I was reading IEP article about C.S Peirce where some of his logical notations for modal logic are presented, and one them seem to make a distinction between a subjective formula and an objective one. It is also indicated that he defined logics for "possible worlds". Was Peirce the first to make the distinction between objective and subjective logical formulas, true under some sort of model (or set of assumptions) but false in others?

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    Could you please reference the article and give a quote, what Peirce means is often sensitive to context. Also, what is the connection between that and possible worlds? Leibniz already introduced them, and he had medieval predecessors like Molina. – Conifold Nov 12 '20 at 18:55
  • What do you mean with "subjective logical formula" ? – Mauro ALLEGRANZA Nov 13 '20 at 7:37
  • See Charles Sanders Peirce, On the algebra of logic: A contribution to the philosophy of notation (*AmJouMath, 1885), page 186: not exactly "possible worlds" but the now standard analysis of conditional (hypothetical proposition) in terms of "state of things" (truth conditions). – Mauro ALLEGRANZA Nov 13 '20 at 7:41
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    Based on the article you referenced, Peirce seems to be using 'subjective' to mean what we would now call 'epistemically possible' which means a proposition that might be true, for a given person, for all that person knows. So, if Bob has no idea where Alice is, it is epistemically possible for Bob that Alice is in New York, but if Charlie knows that Alice is in Amsterdam, it is not epistemically possible for Charlie that Alice is in New York. Several medieval philosophers wrote about epistemic logic, but it wasn't until the 20th century that a modal approach was used. – Bumble Nov 13 '20 at 14:23
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    Peirce simply transferred the distinction from probabilities to (vaguer) possibilities. "Between 1837 and 1842 at least six mathematicians and philosophers, writing in French, English, and German, and working independently of one another, introduced distinctions between two kinds of probability. Although the grounds, contents, and implications of these distinctions differed significantly from author to author, all revolved around a philosophical distinction between "objective" and "subjective" which had emerged ca. 1840." Daston – Conifold Nov 13 '20 at 18:41

Peirce introduces objective and subjective possibilities in the context of describing gamma graphs, an extension of his diagrammatic proof system (existential graphs that are expressively equivalent to the usual predicate calculus) to modal logic. "Subjective" means what we today call epistemic (possibility), while the earlier accounts of modal logic, such as Molina's and Leibniz's, were rather aimed at what we call metaphysical possibility:

"The recto is appropriated to the representation of existential, or actual, facts, or what we choose to make believe are such. The verso is appropriated to the representation of possibilities of different kinds according to its tint, but usually to that of subjective possibilities, or subjectively possible truths. The special kind of possibility here called subjective is that which consists in ignorance. If we do not know that there are not inhabitants of Mars, it is subjectively possible that there are such beings... The verso is usually appropriated to imparting information about subjective possibilities or what may be true for aught we know." [CP 4.573-4]

As for the source of the terminology, it is not hard to guess given Peirce's close involvement with probabilistic reasoning. He uses the same (by then established) terminology for probabilities:

"What is the antecedent probability that matter should be composed of atoms? Can we take statistics of a multitude of different universes? An objective probability is the ratio of frequency of a specific to a generic event in the ordinary course of experience. Of a fact per se it is absurd to speak of objective probability. All that is attainable are subjective probabilities, or likelihoods, which express nothing but the conformity of a new suggestion to our prepossessions..." [CP 2.777]

The probabilistic terminology was established c. 1840 following its spread in the common parlance by 1830, and survives to this day, unlike the modal one, see Daston, How Probabilities Came to Be Objective and Subjective. Although Bernoulli used the terms "objective" and "subjective" for probabilities already in Ars Conjectandi (1713), it was not in the sense that emerged in the 19th century and in use today. The origins of our modern sense had a definite stamp of Kant on them, but it ultimately diverged from his as well:

"A quick survey of major dictionaries in French, English, and German reveals that the words surface again with something like their familiar modern meaning only in the 1830s -- somewhat earlier in German... , and numerous editions of the Grimm brothers' etymological dictionary traced the newer philosophical senses of both objektiv and subjektiv directly to Kant... Just how remote this usage is from our own is made clear by Kant's regular pairing of the "subjective" with the "merely empirical." Yet the usage enshrined by 1840, despite its bows in the direction of the "new philosophy," drew the distinction between an "objective" external reality independent of all minds and "subjective" internal states dependent upon individual minds."

Poisson, Bolzano, Ellis, Fries, Mill and Cournot applied the subjective/objective distinction (not all used this terminology) to probability in 1837-1842. Cournot explicitly related probabilities to possibilities when introducing it:

"The double sense of probability, which at once refers to a certain measure of our knowledge, and also to a measure of the possibility of things independently of the knowledge we have of them... [is to be disambiguated] with the epithets of subjective and objective, which were necessary in order for me to distinguish radically the two meanings of the term probability."


Aristotle? 2,500 years ago?


Yes, the implication φ → ξ is of course not true:

φ → ξ

Now, look here:

(φ → ψ) ∧ (ψ → ξ) ⊢ φ → ξ

Hey presto, the same implication, φ → ξ, now is true, given some assumptions, namely, φ → ψ and ψ → ξ.

One mistaken comment on my answer led me to edit it to replace Aristotle with Theophrastus as the first logician to use what he himself called "hypothetical syllogism". However, my initial answer saying it was Aristotle was in fact correct, even if he didn't use himself the label "hypothetical syllogism".

So, I just correct back to Aristotle.

Of course, neither Aristotle nor Theophrastus would have expressed the idea in terms of "logical formulas" that "may be true under some sort of model (or set of assumptions) but false in others", but the crucial point is that Aristotle's syllogistic was also clearly about hypothetical syllogisms (in all but name) and that hypothetical syllogisms are precisely about logical formulas, for example φ → ξ, that are true or false under some assumptions, for example φ → ψ and ψ → ξ.

I am pleased to see that the Gang of Four downvoted this answer, demonstrating that their logical expertise is at best superficial, to remain polite.

I guess my downvoters don't like the idea that logicians didn't wait for today's mathematical logic to express 2,500 years ago all the fundamentals of logic. Well, they did.

For Theophrastus and hypothetical syllogism, see https://en.wikipedia.org/wiki/Hypothetical_syllogism

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    Aristotle did not use such formulas, or any, and how does this distinguish "subjective" and "objective" ("subject" and "object" did not mean what they mean today before Kant)? – Conifold Nov 12 '20 at 20:29
  • @Conifold Read the question: "Was Peirce the first to make the distinction that logical formulas may be true under some sort of model(or set of assumptions) but false in others?" Aristotle? Yes, you're correct. I'm busy! I corrected. EDIT; No, in fact, I was correct. It was Aristotle who first gave an example of an implication true under some assumptions. – Speakpigeon Nov 14 '20 at 8:07

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