20

Because there was a calculus for one-place predicates, Aristotle's syllogistic, roughly equivalent to monadic predicate calculus. Aristotle does discuss "relatives" in Categories, which refer to multi-place relations, or rather to objects entering them. What will later be called oblique syllogisms involving relatives is mentioned in passing in ...


14

You can see: Irving Anellis, The Genesis of the Truth-Table Device (2004) as well as: Irving Anellis, Peirce's Truth-functional Analysis and the Origin of the Truth Table (2012). Before Bertrand Russell (Harvard logic course: 1914) and Ludwig Wittgenstein (Russell and Wittgenstein's manuscript dated 1912; see also: Tractatus (1921), 4.31 and 4.442 for ...


10

To put it in simple words we have to describe in a couple of words the project of Principia Mathematica, which Russell inherited from Frege: reconstructing mathematics from logic alone. For a broader context see What is the philosophical ground for distinguishing logic and mathematics? Frege himself could not complete the project because Russell discovered ...


7

An n-ary relation gives rise to parameterized unary predicates if one fixes n-1 arguments. Wilfrid Hodges argues that this is what logicians did before the nineteenth century. (There may be other works of his that better explain this.) More concretely, they would re-write the relevant statements by using natural language reasoning so that all relations are ...


6

"Trial and error" applied not to a "reasoning process" but to medical practice, and the name of the practice was derived from Greek ἐμπειρία, experience. The inspiration for the approach apparently came from Greek Pyrrhonism, which recommended permanently suspending judgment on how things are in their own nature. This is what Pyrrho of Elis (c. 360 – 270 BC) ...


5

While it’s not quite a perfect parallel, a related concern about decision problems had been phrased some years before Gödel’s work: see https://en.m.wikipedia.org/wiki/Entscheidungsproblem . David Hilbert was pretty convinced as to the decidability of arithmetic prior to the Incompleteness/Incomputability proofs, but was aware that the problem remained open.


4

Husserl gave an insightful philosophical analysis of the change that occurred in mathematics over the course of 19th and early 20th century, the change some aspects of which he himself foresaw and promoted as a new way for mathematics (and science) in Logical Investigations (1900). Here is a surmise from Formal and Transcendental Logic (1929) §§29-30: "...


4

This is something of a chicken and egg problem. In a sense, the circularity is inherent in the very nature of logic. How are we to justify the rules of good reasoning if not by appealing to their meaning? And how are we to make sense of such appeals if not by following the rules of good reasoning? The crisp semantic/syntactic distinction is recent. It was ...


4

Gareth Evans is arguing that Aristotelean logic is closer to natural language usage and as such introduces fewer unfamiliar logical devices and has fewer counterintuitive features. This is true, but the vast majority of logicians consider this to be a price worth paying to have a much more powerful and expressive logic. Natural languages such as English have ...


3

You wrote... But how did humans develop formal systems and the notion of syntactic consequence in the first place? Wouldn't they have had to develop such systems based on what semantic consequences they agreed with? ... and you are entirely correct: As has been pointed out in the comments, the study of semantic consequence lead to the notion of syntactic ...


3

The formula is used into the third part of Begriffsschrift (1879), regarding the General theory of sequences, not translated in modern symbols by Mendelsohn. You have to look at: G.Landini, Frege’s Notations: What They Are and How They Mean (2012). The basic component of the complex formula is the "gamma-on-beta of f(x,y)" symbol. It is defined in §...


2

The Greeks understood the process of trial and error. You try one way, then you try another. But there is no evidence of which I'm aware that the Greeks elevated trial and error into a methodology of experimentation - a 'reasoning process' explicitly flagged as a proper part of valid scientific inquiry - such as we find in Bacon and later thinkers. In ...


2

The type-token and universal-particular distinctions are probably isomorphic in meaning, but they are used in different domains. Type-token is used in linguistic problems like "They drive the same car". This statement is ambiguous because we are unsure if they drive the instance of a car (token) or model of car (type). I don't see any issues with calling ...


2

The Categories, chap.7 /On Relatives/ contains a remarkable discussion in just a few pages and its author, supposedly Aristotle, might well be the first to have 'studied' asymmetric relations. So Aristotle distinguishes contrariety and reciprocation, adding further consideration on simultaneity. All relatives are 'reciprocated', but no all of them have '...


2

Before Frege, axiomatic systems were not a focus of philosophy, and Goedel is pursuing the immediate upshot of Frege's failure. So, in some sense, no. Nobody cared. Mathematics was grounded in some internal, perfect mental reality and not really based on axioms. Axioms just helped keep things clear. Paradoxes abound throughout the history of philosophy. ...


1

See Charles Sanders Peirce, Collected Papers : Volume 4. The Simplest Mathematics (1933), page 13: [4.12] A Boolian Algebra with One Constant [untitled paper c.1880] Every logical notation hitherto proposed has un unnecessary number of signs. It is by means of this excess that the calculus is rendered easy to use [...]; at the same time, the number of ...


1

See Evert Willem Beth's Aspect of Modern Logic (Springer, original ed.1967), page 10: The argument [above] has a particular property: both premisses are true, but the conclusion is false. We say that we, by a substitution of the terms [...] obtain a counterexample to judge the soundness [read: validity] of the argument. Making use of the concept ...


1

Euclid's Proposition 20 in Book IX of the Elements claim Prime numbers are more than any assigned multitude of prime numbers. David E. Joyce provides the following outline of the proof: Suppose that there are n primes, a1, a2, ..., an. Euclid, as usual, takes an specific small number, n = 3, of primes to illustrate the general case. Let m be the least ...


1

For sure, we can find it in Aristotle : In the proofs for imperfect deductions [the syllogistic figures different from the fist one], Aristotle says that he “reduces” (anagein) each case to one of the perfect forms [the first figure] and that they are thereby “completed” or “perfected”. These completions are either probative (deiktikos: a modern translation ...


1

William Rowan Hamilton's discovery of quaternions in the 19th century may be the first studied non-commutative relations. Here is Wikipedia: In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional ...


1

In the maximally formal sense logics are defined with only one connective and negation. Most often the connective is either "and" or "implies". The reason this is allowed is because all of the other connectives can be constructed out of a series of negations and the chosen connective, you can think of the other connectives as short hand. ...


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