What terms should I use to request (more verbal or informal) explanations like the answers to these questions tagged 'intuition'?

I ask because the following comment implies that 'intuition' may not be the correct term:

On this and several other questions, when you say I sense a deeper, directer, more intuitive explanation, I think you might be erring in what you're seeking equivalences between and intuitions about. Many would argue the implication operator as used in philosophy is not intuitive (and does not always agree with our intuitions about "if") but rather works definitionally, but you're specifically asking for answers that don't use the definitions (while keeping the symbols and inferences that follow from those).

PS: I could not find such explanations here on Philosophy SE, but I found the following 3 that start not with any formal definitions, but instead with simpler notions: https://math.stackexchange.com/q/30437/53259, https://math.stackexchange.com/a/261371/53259, https://math.stackexchange.com/a/129128/53259.


2 Answers 2


On a certain level, it's definitely true that the origin of formal logic is an attempt to formalize certain intuitions. But after the process began, it's basically become a set of rules that dictate a certain set of expectations and outcomes.

Thinking of sentential deductive logic, the basic idea is that if we begin with (a) bivalence, (b) Aristotle's 3 laws (identity, non-contradiction, and excluded middle) and (c) a limited set of operators defined by their truth functions, then we can provide arguments that are valid.

And this has (for those who accept it) a very valuable feature (truth-preserving). When an argument is valid, we no longer need to argue about the legitimacy of the structure of the argument, but can instead merely focus on the truth or falsity of the claims in question.

This system and its cousins and children borrow words from their intuitive origins but aren't actually stuck on those meanings or uses.


You can pick up to three from the beef, the pork, and the fish.

Here, "and" means or in logic. (at least in the most common meaning for the English sentence.)

You can have soup or salad with your main course.

Here, "or" means exclusive or in logic.

"If" , "Only if", etc. are even more distant at times from their natural counterparts.

Often they will coincide with natural usage, but properly translating English to formal logic is precisely correctly understand the truth table the English would match and picking the operator that does the same thing (this is one point for failure when moving from natural language to formal argument).

For instance looking at the "beef, pork, or fish" one, the best way to translate it is to understand which possibilities would make the sentence true and which would make the sentence false, and then pick the logical symbolization that expresses that most closely.

Given the shift from intuitional moorings, it's probably better to think of these words in formal logic (and sometimes more broadly in the hands of philosophers) as having the refined meanings they take on in the development of this logic.

To give an analogy, in the game of chess, the words "King", "Queen", "Rook" / "Castle", "Bishop", "Knight" / "Horsey", and "Pawn" are used. Presumably, at some point, there was some sort of connection between these words and their real world analogues.

But whatever that connection is, knowing about real world queens does not enable us to intuit anything about the piece called "Queen" and knowing about the piece called "Queen" doesn't give us intuitions about real world queens.

Or to give a different analogy, the words in logic and the words in English are semi-"false friends." For example, the word "Gift" in English means a present you give someone whereas the word "Gift" in German means a poison you give someone. The meanings are linked, but knowing the one doesn't give definitive insight into the other.

  • +1. Thank you. About your last paragraph, because I am interested in linguistics and etymology, I can understand that for someone learning 'Gift' in German, knowing the one doesn't give definitive insight into the other; but for this German learner, knowledge of their being cognates can aid memory and deepen their comprehension of both nouns in both languages. In general, 'false friends' (of languages in the same language family) do share something not obvious, but become obvious once the semantic shift is understood. Does this influence your answer?
    – user8572
    Aug 11, 2016 at 3:33

What is intuitive for one person is not always intuitive for another, so permit me to make a guess at what you consider to be "intuition," and we'll see how close I am.

Much of philosophy, and indeed all of learning, is tied up in symbols words and phrases, and the manipulation thereof. Linguistically we call these the "syntax" of our language. Syntax describes what the symbols are and how they are allowed to be manipulated. This is contrasted with semantics, which is the study of the meaning of these symbols.

Sometimes we just understand the manipulation of these symbols at an intuitive level. For instance, most of us on this site can manipulate English nouns to make them plural at an intuitive level. We rarely think "The plural of 'cat' is 'cats' because most plural nouns are a concatenation of the singular noun plus the letter 's'." That statement is true, but most of us would simply say "The plural of 'cat' is 'cats'" because one cat is a "cat" and two or more cats is "cats." We have some intuitive sense of what it means to have many of something, and we intuitively understand what the correct word for it is.

This works great until you come across a word like "octopus," whose plural is not obvious (some say "octopi," others say "octopuses," others say "octopodes"). That's when we break out the "rules of grammar" and point out that "octopus" actually borrows from the Greek "oktopous," not Latin, so it should not be given the 2nd declension nominative plural ending.

But what just happened there? I suddenly shifted gears and started talking about syntax -- the construction of 2nd declension nouns in Latin. In Latin, singular nouns ending in -us are 2nd declension nouns, and the nominative plural of those end in -i. What I just described was syntactical. You don't have to know what it means for something to be in the 2nd declension. You don't have to know that the 2nd declension is most commonly used for masculine things. You can simply apply the rule.

This is where I think my linguistic argument starts to tie into your concept of "intuition." You aren't looking for the syntax of logic, you're looking for the semantics of it. You're looking for the meaning of the symbol, not the syntactic manipulations that are allowed around it.

And therein lies the rub which virmaior wrote to. The meaning of p→q is ¬p ∨ q. Its a definition of the language. It has no deeper meaning, any more than the made up word "blartlfilthygong" has a deeper meaning.

But there's a bit more to it. Individual people do apply meaning to p→q. And it has generally been found that philosophers who are aware of the definition of it as ¬p ∨ q generally come to agree on the meaning of p→q enough for that symbol to facilitate communication.

Accordingly, what I believe you are after is two things:

  • How can I arrive at the agreed upon meaning of this symbol?
  • Why is this meaning so important that philosophers felt the need to give it a symbol?

I would argue your approach to the former, starting a question with "I pursue only intuition; please do not answer with formal proofs or Truth Tables." is tricky because the syntax is a major feature used to capture the meaning. We use syntax in this way because there is no one way to arrive at an intuitive sense of the meaning of the word. I program computers; there are plenty of arguments which make intuitive sense to me which would not make intuitive sense to an artist. In person, we can tailor our discussion of the symbols to fit what we know about the person we are talking to. Lacking knowledge of the person, the best we can do is fall back on syntax. How one can arrive at a meaning is simply out of scope in many cases, because everyone does it their own way.

The latter half, however, may prove more valuable in a forum such as this. There are typically many answers for why certain meanings resonated better than others. Again, stealing from virmaior's answer, if you are considering bivalence, Aristotle's 3 laws, and a set of operators defined by their truth functions, meanings such as "implication" naturally resonate and are worth capturing in a symbol such as . In fact, in this case, it resonates so strongly that propositional logic has two related symbols for this: and . The former is fully defined within propositional logic via the definition above, while the latter is considered a "meta symbol" whose meaning is not fully defined within propositional logic. The meaning of is far more subtle, and one way to capture this meaning is to look how it has been used historically, and why philosophers felt it was needed.

I think the latter concept may be something which can be explored in a setting such as StackExchange, because the question inherently has a historical context to it, and StackExchange is very good at topics that are tightly tied to history.

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