpretation: discusses different forms of propositions and their relations, including modalised statements.
3. Prior Analytics: contains his deductive logical system, the syllogisms.
4. Posterior Analytics: deals with the first theory of science.
Topics: probably the oldest part of the Organon, collecting an early, informal logic for practical, rhetoric purposes.
5. On Sophistical Refutations: discusses fallacies, i.e., invalid arguments.

Q: Aristotle's definition of a deduction bears some important differences to its modern-day sibling:
A: 1. A conclusion must be different from the premisses in Aristotle’s
definition ⇒ this is not required in modern logic.
2. The plural “certain things having been supposed” was understood
by some ancient commentators to rule out arguments with only one
premiss.
3. The expression “because of their being so” has sometimes been
understood as excluding arguments in which the conclusion is not
“relevant” to the premisses, e.g.,
  - arguments with inconsistent premisses,
  - arguments with conclusions that would follow from any premisses whatsoever, or
  - arguments with superfluous premisses.

Q: By an assertion, Aristotle understands a sentence which is either [...] having the following structure:
- [...]
A: true or false; it must contain a subject and a predicate and must either affirm or deny the predicate of the subject.

Q: [In Aristotle's syllogistic] An assertion is either universal or particular
- [...]
A: “all” and “no” added to the subject refer to universality,
- while “some” and “some not” or “not all” refer to particularity.

Q: Aristotle calls subjects and predicates of assertions [...].
A: terms

Square of opposition

Q: A syllogism is an inference pattern of the form if α and β, then γ where:
- α, β, γ are distinct assertions of the form a, e, i, o
  - α, β are the premisses and γ is the conclusion.
- The assertions involve three terms s.t. [...]
- α contains [...]
A: α and β share exactly one term in common, called the middle term, and γ contains the other two terms not shared by the premisses.
  - The first term in γ is the minor term and the second one the major term.; the major and middle term and β contains the minor and middle term.
  - Accordingly, α is called the major premiss whilst β is the minor premiss.

Q: Since the order of terms in the conclusion is fixed, there are just four possible ways of permuting the two terms in the two premisses:
- I. if Mω1 P and Sω2 M, then Sω3 P;
- II. if Pω1 M and Sω2 M, then Sω3 P;
- III. if Mω1 P and Mω2 S, then Sω3 P;
- IV. if Pω1 M and Mω2 S, then Sω3 P.
(ωi ∈ {a, e, i, o}, i = {1, 2, 3}.)
- These arrangements are called [...].
A: figures

Q: The specific distribution of values for ω1 , ω2 , ω3 for a given syllogism is called the [...] of this syllogism.
- A syllogism is thus uniquely characterised by [...].
A: mood (“modus”); its mood and its figure

Q: Lebiniz's characteristica universalis refers to his project of [...].
A: an alphabet of human thought
Extra: Page 12/37.

Q: Leibniz's Plus ultra refers to [...].
A: an encyclopedic form of all human knowledge
Extra: Page 14/37.

Q: The idea of an alphabet of human thought rests on two ideas:
A: 1. all concepts are either simple or complex;
2. the latter are composed out of the former.

Q: The basic idea of the approach [an arithmetical calculus of the syllogistic] is to [...]
A: correlate variables of the syllogistic with ordered pairs of natural numbers subject to certain conditions.
Extra: Page 20/37.

Q: SaP is true iff [...].
A: p1|s1 and p2|s2, i.e., s1 is divisible by p1 and s2 is divisible by p2
Extra: Page 20/37.

Q: SiP is true iff [...].
A: gcd(s1 , p2) = 1 and gcd(s2 , p1) = 1, i.e., s1 and p2 are relatively prime and s2 and p1 are relatively prime
Extra: Page 20/37.

Q: Modus Barbara if MaP and SaM, then SaP is verified by this interpretation:
A: - Let (m1 , m2), (p1 , p2), (s1 , s2) be the associated pairs of M, P, and S.
- Assume that MaP and SaM are true, i.e., pi|mi and mi|si , for i = 1, 2.
- By the transitivity of divisibility, we obtain pi|si, for i = 1, 2, i.e., SaP is true.

Q: Approaches to Formal Reasoning: 2nd Version. Principles of the calculus:
A: - Anything included in certain undetermined letters is to be understood as concluded in any others whatsoever which are subject to the same conditions.
  - E.g., since AB is A is true, BC is B is also true.
- Transposition of letters in the same term makes no difference,
  - i.e., AB is equivalent to BA.
- Repetition of the same letter in the same term is superfluous
  - e.g., in B is AA, it is enough to say B is A.
- A proposition can be made from any number of propositions by combining all the subjects into one subject and all the predicates into one predicate,
  - i.e., from A is B and C is D and E is F we can get ACE is BDF.
- From any proposition whose predicate is composed from several terms it is possible to make several propositions each of which has the same subject as the original but a predicate which is part of the predicate of the original,
  - i.e., given A is BCD we get A is B and A is C and A is D.

Q: Perfect syllogisms are the moods [...] of Figure I (which are the moods [...]).
A: aaa, eae, aii, and eio; Barbara, Celarent, Darii, and Ferio

Q: Aristotle’s method to determine valid patterns is to [...]
and then [...].
A: accept some as “evidently true”,
- these are the perfect syllogisms, or axioms in modern terminology,; using certain principles to deduce the valid patterns amongst the remaining figures

Q: Indirect reduction is basically an indirect proof, using one of the following lines of reasoning:
A: - if α and β entail γ, then α and not γ entail not β, or
- if α and β entail γ, then β and not γ entail not α.

Q: A direct reduction is [...]
A: a series of steps leading from the premisses to the conclusion
- each of which is either a conversion of a previous step or an inference from two previous steps relying on a perfect syllogism.

Q: Aristotle uses the following conversions:
A: - MeP ⇒ PeM (no M is P entails that no P is M) [“simple conversion of e”];
- MiP ⇒ PiM (some M is P entails that some P is M) [“simple conversion of i”];
- MaP ⇒ PiM (all M are P entails that some P is M) [“conversion per accidens”].

Q: But Aristotle’s theory of syllogisms makes the stronger assumption that any term is [...].
A: non-empty, i.e., that every property has instances

Q: The general picture is as follows:
- Among the [...] possible syllogistic patterns, [...] are valid.
- Among the latter, [...] require additional existential assumptions.
A: 256; 24; nine

- Boole was influenced by the works of Sir William Hamilton (1788-1856) and Augustus De Morgan (1806-1871).
- Two points of their development are important:
  - They went beyond the traditional a, e, i, and o statements, having statements like
    - “all S is all P” (= all S is P and all P is S) and
    - “all S is some P” (= all S is P and some P is not S).
- A change from the intentional to the extensional interpretation of letters in an argument took place.
  - As done by Leibniz, variables were traditionally interpreted as attributes, i.e., signs of qualities,
  - but now they were interpreted as being signs of the things themselves having the qualities.

Q: Boole uses lowercase letters like x, y to stand for [...], called [...].
A: classes; elective symbols (because they “elect”, i.e., select, certain things)

Q: Furthermore, two special classes are introduced:
A: - The universe class 1 and the null class 0.
  - ⇒ it holds that 1x = x and 0x = 0 as in ordinary algebra.

Q: - Although intersection of classes can be thought of as their logical product, there is one important peculiarity in the system of classes, viz. [...].
A: idempotency: xx = x

Q: - The addition x + y of classes signifies [...], providing [...].
  - The proviso of disjoint classes has been [...] by successors of Boole, giving rise to the assertion that x + x = x, and thus to a principle of duality for union and intersection.
A: the class of things belonging to one of x or y; x and y are mutually exclusive, i.e., if xy = 0; removed

Q: Boole writes [...] for [$]1 - x[/$], denoting [...].
A: [$]\overline{x}[/$]; the complement of [$]x[/$]

The current system of notation is sufficient to express the traditional a, e, i, and o propositions:

Q: The basic principles of Boole's algebra are

Propositions (1)-(7) are similar to [...], while [...] distinguishes Boole’s system.
A: rules of ordinary numerical algebra; (8)

Q: A fundamental process in the formal elaboration of Boole’s system is what he calls [...].
A: development

Q: - The power (“Mächtigkeit”), or cardinal number, [$]\overline{\overline{S}}[/$], of a set [$]S[/$] can then be introduced as
  - [...]
- According to Cantor:
  - [...]
A: that which it has in common with all equivalent sets but with no others.; “the general concept which [...] results from a set when we abstract from the nature of its various elements and from the order of their being given”.

Operations by which sets may be constructed out of sets (finite or infinite):
- [$]S + T[/$], called the logical sum (or union or join) of [$]S[/$] and [$]T[/$]
- [$]ST[/$], called the inner product (or intersection or meet) of [$]S[/$] and [$]T[/$]
- [$]S \times T[/$], called the outer (or Cartesian or cross) product of [$]S[/$] and [$]T[/$]
- [$]T^S[/$], sometimes called the insertion set of [$]T[/$] into [$]S[/$], is the set of all single-valued functions by which elements of [$]T[/$] may be assigned to all the various elements of [$]S[/$], when
  - [$]S[/$] is not null
  - one element of [$]T[/$] may be assigned to more than one element of [$]S[/$], and
  - two functions are different if there is at least one element of [$]S[/$] to which they assign different elements of [$]T[/$].

Q: If [$]S[/$] and [$]T[/$] are mutually exclusive, then
A: - [$]\overline{\overline{S}} + \overline{\overline{T}} = \overline{\overline{S + T}}[/$],
- [$]\overline{\overline{S}} \times \overline{\overline{T}} = \overline{\overline{S \times T}}[/$],
- [$]\overline{\overline{T}}^{\overline{\overline{S}}} = \overline{\overline{T^S}}[/$].

Q: An immediate corollary of Cantor’s Theorem is the following:
A: - [$]2^c > c[/$] for any cardinal [$]c[/$] (finite or transfinite).
  - Since [$]\overline{\overline{\mathcal{U}S}} > \overline{\overline{S}}[/$] and [$]\overline{\overline{\mathcal{U}S}} = 2 ^ {\overline{\overline{S}}}[/$], whatever [$]S[/$] may be.

Q: Continuum hypothesis (CH): Cantor conjectured that
A: [$]\aleph_1 = 2 ^ {\aleph_0}[/$], respectively that [$]\aleph_{\alpha + 1} = 2 ^ {\aleph_{\alpha}}[/$] (generalized CH).

Q: Frege’s logicist view: arithmetic is [...].
A: an elaboration of logic

Q: Frege rejects [...].
A: distinction between subject and predicate

Q: If a simple or complex symbol occurs in one or more places in an expression and it is replaceable by another at one or more occurrences, then
- the invariant part of the expression is called [...],
- and the replaceable part [...].
A: the function; the argument of the function

For deriving theorems from these axioms:

Q: - Analytic philosophy
A: Eliminate meaningless and incoherent assertions in philosophy for the sake of clarity and precision in argument by the use of exact language, breaking down propositions into their simplest components.

Q: Logicism is the view [...].
A: that mathematics can be reduced to logic

Q: - This is often considered to entail the following claims:
  1. Language Logicism: [...]
  2. Truth Logicism: [...]
A: The language of mathematics consists of purely logical expressions.; Mathematical truths are true as a matter of pure logic.

Q: - These can be made more precise as follows:
  - Assume a mathematical language [$]L[/$] with an intended model [$]M[/$].
  - A paraphrase-function, [$]\cdot^*[/$], is a function such that [...]
- Language logicism:
  - [...]
- Truth logicism:
  - [...]
A: for any sentence [$]\phi[/$] in the domain of [$]\cdot^*[/$], [$]\phi^*[/$] is a paraphrase of [$]\phi[/$].; There is a paraphrase-function [$]\cdot^*[/$] such that, for any sentence [$]\phi[/$] of [$]L[/$], [$]\phi^*[/$] contains no nonlogical vocabulary.; There is a paraphrase-function [$]\cdot^*[/$] such that, for any sentence [$]\phi[/$] of [$]L[/$], which is true (resp., false) according to [$]M[/$],
    - [$]\phi^*[/$] (resp., its negation) is a logical truth (semantic version),
    - [$]\phi^*[/$] (resp., its negation) is derivable from the empty set on the basis of purely logical axioms and rules of inference (syntactic version).

Q: The main idea of logicism was put forward by [...] already in [...]
A: Leibniz; the late 17th century but it was not until Frege had developed his general logic that it became technically plausible.

Russell’s Theory of Logical Types
- Russell published his paradox in 1903 in an appendix to his Principles of Mathematics.
  - But he remained a logicist (i.e., hoped for a satisfactory demonstration of the identity of arithmetic and logic) and tried to overcome the problem.
  - Problem:
    - vicious circles, e.g., a definition of a concept referring to the totality of all possible objects under the concept.
- Attempt to overcome contradictions: Russell’s Theory of Logical Types published in:
  - an article of 1908 (Mathematical Logic as based on the Theory of Types), and in
  - the introduction of Principia Mathematica (PM).

Russell’s Theory of Logical Types (ctd.)
- No function can have among its values anything which presupposes the function.
- This is a particular case of the vicious-circle principle:
  - No totality can contain any members involving the totality.
  - “Whatever involves all of a collection must not be one of the collection”.

- A type may be defined as a range of significance,
  - i.e., as the collection of arguments for which a given propositional function has values, or
  - according to the entities which it admits as proper arguments.
- Example:
  - Consider men as individuals or entities of type 0.
    - Wisdom is of type 1 (asserted, e.g., significantly to Socrates)
    - Cardinal virtue is of type 2 (asserted significantly to wisdom).

Q: Russell’s Theory of Logical Types (ctd.)
- For attributive functions, we may think of types arranged hierarchically, but for relational functions, types have a much greater complexity which cannot be represented by ordinal numbers alone.
- The general rule is:
  - [...]
- Additionally, Russell uses his "no classes" theory:
  - [...]
A: Propositional functions take arguments from appropriate lower type(s) only!
  - This theory excludes the paradox of the property of being a property which does not exemplify itself (like the class of all classes which are not members of themselves).; Designations of classes are incomplete symbols, i.e., symbols which cannot be defined alone but may be defined in company.
  - "A proposition about a class is always to be reduced to a statement about a function which defines the class."

Q: Russell’s Theory of Logical Types (ctd.)
- In order to avoid other (semantic, resp., linguistic) paradoxes, e.g., the Liar paradox, a further development is necessary.
  - [...]
- For simplicity, consider only functions that take individuals as arguments.
  - [...]
- A function is called predicative iff [...]
A: This is the ramified theory of types, i.e., the distinction of orders within types.; First order: It can be defined without the application of quantifiers to any variables other than individual variables.
  - Second order: It can be defined quantifying only over individual variables and/or first-order functional signs.
  - ...; it is of the lowest order compatible with its arguments.

Russell’s Theory of Logical Types (ctd.)
- This approach results in drastic limitations and enormous complication of logic and numbers.
- E.g., the theorem that any non-empty set of real numbers with an upper bound has a least upper bound cannot be formulated.
  - Introduction of the axiom of reducibility, i.e., the assumption that for every propositional function there is a predicative function that is satisfied by exactly the same arguments.
- The definition of the infinity of natural numbers (in the style of Frege) is impossible.
  - Introduction of a special axiom of infinity.

Principia Mathematica
Bertrand Russell and Alfred Whitehead (1861-1947) started their joint work on foundations in 1900:
- Aim: the seamless development of mathematics from the view of clearly stated axioms and rules of inference in pure logic.
- “All pure mathematics follows from purely logical premises and uses only concepts definable in logical terms” (Russell).
- Opted for the more modern notation of Peano instead of that of Frege’s Begriffsschrift.
- Approach was essentially that of Frege, but avoided paradoxes by Russell’s Theory of Types.
- Three volumes, almost 2000 pages, appeared in 1910, 1912, and 1913.
- A fourth volume on geometry was planned, but never appeared.

Q: Principia Mathematica (ctd.)
- Vol. I:
  - [...]
- Vol. II:
  - [...]
- Vol. III:
  - [...]
A: Introduction: explanations of ideas, notation, theory of types;
  - Part I: “Mathematical Logic” including axioms and rules of inference; elementary results on classes and binary relations;
  - Part II: “Prolegomena to Cardinal Arithmetic”, definition of numbers;; Part III: “Cardinal Arithmetic”, definitions and arithmetic properties of cardinal numbers;
  - Part IV: “Relation Arithmetic”, arithmetic of binary relations;
  - First half of Part V: “Series” including linear orderings, limit points, continuous functions;; Second half of Part V: including well-orderings, finite and infinite series and ordinals;
  - Part VI: “Quantity” incl. measurement (modulo a quantity).

Q: Formalism
- The central doctrine of symbolic formalism is that
  - [...]
- This doctrine is itself composed of two key elements:
  - [...]
A: the instrumentalist conception of language allows for purely symbolic uses of signs in our reasoning,
  - uses that do not depend in any essential way on the semantic content of the signs involved or on their even having such content.; Creative element: the idea that the mathematician is free to “create” methods out of considerations of convenience or efficiency as distinct from evaluation of content.
  - Symbolic element: that nonsemantical uses of signs may, at least on occasion, constitute such conveniences.

Q: Hilbert’s Program
- The Principa Mathematica provided the required logical basis for addressing these issues.
- In the early 1920s Hilbert put forward a new proposal for the foundation of classical mathematics:
  - [...]
- Hilbert’s finitary standpoint restricts mathematical thought to:
  - [...]
A: Formalisation of all mathematics in axiomatic form, together with a proof that this axiomatisation is consistent.
  - The consistency proof is to be carried out using only finitary methods.; Objects which are “intuitively present as immediate experience prior to all thought”, and
  - operations and methods of reasoning which do not require abstract concepts, in particular completed infinite totalities.

Q: - However, Brouwer was the first to draw the consequences of
A: - not allowing any infinity except the potential infinity of a sequence that can be continued ad libitum.

Q: Intuitionism (ctd.)
- Maintains Kant’s view that arithmetic must be derived from [...].
- From this intuition we get
  - [...]
- There are no sets except the denumerable, and so
  - [...]
A: the intuition of time; the notion of sequence of ordinal numbers,
  - the notion of the linear continuum, i.e., “of the betweenness which is not exhaustible by the inter-position of new units and therefore can never be thought of as a mere collection of units”.; there are no transfinite cardinal numbers other than [$]\aleph_0[/$];
  - there is no meaning to be given to any phrase such as “the set of all real numbers between 0 and 1”.

Q: Intuitionism (ctd.)
- Even the sequence of natural numbers is to be conceived as [...]
  - [...]
- Intuition is the mind’s clear apprehension of what it has itself constructed.
- Denial of the dependency on any special kind of language and the technique of formalisation.
  - [...]
A: an open manifold, always in growth and never brought to finish;; we can only comprehend it by understanding the law of its construction.
  - All satisfactory proofs in mathematics are constructive!; Mathematics does not presuppose a system of logic.
  - It is rather a source of logical principles that may be generalised only after their validity has been established by an appropriate intuition.

Intuitionism (ctd.)
- In practice, the most important requirement for a constructive proof is that no existential statement is admitted unless it can be demonstrated by an instance.
- Brouwer: Anyone who fails to obey to this principle has to blame himself if he falls into paradox.
- Consider an existential statement
[$](\exists x)[\phi x. \sim \psi x], \qquad (1)[/$]
and the universal statement
[$](x) \sim [\phi x. \sim \psi x], \qquad (2)[/$]
which is ordinarily considered the contradictory of (1). (Here, “.“ denotes conjunction, “∼” negation, and “(x)” universal quantification.)
  - If the set of all objects satisfying φ, denoted by [$]\hat{x} \phi x[/$], is finite we can in principle examine all the members and decide in favour of one or the other.
  - If the set [$]\hat{x} \phi x[/$] is denumerable infinite, however:
    - Systematic enumeration is still satisfactory for proving (1) since it leads to an instance in a finite number of steps if such an instance exists.
    - But it is no longer effective for (2) since it is senseless to go through all the members.
- But how to prove a universal statement of this kind?
  - Brouwer is prepared to concede that one way is to show, if we can, that the existential statement involves a contradiction and must therefore be rejected, i.e., the following is intuitionistically accepted:
  - [$]\sim (\exists x)[\phi x. \sim \psi x] \supset (x) \sim [\phi x. \sim \psi x].[/$]
  - However the following is not admitted:
[$]\sim (x) \sim [\phi x. \sim \psi x] \supset (\exists x)[\phi x. \sim \psi x], \qquad (3)[/$]
since if we succeed in refuting the universal statement by showing it involves a contradiction, we should be no nearer to discover a thing a of which we can assert [$]\phi a. \sim \psi a[/$].
- The rejection of (3) amounts to the abandonment of the principle of excluded middle and the associated principle for the elimination of double negation!
- For Brouwer, the principle of excluded middle is equivalent of claiming to be in the possession of a universal method for solving mathematical problems, which he rejected.
- Serious consequences in mathematics:
  - Non-constructive methods very common, in particular in the theory of real numbers.
  - E.g., in Dedekind’s theory a real number is considered to be an actual infinite set of rational numbers, and even the simplest theorems are proved using the principle of excluded middle.
- Brouwer and his followers aimed to reconstruct mathematics in accordance to their stricter notion of a proof.
  - However, this undertaking results in structures which are inevitably more complicated than their classical counterparts.

Q: In the 1930s [...] developed [...].
A: Arend Heyting (1898-1980), Brouwer’s assistant,; intuitionistic logic and formal intuitionistic number theory

Q: The system [Arend Heyting's system] consists of modus ponens as a rule of deduction but
1. [...]
2. the various logical axioms are abandoned in favour of the following eleven axioms:

3. Similar in appearance to theses in classical logic (Heyting uses ¬ instead of ∼).
4. Allows for many of the ways of reasoning in classical logic but it is not possible to derive [...].
A: all logical signs are taken as undefined primitives;; p ∨ ¬p and ¬¬p ⊃ p

Q: Intuitionistic Logic (ctd.)
- Heyting’s calculus should be interpreted as [...]
- The main idea is as follows:
  - [...]
- According to this interpretation, it is not a system of logic in the strict sense, because [...]
- Later model-theoretic approaches provided precise semantic notions of truth and logical validity:
  - First outlined by Evert Willem Beth in 1947 in the context of intuitionistic mathematics and later formally developed by Saul Kripke in 1965.
- Their underlying idea is the same:
  - Instead of a single model one has a partially ordered collection of nodes each of which might be thought of as a model in its own right.
  - The partial order among nodes is called accessibility relation.
  - Truth of compound statements may well depend upon truth values of components at accessible nodes.
A: an axiomatic theory of the notion of provability by constructive methods.
  - In mathematics, something being true means that it is provable.
  - Intuitionistic logic is the result of applying this principle to the semantics of the connectives and quantifiers.; p.q is proved in a situation iff the situation proves p and q;
  - p ∨ q is proved in a situation iff the situation contains evidence indicating that either p or q will eventually be proved;
  - p ⊃ q is proved in a situation iff the situation contains a method for converting any proof of p into a proof of q;
  - ¬p is proved in a situation iff the situation contains evidence that a proof of p can be turned into a proof of contradiction.; it does not fix the sense of the logical signs.

Q: Approaches to Formal Reasoning: 3rd Version (ctd.)
- What is important to note is that Leibniz showed interest in the abstractness of his calculus, i.e., in the possibility of interpreting it in various ways.
- He thought primarily of the combination of attributes, but he illustrated many of this results in terms of classes.
- These two interpretations correspond to [...]:
  - [...]
  - [...]
  - For instance, man = rational ⊕ animal:
    - the concept of a man includes the concept of an animal whilst the class of animals includes the class of men.
- In accord to being mainly interested in attributes, he favours the intensional interpretation.
A: an intensional and an extensional interpretation of his letters; in an intensional interpretation, A ⊕ B = L means that the concept of L includes the concept of A,; while an extensional interpretation of A ⊕ B = L implies that class A includes class L.

Approaches to Formal Reasoning: 3rd Version
Def. 1. Terms are the same or coincident which can be substituted one for another wherever we please without altering the truth of any statement. [$]A = B[/$] signifies that [$]A[/$] and [$]B[/$] are the same. [N.B. This is Leibniz’s famous substitution principle “salva veritate”.]
Def. 2. Terms which are not the same, i.e., terms which cannot always be substituted one for another, are different. [$]A \neq B[/$] signifies that [$]A[/$] and [$]B[/$] are different. [N.B. Leibniz used [$]\infty[/$] and “non [$]\infty[/$]” for “[$]=[/$]” and “[$]\neq[/$]”, respectively.]
Prop. 1. If [$]A = B[/$], then also [$]B = A[/$].
For since [$]A = B[/$] (by hyp.), it follows (by Def. 1) that in the statement [$]A = B[/$] (true by hyp.) [$]B[/$] can be substituted for [$]A[/$] and [$]A[/$] for [$]B[/$]; hence, we have [$]B = A[/$].
Prop. 2. If [$]A \neq B[/$], then also [$]B \neq A[/$].  Otherwise we should have B = A, and in consequence (by the preceding prop.) A = B, which is contrary to hypothesis.
Prop. 3. If A = B and B = C, then A = C.  For if in the statement A = B (true by hyp.) C be substituted for B (by Def. 1, since B = C ), the resulting statement will be true.
Prop. 4. If A = B and [$]B \neq C[/$], then [$]A \neq C[/$]. For if in the proposition [$]B \neq C[/$] (true by hyp.) A be substituted for B, we have (by Def. 1, since A = B) the true proposition [$]A \neq C[/$].
Def. 3. A is in L, or L contains A, is the same as to say that L can be made to coincide with a plurality of terms taken together of which A is one. B ⊕  N = L signifies that B is in L and that B and N together compose or constitute L. The same thing holds for a large number of terms.
Axiom 1. B ⊕ N = N ⊕ B.
Postulate. Any plurality of terms, as A and B, can be added to compose as single term A ⊕  B.
Axiom 2. A ⊕ A = A.
Prop. 5. If A is in B and A = C, then C is in B.
For in the proposition A is in B (true by hyp.) the substitution of C for A (by Def. 1, since by hyp. A = C ) gives C is in B.
Prop. 6. If C is in B and A = B, then C is in A.
For in the proposition C is in B the substitution of A for B (since A = B) gives C is in A.
Prop. 7. A is in A.
For A is in A ⊕  A (by Def. 3) and A ⊕ A = A (by Ax. 2). Therefore (by Prop. 6) A is in A.
Leibniz continues with further results which we omit here, except for the following:
Prop. 15. If A is in B and B is in C, then A is in C.
For A is in B (by hyp.), hence A ⊕ L = B (by Def. 3). Similarly, since B is in C, B ⊕ M = C, and putting A ⊕ L for B in this statement (since we have shown that they are coincident), we have A ⊕ L ⊕ M = C . Therefore (by Def. 3) A is in C.

Approaches to Formal Reasoning: 2nd Version
In his second attempt, discussed in his Specimen Calculi Universalis and its Addenda, Leibniz proposes the following:
- Propositions true of themselves:
  - A is A;
  - AB is A;
  - A is not not-A;
  - not-A is not A;
  - whatever is not A is not-A;
  - whatever is not not-A is A.
- Consequentia true of itself:
  - A is B and B is C, therefore A is C.
Principles of the calculus:
- Anything included in certain undetermined letters is to be understood as concluded in any others whatsoever which are subject to the same conditions.
  - E.g., since AB is A is true, BC is B is also true.
- Transposition of letters in the same term makes no difference,
  - i.e., AB is equivalent to BA.
- Repetition of the same letter in the same term is superfluous
  - e.g., in B is AA, it is enough to say B is A.
- A proposition can be made from any number of propositions by combining all the subjects into one subject and all the predicates into one predicate,
  - i.e., from A is B and C is D and E is F we can get ACE is BDF.
- From any proposition whose predicate is composed from several terms it is possible to make several propositions each of which has the same subject as the original but a predicate which is part of the predicate of the original,
  - i.e., given A is BCD we get A is B and A is C and A is D.

Approaches to Formal Reasoning: 2nd Version
For the further development of his calculus, Leibniz expressed Aristotle’s four types of propositions in terms of two schemas:

- Here, ens and non-ens may be rendered as “something” and “nothing”, respectively, and “est” is latin for “is”.
- However, the use of ens as a term involved in all others is problematic, as it is of a second-order logic character!
- When he returned to the subject some years later, in his paper Non Inelegans Specimen Demonstrandi in Abstractis (“A not Inelegant Specimen of Abstract Proof”), Leibniz abandoned all reference to existential propositions and made a small change of notation:
  - instead of juxtaposition, he used “+” to refer to the combination of concepts.
- This suggests the possibility of introducing the inverse operation of subtraction:
  - He stipulates A − B = C iff A = B + C and B and C have nothing in common.
  - The second condition is important because otherwise subtraction would not give a unique result.
    - E.g., in such a case, we would have not only A − (L + M) = N but also A − (L + M) = M + N since L + M + N = L + M + M + N, as repetition of a letter in a complex term makes no difference.
- Subtraction is not the same as negation:
  - E.g., Man not Rational = Impossible, but Man − Rational = Brute.
- Using + and − in a calculus where A + A = A holds proved unfruitful, so Leibniz abandoned it for his last and most highly developed calculus, taking instead the symbol ⊕.
- We describe this calculus in some detail next, reproducing parts of Leibniz’s original formulation.