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I'm interested in applying logic to day-to-day reasoning. The problem is that formal logic seems really restrictive to limit inductive arguments to be only universal ("all swans are white"). Few things in the world are true across all instances.

My question is if it's fine to make statements like "all swans are things that are probably white". Because it also seems like this is the same thing as saying "some swans are white".

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You have asked about using the term 'probably' in this post. Yes, you can use any word at all in a proposition if it's meaningful. There are different types of logic such as syllogistic, sentential, FOPC, and modal for starters, and some are more sophisticated than others. Ultimately, when analyzing language, one chooses a logic depending on what is important. Necessity, probability, possibility, and obligation are different aspects of language related to linguistic ideas such as tense, evidentiality, conditionals, and so on, and basic logics often lack symbols to represent them.

Additionally, it seems what you are looking at is how to make logic fit natural language and ordinary practice. If you're interested in improving inductive logic, start with that article. There's a big difference between purely Aristotelian syllogisms of Logic 101 and informal logic. Conifold has pointed you towards two entries: logic and probability and generalized quantifiers. There is no shortage of logics, so another interesting variant is fuzzy logic; as well, explore the nature of concepts themselves. Traditional logics presume a nature of definition (generally necessity and sufficiency) that some considered a special case of how concepts function. Wittgenstein's passage on family resemblance moved a lot of thinkers in a different direction. Lastly, if you're interested on improving propositional structure, look at grammar. Since Chomsky's work, the notion of surface and deep structure has provoked philosophers of language to explore issues of syntax and semantics from a number of perspectives that give insight on what goes on in ordinary language use with a philosophical bent.

  • The question asks not about the divisions of logic, nor about "natural logic and ordinary practice," but instead asks whether the statements "all swans are things that are probably white" and "some swans are white" are equivalent; and too, whether the statement "all swans are things that are probably white" is acceptable. – user96931 Nov 23 '19 at 19:26
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    Thanks - as a beginner this is a very useful outline. – SingularJon Nov 23 '19 at 21:44
  • @user96931 True, but such a question is a clue that the OP is just beginning, and it's generally better to teach a man to fish than to merely provide him one. One of the fascinating things about inference is that most of the time logic isn't the formal sort, and the fact the OP has recognized so quickly how artificial formal logic is is a good sign he'll grow in leaps in bounds. But I'll amend to include a prefatory remark. – J D Nov 23 '19 at 22:01
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My question is if it's fine to make statements like "all swans are things that are probably white". Because it also seems like this is the same thing as saying "some swans are white".

All swans are things that are probably white" and "some swans are white" differ in quantity -- "all" vs "some." Moreover, the predicate-terms "things that are probably white" and "white [things]" differ greatly: "things that are probably white" denotes a thing whose quality of whiteness depends on an undefined "probability" (whose lack of definition makes this an unacceptable proposition) -- which we impute to all swans; Whereas "white [things]" denotes things that are white -- that is, we are imputing whiteness to some swans.

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    "An unacceptable proposition" according to who? In logic one is usually only interested in the logical structure and treats all predicates as if they picked out well-defined properties, whether or not it would be straightforward to decide where every entity has the property or not in practice. For example, Bertrand Russell had a whole extended analysis of how to translate "the present king of France is bald" into formal logic, but never worried about there being no exact definition of how many hairs a "bald" person can have on their head. – Hypnosifl Nov 23 '19 at 20:05
  • Right, if we defined "probably" as, say, having the quality of being a 75% chance or more, would it be a valid proposition then? – SingularJon Nov 23 '19 at 21:47
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    @SingularJon In logic, propositions are held to be true or false classically, while arguments are characterized as possibly valid and sound. IEP: Validity and Soundness. One doesn't characterize propositions as valid or not. As far as unacceptable or not, to a logician, generally any proposition which is gramatically sound is acceptable. All language has ambiguity, and lack of precision of terms or probabilities is not grounds to consider a proposition unacceptable. – J D Nov 23 '19 at 23:05
  • @JD thanks for the help - it does clear things up. – SingularJon Nov 24 '19 at 7:17
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(1) All the swans observed until today were white.

(2) Therefore, all swans are white.

This is a strong inductive reasoning ( although the conclusion is false).

When I reason like this, I'm not looking for the "probable color" of swans, but about the color of swans.

I mean I am not looking for a probability.

It seems to me that the probability is not part of the conclusion itself, but belongs to the relation between the premises and the conclusion in an inductive inference.

Trying to infer probability statements ( statements regarding the probability for a given proposition to be true) may be the job of deductive reasoning ( rather than of inductive reasoning).

If I introduce the concept of probability in the conclusion itself, the reasoning is arguably no longer an inductive one.

(1) If bought a lotery ticket 300 times.

(2) I never won.

(3) Therefore, there is a high probability me to lose the next time I by a ticket.

Suppose this reasoning is inductive. It means that it is not deductive, and therefore, not deductively valid.

But how could it be the case (1) the premises to be true and (2) the conclusion to be false?

I mean, could it be the case that (1) I've lost 100% of times I have bought a ticket and (2) there is not a high probability me to lose the next time I'll by one?

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