A repo that keeps track of the history of logic. It chronicles the eminent personalities, schools of thought, ideas of each epoch.
The period roughly between 8th century B.C. to 6th century A.D
Argumentation had methods that resembled reduction ad absurdum
A/this if B/that. But B/that is impossible, hence A/this is impossible.
Distinguished between wish, question, answer, and command
Attempt at formulating principles of scientific research. This is said to be related to logic of induction.
Pupil of Gorgias
Distinguished between phasis (assertion) and apophasis (denial), and question and address (prosagoreusis)
Defined a sentence as that which indicates a thing was or is.
Also known as Dialexis.
Book of unknown authorship
Perhaps the earliest surviving piece of logical arguments. One of the key themes of the book seems to be that in order to act, one must be persuaded to choose one side or the other. This process is said to be accomplished through dissoid logoi
The book is also said to be one of the primary records that shows the awareness that self-reflexive/self-referential use of the truth predicate can be problematic as seen in Liar’s paradox.
Distinguished between syntax (What is a statement?) and semantics (When is it true?).
Something was considered to be a statement when it specifies a subject and says something about it. That is in other words, both the onoma and rhema (or subject and predicate) are present. Statements with empty subject-predicate relationships are excluded as statements.
Plato is said to have produced a deflationist theory of truth by saying that when a statement is true when it says what is and false when it is not (Theatetus is flying).
Negation is said to have a narrower scope than affirmation in that negation negates the predicate not whole sentences.
Plato is said to have grappled with the ideas of multiple meanings of the word ‘is’: 1/ to represent being (Example: There is a ball) 2/ to represent equality (Archytas is the King of Greece)
Think ‘is’ can have one more which is predication (The ball is red)
He is also said to have anticipated the law of non-contradiction
Deepening the studies on the structure of argumentation lead to reflection upon patterns of argumentation. Valid and invalid arguments were studied which resulted in identifying ways in which valid inferences can be made.
Plato brought structure and rules to the argumentation that was conducted in antiquity.
Analyzes simple statement as containing verb (rhêma / ῥῆμα) which indicates the action and noun (onoma / ὄνομᾰ).
Contains a large number of paradoxes
In Aristotle’s work, a subject (onoma) is linked to a verb (rhema), together which becomes a proposition (logos).
Initiated work on modal logic.
Deduction and proof developing in Greek mathematics is said to have influenced Aristotle (Verify/validate, how exactly he was influenced by it).
This title means predications. It can mean one of:
A declarative sentence (apophantikos logos) or declaration (apophansis) is different from other kinds of discourse like command or question. These can be judged for their truth value. They signify thoughts which are shared by humans and through these, indirectly, things. Written sentences signify spoken ones.
Subjects and predicates are connected by a copula.
Names can be singular terms or common nouns, both can be empty. Singular terms can take subject position. Verbs co-signify time.
A declarative sentence can be affirmed or negated depending on its predicate.
In terms of quantity, a sentence can be thought of as singular, particular, indefinite, universal.
Includes four forms of categorical statments: Every S is P No S is P Some S is P Some S is not P
And the relations between them
Based on this theory, he created the doctrine of the categorical syllogism, in Prior Analytics.
Contains variable letters A, B, C etc. to stand in place of terms Develops the syllogistic as a system of deductive inference
By the theory of reduction, the syllogistic moods are shown to be interconnected so that all can ultimately be reduced to two: Barbara and Celarent.
Books (2 – 7) about how to find procedures/rules (topoi) to find an argument to establish/refute a thesis. 4 kinds of relationship a predicate may have to subject known as predicables. It may grant it a:
Systematical classification of logical fallacies
Syllogism as an argument (logos) in which, certain things having been laid down, something different from what has been laid down follows of necessity because these things are so.
The definition appears to require i) The syllogism consists of at least 2 premises (for Sorites, I think there can be more than 2 premises) and a conclusion. ii) The conclusion follows of necesitty from the premises iii) The conclusion differs from the premises
Aristotle’s syllogistic is said to cover only a small part of all arguments that satisfy these conditions.
Admissible truth-bearers are now defined as each containing two different terms (horoi) conjoined by a copula, of which one (the predicate terms) is said of the other (the subject term).
Aristotle is said to have not clarified whether the terms are things or linguistic expressions for these things.
Universal and particular sentences are said to be discussed and singular sentences are said to be excluded and indefinite sentences mostly ignored.
Syllogistic also features the use of letters in place of terms. Thus initiating the tradition of using short symbols for variables.
The four types of categorical sentences used by Aristotle are:
A is precidated of all B (AaB) A is predicated of no B (AeB) A is predicated of some B (AiB) A is not predicated of some B (AoB)
In Greek, I think these can be expressed in a different way rather than the asymmetrical way in which the not appears at the beginning of AoB.
Syllogistic propositions consist of premises which share one common term called the middle term which disappears in the conclusion and only the other two terms, sometimes called the extremes are retained. Based on the position of the middle term, Aristotle classified all possible premise combinations into 3 figures (schêmata):
I Subject - Middle Term Middle Term - Predicate
II Middle Term - Subject Middle Term - Predicate
III Subject - Middle Term Predicate - Middle Term
Subject here is sometimes called the major term and Predicate the minor term.
Each figure can be further classified based on whether the premises are universal or not.
This is said to give 58 possible premise combinations (how?) and Aristotle showed that only 14 have a conclusion following of necessity from them, i.e. are syllogisms.
He assumed that syllogisms of the first figure are complete and not in need of proof, since they are evident. He proves this by reducing the figures in the second and third to the first thereby ‘completing’ them. He uses: i) conversion (antistrophê): A sentence converted by interchanging its terms. AeB infer BeA AiB infer BiA AaB infer BiA
ii) reductio ad impossible (apagôgê): Contradictory of an assumed conclusion together with one of the premises is used to deduce by a first figure syllogism a conclusion that is incompatible with the other premise. Thus using this semantic relation network, the earlier assumed conclusion is established
iii) exposition or setting-out (ekthesis): Choosing or setting out some additional term that falls in the non-empty intersection delimited by two premises and using it to justify the inference from the premise to a particular conclusion.
It is debated whether this newly introduced term represents a singular or a general term and whether exposition constitutes proof.
| First Figure | Second Figure | Third Figure |
|---|---|---|
| Barbara | Camestres | Darapti |
| Celarent | Cesare | Felapton |
| Darii | Festino | Disamis |
| Ferio | Baroco | Datisi |
| Bocardo | ||
| Ferison |
These names are mnemonics: each vowel or the first three in cases where the name has more than three, indicates the order in which the first and second premises and the conclusion (a, e, i, or o).
Aristotle implicity recognized that by using the conversion rules on the conclusion we obtain eight further syllogisms and that of the premise combinations rejected as non-syllogistic, some five ones which yield a conclusion in which the minor term is predicated of the major.
Aristotle reduced Darii and Ferio into Barbara and Celarent and later on in the Prior Analytics he invokes a type of cut-rule by which a multiple-premise syllogism can be reduced to two or more basic syllogisms.
From a modern perspective, Aristotle’s system can be understood as a sequent logic in the style of natural deduction and as a fragment of first-order logic. It has been shown to be sound and complete (see Corcoran’s Completeness of an Ancient Logic), if one interprets the relations expressed by the categorical sentences set-theoretically as a system of non-empty classes as: AaB iff A contains B (all) AeB iff A and B are disjoint (none) AiB iff A and B are not disjoint (some) AoB iff A does not contain B (some are not)
It is generally agreed that Aristotle’s syllogistic is a kind of relevance logic rather than classical.
Aristotle is also considered as the originator of modal logic.
Apart from quality (affirmation or negation), quantity (singular, universal, particular, or indefinite), he also took categorical sentences to have a mode (actually, necessarily, possibly, contingently, impossibly). The latter four are expressed by modal operators that modify the predicate: It is possible for A to hold of some B; A necessarily holds of every B.
In De Interpretatione, Aristotle concludes that modal operators modify the whole predicate (or the copula), not just the predicate term of a sentence.
He is said to equate between possible and contingent.
In Prior Analytics, he develops his modal syllogistic and develoops tests for all possible combination of premise pairs of sentences with necessity, contingency, or no modal operator: NN, CC, NU/UN, CU/UC, and NC/CN. Syllogisms with the last 3 types of premise combinations are called mixed modal syllogisms.
Aristotle is said to have wavered between one-sided interpretation (where necessity implies possibility) and a two-sided interpretation (where possibility implies non-necessity).
It is noted that appart from the NN category which maps on to unmodalized syllogisms (categorical syllogisms?), all categories contain dubious cases.
Both universal and particular sentences contain a quantifier and both universal and particular affirmatives were taken to have existential import. This was not so the case for negations, as it is reflected in some of the work Sturm has done and Leibniz has said not to be in line with Aristotle’s work.
Contrary sentences cannot both be true Contradictory of a universal affirmative is the corresponding particular negative and conversely, for the universal negative.
Student of Aristotle.
Made the theory of modal syllogisms consistent by peiorem rule
Developed a theory of wholly hypothetical syllogisms based on a prototype of syllogisms composed of three conditional propositions. He thought these were reducible to categorical syllogisms though the method is said to have not survived.
Wrote more logical treatises than Aristotle.
Settled on a one-sided possibility, so that possibility no longer entails non-necessity.
Theophrastus and Eudemus introduced the principle that in mixed modal syllogisms the conclusion alwasy has the same modal character as the weaker of the premises, where possibility is weaker than actuality, and actuality than necessity.
Prosleptic syllogisms have the form:
For all X, if f(X), then g(X)
where f(X) and g(X) stand for categorical sentences in which the variable X occurs in place of one of the terms
Statements like these are:
- A holds of all of that of all of which B holds.
ForAll(X) (BaX -> AaX)
- A holds of none of that which holds of all of B
ForAll(X) (XaB -> AeX)
Prosleptic syllogisms are composed of a prosleptic premise and the categorical premise obtained by instantiating a term (C) in the antecedent ‘open categorical sentence’ as premises, and the categorical sentences one obtains by putting in the same term (C) in the consequent ’open categorical sentence’ as conclusion:
A holds of all of that of all of which B holds B holds of C Therefore, A holds of all C
Theophrastus distinguished there figures of these syllogisms, depending on the position of the indefinite term.
Theophrastus held that certain prospleptic premises were equivalent to certain categorical sentences, but not all of them. Thus prosleptic syllogisms increased the inferential power of Peripatetic logic.
Theophrastus and Eudemus considered complex premises which they called ’hypothetical premises’ and which had one of the following two (or similar) forms:
If something is F, it is G Either something is F or it is G (with exclusive or)
They developed arguments with them which they called ‘mixed from a hypothetical premise and a probative premise’.
These were forerunners of modus ponens and modus tollens.
Theophrastus considered quantified sentences such as those containing ’more’, ‘fewer’, and ’the same’.
If [something is] A, [it is] B If [something is] B, [it is] C Therefore, if [something is] A, [it is] C
At least one of these syllogisms were regarded as reducible to Aristotle’s categorical syllogisms.
In later antiquity, after some intermediate stages, and possibly under Stoic influence, the wholly hypothetical syllogisms were interpreted as propositional-logical arguments of the kind
If p, then q. If q, then r. Therefore, if p, then r.
Followers of Euclid of Megara
Pupil of Euclid of Megara
Main logicians of this school were Philo and Diodorus. They are considered to be influenced by Eubulides of Miletus.
A conditional (sunêmmenon) is a non-simple proposition composed of two propositions and the connecting particle ’if’.
Definitions of modalities by both Diodorus and Philo satisy the following standard of requirements of modal logic:
i) Necessity entails truth and truth entails possibility ii) Possibility and impossibility are contradictories, and so are necessity and non-necessity iii) Necessity and possibility are interdefinable iv) Every proposition is either necessary or impossible or both possible and non-necessary
Main achievement of stoics was the development of a propositional logic: a system of deduction in which the smallest substantial unanalyzed expressions are propositions, or rather, assertibles.
The Stoics defined negation as assertibles that consiste of a negative particle and an assertible controlled by this particle.
Non-simple assertibles were defined as assertibles that consist of more than one assertible or of one assertible taken more than once and that are controlled by a connective particle. Both definitions can be understood as being recursive and allow for assertibles of indeterminate complexity.
What type of assertible an assertible is, is determined by the connective or logical particle that controls it, i.e. that has the largest scope. This is similar in some ways to the Polish notation.
Stoic negations and conjunctions are truth-functional. Stoic disjunction is exclusive and non-truth functional. Later Stoics introduced a non-truth functional inclusive disjunction.
TODO: Find out how truth-functional and non truth-functional distinction was made here.
Arguments are compounds of assertibles. They are defined as a system of at least two premises and a conclusion.
An argument is valid if the Chrysippean conditional formed with the conjunction of its premises as antecedent and its conclusion as consequent is correct. An argument is sound when in addition to being valid it has true premises.
The Stoics defined so-called argument modes as a sort of schema of an argument. The mode of an argument differs from the argument itself by having ordinal numbers taking the place of assertibles.
If the 1st, the 2nd But not the 2nd Therefore not the 1st
This is similar to figures/moods in Aristotelian logic.
In terms of modern day logic, Stoic syllogistic is best understood as a substructural backwards-working Gentzen-style natural deduction system that consists of five kinds of axiomatic arguments (the indemonstrables) and four inference rules, called themata.
An argument is a syllogism precisely if it either is an indemonstrable or can be reduced to one by means of the hemata. Thus syllogisms are certain kinds of formally valid arguments.
Stoics explictly acknowledged that there are valid arguments that are not syllogisms; but assumed taht these could be somehow tarnsformed into syllogisms.
A conditional propositional is true if it neither was nor is possible that its antecedent is true and its consequent false. This is reminiscent of strict implication.
Philo and Diodorus considered four modalities: possibility, impossibility, necessity, and non-necessity. These were conceived of as modal proporties or modal values of propositions, not as modal operators.
Possible is that which either is or will be true; impossible is that which is false and will not be true; necessary that which is true and will not be false; non-necessary that which either is false already or will be false.
Diodorus’ definition of possibility rules out future contingents and implies the counterintuitive thesis that only the actual is possible.
i) Every past truth is necessary ii) The impossible does not follow from the possible iii) Something is possible which neither is nor will be true
Some affinity with the arguments for logical determinism in Aristotle’s De Interpretatione 9 is likely.
Diodorus held that no linguistic expression is ambiguous. He supported this dictum by a theory of meaning based on speaker intention. Speakers generally intend to say only one thing when they speak. What is said when they speak is what they intend to say. Any discrepancy between speaker intention and listener decoding has its cause in the obscurity of what was said, not its ambiguity.
Gave a truth-functional definition of the connective ‘if…then…’
A conditional is false when and only when its antecedent is true and its consequent is false
The Philonian conditional resembles material implication, except that — since propositions were conceived of as functions of time that can have different truth-avlues at different times — it may change its truth-value over time.
Diodorus’ conditional is Diodorean-true now if and only if it is Philonian-true at all times.
Possible is that which is capable of being true by the proposition’s own nature… necessary is that which is true, and which, as far as it is in itself, is not capable of being false. Non-necessary is that which as far as it is in itself, is capable of being false, and impossibel is that which by its own nature is not capable of being true.
Systematized propositional logic inspired from Megarian logic.
Used variables where the propositions themselves were variables not terms.
Both Zeno and Cleanthes held knowledge of logic as a virtue and held it in high esteem
And the signs employed for these were ordinal numbers (the first, the second) and not letters.
TODO: It is said to be independent of Aristotle’s term logic, but verify this claim.
The subject matter of Stoic logic is the so-called sayables (lekta): they are the underlying meanings in everything we say and think., but like Frege’s senses, also subsist independently of us.
They are distinguished from spoken and written linguistic expressions: what we utter are those expressions, but what we say are the sayables.
Complete and deficient sayables. Deficient sayables, if said, make the hearer feel prompted to ask for a completion. E.g.: The sayable ‘writes’ makes the hearer prompted to enquire ‘who?’. Complete sayables, if said, do not make the hearer ask for a completion.
Assertibles (axiômata)
They include assertibles (Stoic equivalent of propositions0, imperativals, interrogatives, inquiries, exclamatives, hypotheses or suppositions, stipulations, oaths, curses, and more.
According to Stoics, when we say a complete sayable, we perform three different acts: we utter a linguistic expression; we say the sayable; and we perform a speech-act.
Truth is temporal and assertibles may change their truth-value.
Stoic principle of bivalence is hence temporalized.
Truth is introduced by example: the assertible ‘it is day’ is true when it is day, and at all other times false. This suggest some kind of deflationist view of truth, as does the fact that the Stoics identify true assertibles with facts, but define false assertibles simply as the contradictories of true ones.
Assertibles are simple or non-simple. A simple predicative assertible like ‘Dion is wakilng’ is generated from the predicate ‘is walking’, which is a deficient assertible since it elicits the question ‘who?’, together with a nominative case, which the assertible presents as falling under the predicate.
Thus there is no interchangeability of predicate and subject terms as in Aristotle; rather, predicates — but not the things that fall under them — are defined as deficient, and thus resemble propositional functions.
Some stoics took the Fregean approach that singular terms had correlated saymables, others anticipated the notion of direct reference.
Concerning indexicals, the Stoics took a simple definite assertible like ‘this one is walking’ to be true when the perso pointed at by the speaker is walking. When the thing pointed at ceases to be, so does the assertible, though the sentence used to express it remains.
A simple indefinite assertible like ‘someone is walking’ is said to be true when a correspending definite assertible is true3
Aristotelian universal affirmatives (Every A is B) were to be rephrased as conditionals:
If something is A, it is B.
Negations of simple assertibles are themselves simple assertibles.
The Stoic negation of Dion is walking is It is not the case that Dion is walking and not ‘Dion is not walking’.
The latter is analyzed in a Russelian manner as ‘Both Dion exists and not the case that Dion is walking’
There are present tense, past tense, and future tense assertibles. The temporalized principle of bivalence holds for all of them. The past tense assertible ‘Dion walked’ is true when there is at least one past time instance in which ‘Dion is walking’ was true.
Studied with diodorus
Tried to solve the Master Argument by denying that every past truth is necessary
Wrote books on paradoxes, dialectics, argument modes and predicates.
Wrote over 30 books on logic including speech act theory, sentence analysis, singular and plural expressions, types of predicates, indexicals, existential propositions, sentential connectives, negations, disjunctions, conditionals, logical consequence, valid argument forms, theory of deduction, propositional logic, modal logic, tense logic, epistemic logic, logic of suppositions, logic of imperatives, ambiguity, and logical paradoxes.
Was aware of the use-mention distinction. He seems to have held that every denoting expression is ambiguous in that it denotes both its denotation and itself
Like Philo and Diodorus, Chrysippus distinguished four modalities and considered them modal values of propositions rather than modal operators; they satisfy the same standard requirements of modal logic.
An assertible is possible when it is both capable of being true and not hindered by external things from being true. An assertible is impossible when it is [either] not capable of being true [or is capable of being true, but hindered by external things from being true]. An assertible is necessary when, being tru, it either is not capable of being false or is capable of being false, but hindered by external things from being false. An assertible is non-necessary when it is both capable of being false and not hindered by external things [from being false].
Chrysippus’ modal notions differ from Diodorus’ in that they allow for future contingents and from Philo’s in that they go beyond mere conceptual possibility.
Developed a theory of inductive inference
Cantor had disputes against Origen’s views
Roman author and Grammarian
Tried to synthesize the logic of Stoics and Aristotle/Peripatetics
Said to have introduced relational syllogism
Commentator on Aristotle
Commentator on Aristotle
Commentator on Aristotle
Commentator
Middle Ages are roughly the period between 6th century and 14th century
Medieval Foundations of the Western intellectual Tradition (1976)
Articulating Medieval Logic (2014)
Because they put emphasis on term logic
As Pope known as John XXI
Nominalistically oriented Conducted semiotic inquiries into the soc-acalled proprietates terminorum
Modality linked with time
Developed theory of modality
Took shape by 12th century, then neglected for a while until Burley in England and Buridan in France revived it in the 14th century. Was the main vehicle for semantic analysis in the middle ages.
A trend as new renaissance logic emerges is the start of sustained tension and critique between new forms of logic and that of traditional scholastic logic. Certain schools such as Oxford (in the form of individuals like Aldrich, Whately) gives support to the traditional logic, while alternatives cast in the tradition of Cartesian and Lockean modes of epistemology begins to criticize syllogisms and a mutual dialogue between them can be seen in this period.
Nicholai Ivanovich Styazhkin identifies three main branches that spun off during Renaissance. The Traditional logic that remained conserved in religious institutions, reform logic of Melanchthon and Ramus, and mystical tradition beginning from Ramón Llull.
14th Century thinkers from Oxford Merton School
Book on dialectic Contains square of oppositions
This book earned him the nickname The Calculator.
Co-founder of Protentanism
De censura veri et falsi
Replacement for the Protestant logic of Melanchthon
Known as a Scholastic Cartesian Attempted to develop a Cartesian Logic
Followed Ramus’ outline of concept, judgement, argument, and method. Had explicit distinction between comprehension and extension.
Influential discussion of definitions that was inspired byt he work of French mathematician and philosopher Blaise Pascal.
La logique, ou l’art de penser: contenant, outre les règles communies, plusieurs observations nouvelles propres à former le jugement (1662)
Tried to reinstitute the rich detail of scholastic logic
Universalia Euclidea. Accedunt ejusdem XII. Novi Syllogizandi Modi in propositionibus absolutis, cum XX. aliis in exclusivis, eâdem methodo Geometricâ demonstrates. Hagæ-Comitis (1661)
Used diagrams like Euler circles
Pharus Intellectus, sive Logica electiva Gründliche Anweisung zur Logica
Son of Jakob Thomasius
Introductio ad philosophiam aulicam, seu lineae primae libri de preduntia cogitandi et ratiocinandi (1688)
Extensively used Euler circles
700 page commentary on the Weise’s handbook of logic. These comments contain numerous diagrams.
Uses circle diagrams together with tree diagrams to represent the relation of concepts in jurisprudence.
Wolfgang Lenzen (1987)
Published a work on parallels of logic and algebra. Gave algebraic examples of categorical statements.
Logic: Or the Right Use of Reason in the Enquiry after Truth, with a Variety of rules to guard against Error, in the Affairs of Religion and Human Life, as well as in the Sciences (1725)
The Improvement of the Mind, or a Supplement to the Art of Logick: Containing a Variety of Remarks and Rules for the Attainment and Communication of useful Knowledge, in Religion, in the Sciences, and in common Life.
Founder of Methodism
Was influenced by Wolff’s exposition of Leibniz
Introduced Euler circles
Helped establish the Aberdeen Philosophical Society in 1785
An Essay on Quantity; occasioned by reading a treatise in which simple and compound rations are applied to virtue and merit
Outlines of Philosophical Education, illustrated by the Method of Teaching the Logic, or First Class of Philosophy, in the University of Glasgow (1818)
Elements of Intellectual Philosophy or an Analysis of the Powers of the Human Understanding: tending to ascertain the Principles of a Rational Logic
St. Andrews Logician
Teacher of Hegel. Developed squares to represent syllogisms.
Used squares for logic diagrams
Two volumes
Worked on creating an intensional calculus of logic in connection with the tree of Porphyry.
He introduced relative product which can be used to compose relations together to attain transitivity: X “`is a friend of“` Y “`is a friend of“` Z giving X “`is a friend of“` of Z.
He also introduce ideas like newConcept = fn :: anotherConcept where :: means a function fn applied to another concept. This is said to have possibly influenced Frege’s work.
Six Attempts at a Symbolic Method in the Theory of Reason
Intensional system with terms standing for concepts instead of individuals.
Neues Organon oder Gedanken über die Erforschung und Bezeichnung des Wahren und dessen Unterscheidung vom Irrtum und Schein (1764)
Popularly known as Neues Organon
Took an extensional standpoint
Abhandlung über die Mathematik, die allgemeine Zeichenkunst und die Verschiedenheit der Rechnungsarten (1764)
Produced a compilation of logic lectures of Kant including the logical diagrams used
Used linear diagrams
Used linear diagrams
Used triangles in his logic based on Lambert’s line diagrams
Used circle diagrams to illustrate rules of conversion
Linked Kant’s The False Subtlety of the Four Syllogistic Figures with Euler’s and Lambert’s logic
Elements of General Knowledge, Introductory to Useful Books in the Principal Branches of Literature and Science. With Lists of the Most Approved Authors; including the Best Editions of the Classics. Designed Chiefly for the Junior Students in the Universities, and the Higher Classes in Schools (1802)
Logic Made Easy; or, A Short View of the Aristotelic System of Reasoning, and its Application to Literature, Science, and the General Improvement of the Mind. Designed Chiefly for the Students of the University of Oxford (1809)
Translated Aristotle’s work and gave a nominalist defense of his logic.
Wrote a Leibnizian nonsymbolic logic in 1837
Work on theory of extension
Brother of Hermann Grassmann
Introduced German speakers to the work of Boole, De Morgan, and Jevons
Tutor of Richard Whately
Whately’s work is said to have arose from the tradition of Henry Aldrich, Isaac Watts, and John Wesley.
Credited by De Morgan as single handedly bringing about a revival in English school of logic
Richard Whately and the Revival of Syllogistic Logic in Great Britain in the Early Nineteenth Century – Calvin Jongsma (1982)
Available in Discussions on Philosophy and literature (1861)
Published posthumously.
Contains a brief section on the history of ancient logic and metaphysics.
Outline of a new system of logic, with a critical examination of Dr. Whately’s ‘Elements of Logic’ (1827)
First published work of Boole in mathematics was on analysis. Fellow of Trinity College in Cambridge called Gregory would help Boole to publish this paper in Transactions of The Cambridge Philosophical Society in 1844. This paper would go on to win the society’s gold medal thereby establishing Boole’s reputation. The paper is a work on calculus of operations.
Boole was inspired to take up work in logic after seeing a dispute between De Morgan and Hamilton in one of the periodicals. He would go on to apply this calculus of operators to logic, thereby algebraizing logic and creating the domain of algebra of logic.
Boole’s logic on deductive logic is but one half with inductive logic and probability theory being the other half of his works in 1847 and 1854.
What Boole started could be thought of as algebraizing logic rather than the algebraic structure that bears his name in modern logic. This is a way of constructing a formal system from which logical deductions could be drawn out by algebraic manipulations.
TODO: Trace if these letters of dispute between De Morgan and Hamilton is available.
His first pamphlet was called a mathematical analysis of logic, that was published on 29th October, 1847. He first sent it to Rev. Charles Graves at the Cambridge University, who approved it and is said to make some ingenious additions.
TODO: Trace what these additions where.
A detailing of this work is available here: https://www.math.uwaterloo.ca/~snburris/htdocs/MAL.pdf
A good introduction into the life of George Boole.
Michael Schoeder (1997)
Arthur Prior, 1949
Presented an extensional logic
- Created the first known variant of propositional calculus (Verify).
- Influenced C. I. Lewis in his modal logic.
- Explored pluralistic logical systems.
- Worked on the minimization problem
Contains a section on Axioms,
Peirce tried to unite the algebraic logic of Boole and quantified relational inferences of De Morgan. Peirce moved away from equational logic and tried to bring a logic of relatives.
Peirce introduced quantifiers briefly in 1870.
Schröder borrowed quantifiers from Peirces for his influential treatise on the algebra of logic which was adopted by Peano from Schröder.
Equational algebraic logic influenced by Boole and Grassmann
Some views of Russell and Russell’s logic by his contemporaries, with particular reference to Peirce
Cantor was working with calculus and the attempt to formalize the idea of inifinitesimals and infinities lead to the creation of set theory formalism. This helps with modelling logic when classes themselves have other classes inside them.
Introduced first-order predicate logic.
Predicates are described as functions, suggestive of the technique of Lambert.
Shows the dynamical aspect of the meaning of quantifiers
An axiomatic system can be thought of as a system that describes a part of the world by employing a set of models. These models can be thought of as interpretations of the system in which all the axioms are tautological.
A proposition specifies a class of models.
This is known in different names as:
Term logic can be expressed as monadic predicate calculus. It is completely decidable.
Allows predicates of higher arity in formulas, but restricts second-order quantification to unary predicates.
In first order logic, to be is ambiguous. It can stand for:
Tomato is red IsRed(tomato)
Tomato is pomodoro in Italian tomato = pomodoro
Tomato is / Tomato exists ThereExists x : x = Tomato
Tomato is a fruit ForAll(x) : Tomato(x) subsetOf Fruit(x)
First order logic cannot express all the concepts and modes of reasoning in mathematics such as:
- Equinumerosity / Equicardinality
- Infinity
Hilbert’s Program
The distinction between logical systems in which quantification is allowed over higher-order entites and first order was accomplished largely by the efforts of David Hilbert and his associates. It is expounded in Grundzüge der Theoretischen Logik.
Gödel proved that the negation of continuum hypothesis cannot be proved in ZF set theory.
Functions that are effectively calculable are called general recursive
It was assumed that descriptive completeness and deductive completeness coincide. This was a central assumption in the metalogical project of proving the consistency of arithmetic.
James Bymie Shaw, 1916
Also, published as a book
Used a higher order logic. Intended to reduce arithmetic to logic. Defined number of a class as the class of classes equinumerous with it. A number as all the classes of objects with the same cardinality. This is quoted often as said by Quine.
Following arithmetization of analysis, a lot of mathematics was shown to be linked with arithmetic.
Bernard Linsky
H. C. Kennedy
Founding member of Institute of Advanced Study
Showed how Principia Mathematica could be casted differently by taking a purely extensional view of higher order objects such as properties, relations, and classes.
Has written a survey on symbolic logic.
Proposed to extend classical logic with modalities.
Article on completeness theorem: http://citeseerx.ist.psu.edu/viewdoc/download;jsessionid=16D208C94D8F16914F1921B765820E42?doi=10.1.1.89.2460&rep=rep1&type=pdf
Proved that continuum hypothesis cannot be proved in ZF.
Contributed to the view of intuitionism as “calculus of problems”
Proved the lattice of Muchnik degrees is Brouwerian.
Demonstrated that Boolean algebra without quantifiers were inadequate.
Introduced object language vs. metalanguage distinction
Established connection between modal logic S4 and topological and metric spaces.
Presented an improved version of the completeness proof of first order logic
Formalized the distinction that all classes can mean all classes that are composable of their members or all the classes definable in a given language.
Presented a systematic theory of semantics
Begründung Der Elementaren Arithmetik Durch Die Rekurrierende Denkweise Ohne Anwendung Scheinbarer Veranderlichen Mit Unendlichem Ausdehnungsbereich (1923)
As in Model Theory
If a formal logic that the axiomatization uses is semantically complete, deductive completeness usually coincides with descriptive completeness.
If adding new elements to one of its models leads to a violation of the other axioms.
Formulated relational semantics for modal logic.
- Frames
- Kripke Semantics
Encouraged the development of noncompositional truth definitions, initially formulated in terms of a selection game.
The paradigm of using games to define truth eventually lead to the development of game-theoretic semantics.
A first order theory Eliminated comprehension principle and brought in construction of set using iterative method.
Two sets are identical if they have the same members
There exists a set with no elements. For any a and b, there exists sets with just a and be as their elements.
A set can be partitioned based on the property it has.
If S is a set, there is another set PowerSet(S) that contains all and only the subsets of S.
If S is a set of sets, then there is another UnionOfChildren(S) which contains a set with all the elements of the subsets of S.
If S is a non-empty set with mutually exclusive sets, then there is a set that contains pairs of members from each sets of S.
There is at least one set that contains an infinity number of members.
Invented dialogical logic with Kuro Lorenz. Influenced semantical tableaux method
Introduced semantical tableux method in 1955. Found out that Gentzen-type proofs could be interpreted as frustrated counter-model constructions.
A good reading list of Martin Löf’s papers is available here.
Lead to making equinumerosity, infinity, and truth being expressible in first-order language
There seems to be a link between how Kempe influenced Peirce, both influenced Royce, which ends up influencing Sheffer in arriving at his “notational relativity” programme.
C. I. Lewis was the student of Royce, whose book Post reads and becomes an aid in formulating at his linguistic approach to logic to arrive at string rewriting systems.
Chomsky learns of Post’s work via Rosenbloom’s book.
A multivolume series with scholarship in the history of logic
Jósef Maria Bocheński
Jósef Maria Bocheński
J. Lukasiewicz
Piotr Kulicki
B. Mates
G. Patzig
J. Corcoran (ed.)
Michael Frede
Michael Frede
in Ancient Philosophy (1987)
Michael Frede
in Ancient Philosophy (1987)
P. H. and E. A. De Lacy
J. Brunschwig
M. Burnyeat
in Science and Speculation (1982)
D. Sedley in Science and Speculation (1982) by J. Barnes and others (eds.)
T. S. Lee
Karlheinz Hülser
P. M. Huby
Hans Reichenbach
I. M. Bochenski
Originally published as: Formale Logik
Negation as a rotation of polygons/polyhedra. Also gives a brief survey of different kinds of logic systems and the kind of group actions implicit in their structures.
Links logic with category theory and adjointness
Ernest Nagel
Ernest Nagel (1935)
A book released in Russia where it interprets the old work in medieval logic and interprets it in the light of modern logical concepts.
A sourcebook which contains a curation of the significant papers that influenced the course of history of logic.
Leila Haaparanta
D. P. Henry
William of Sherwood
Paul of Venice
Publisher: Felix Meiner Verlag
Ivan Boh
D. Macbeth (2014)
The Kant-Eberhard-Controversy
F. A. Lange (1894)
I. Denzinger (1824)
Hamilton, W. (1860) Documents the work of Alsted on logical diagrams
Contains details on the work of Vives
Leinsle U. G. (1988)
Drobisch, M. W. (1827)
Ueberweg, F. (1857)
Logik, Mathesis universalis und allgemeine Wissenschaft: Leibniz und die Wiederentdeckung der formalen Logik
Peckhaus, V. (1997)
Bullynck, M.
H. Lu-Adler (2012)
Randolph, J.F. 1965
Der junge Leibniz I: Die wissenschaftstheoretische Problemstellung seines ersten Systementwurfs (1978)
Moll, K.
Die Quantität des Inhalts: Zu Leibnizens Erfassung des Intensionsbegriffs durch Kalküle und Diagramme (1979)
Thiel, C.
Mahnke, D. 1925
Gabriel Nuchelmans (1983)
Apparently a third part. TODO: Seek out the previous two volumes.
Schopenhauer 1913
Ueberweg 1857
Kratochwil, S.
E. VVeigelus
E. Weigelius
Stekeler-Weithofer 1986
Robert Blakely
Nolan 2002
Embodiments of Will. Anatomical and Physiological Theories of Voluntary Animal Motion from Greek Antiquity to the Latin Middle Ages, 400 B.C.–A.D. 1300
Frampton, M. 2008
Venn, J. 1880
E. Coumet 1977
Stanley Burris (2001)
Stanley Burris (2001)