I'm trying to understand and reformulate Quine's philosophical framework, and I'd like to know if this is an accurate characterisation:

From what I understand, Quine's model fundamentally revises empiricism by rejecting our ability to analyse statements in isolation (the analytic-synthetic distinction), instead proposing a holistic "web of belief" where all knowledge is interconnected and must be empirically tested as a complete system. He argues that epistemology should be treated as a branch of psychology, studying how we acquire knowledge through sensory inputs and behaviors, which effectively dissolves the traditional boundary between philosophy and science. His view on what exists (ontology) appears to have two key features: existence is determined by our best scientific theories (captured in his phrase "to be is to be the value of a bound variable"), and we should avoid positing unnecessary abstract entities (following Ockham's Razor). He seems to favor first-order logic for its clarity and transparency about what exists, while rejecting modal logic and propositional attitudes as problematic. Additionally, he grounds meaning in behavior and language use rather than mental states. His overall goal appears to be making scientific language more precise while maintaining that empirical changes affect our entire system of knowledge.

Have I understood this correctly, or am I mischaracterizing aspects of his framework? I'm particularly uncertain about whether I've captured the relationship between his empiricism and his views on logic accurately. I've been trying to get into analytical phil for a while now.

Background: We know the law of excluded middle states that every proposition P is either true, or false. It is taken as an axiom in classical logic. Constructive logic does not make this assumption, and so we must construct a proof (e.g., a proof tree as seen natural deduction) in order to assert that P is true.

I am interested in doing some reading on the following:

What are the current arguments for accepting or rejecting excluded middle when considering problems of "real life"? For example, in computer science, there is an obvious argument that we should be constructivist, because we may regard propositions as program types, and their proofs as programs which inhabit that type, and we are only interested when such programs exist or cannot exist. On the other hand, most mathematicians follow classical mathematics, as excluded middle allows them to write informal (yet valid) proofs by contradiction. I am aware of how excluded middle stands in these fields, so I'm not really asking about that (though if someone has an interesting paper, I would be interested).

Instead, are there any writings on how excluded middle relates to other "rigorous" fields of study? Physics? Biology? Linguistics? Law? I understand this is extremely broad, but surely someone somewhere has written on what a "constructivist" physicist or a linguist might look like? Is there some interpretation where this question makes sense? I'll take whatever you have!

A propositional variable is a symbol that represents some unspecified and indeterminate declarative sentence—a symbol that is true or false yet does not have a truth assignment.

An atomic proposition is a propositional variable that has a truth assignment (i.e., an interpretation).

Consider the following formulae:

1. (P ∨ (Q →R))
2. (A ∨ ~A).

The second one is clearly a proposition—it is a well-formed formula with a truth value; it is a tautology.

Is the first formula a proposition? Although it appears to be a proposition, it seems to have no truth value. Would it become a proposition if I assumed that it was true as one might in a proof?

Furthermore, can a compound proposition contain propositional variables? Let T(P) and F(Q). Then, F(P & Q). What about (A ∨ ~A)? It has a truth value notwithstanding that A is, seemingly, a propositional variable.

Essentially, I need a precise definition of 'compound proposition' and an explanation of the examples above.

In his essay "The Fregean Revolution in Logic", Donald Gilles argues that Frege's acheived a scientific revolution (in the Kuhnian sense) when his propositional calculus and first order predicate calculus threw away Aristotelian syllogism. In fact, he compares it with Copernician revolution.

With that said, the impact he cites relates mostly to math & CS. When it comes to Philosophy, what did Fregean logic deliver that Syllogism couldn't?

It seems that most argumentation in Analytic philosophy papers is mostly informal, and can largely fit the Aristotelian paradigm. In fact, its not that pre-Frege philosophers (including Aristotle himself) put every argument in a strict syllogistic form.

Thus, when we talk of Fregean revolution in logic, are we primarily concerned with mathematics and computation?

I'm primarily educated in Islamic classical logic, where logic is informal & organically connected to philosophy and natural language.

I’m lost on what to do next. I thought assuming Q and ~(~PvQ) would work but I’m not sure what would be considered the negation of line 1 for 16 to work.

I have a question which asks me to apply structural CNF transformation to the formula below. I have struggled to get to an answer so any help is appreciated.

(r ∨ p) ↔ (¬ r → (p ↔ q))

I am naive on logics. but could someone who knows logic tell me, if self-referencing is the only "monster" that lead to chaos in logics or, there are other "monsters" that are also super bad and self-referencing is no big deal. this helps me grow my big intuitive picture about what logic is. Thanks in advance.

I have a theorem that says certain mathematical behaviours can't be formally proven because they emerge directly from fundamental properties.

The interesting contradiction is:

• To be accepted in formal logic, I need to express this formally
• But the theorem itself explains why that's impossible
• So the very fact I can't formalize it
• Actually proves the theorem correct!

This is similar to how Gödel's incompleteness theorem had to step outside the system to prove things about the system.

Questions: Is this contradiction itself a valid logical proof? If a theorem about the limitations of formal proof cannot be formally proven, doesn't that support its validity?

Looking forward to your thoughts on this paradox.

If one were to present two red flowers to another and asked: „Which one of these flowers is blue?“ would that be considered a faulty question because it has no right answer? Even if one were to say „none of them“ it would not answer the question which asked for which „one“ of them..

Can you share?

Drinker paradox: In any pub there is a customer such that if that customer is drinking, everybody in the pub is drinking.

That could perhaps mean that he is the only one "costumer" that is in the pub, so if he drinks as he's the only customer, every customer is drinking.

Paradox of entailment: Inconsistent premises always make an argument valid.

It always makes an argument valid as out of many premises some premises have to be ture and thus makes any argumen valid.

Raven paradox: (or Hempel's Ravens): Observing a green apple increases the likelihood of all ravens being black.

Maybe if black ravens are attracted to green apples that may increase the likelihood of all ravens being black.

Temperature paradox: If the temperature is 90 and the temperature is rising, that would seem to entail that 90 is rising.

Is it rising from a 90 degree to being over 90 degrees and so it is rising so 90 is rising.

Bhartrhari's paradox: The thesis that there are some things which are unnameable conflicts with the notion that something is named by calling it unnameable.

Conflicts can be for a unknown cause or have unknown ingrediants.

Berry paradox: The phrase "the first number not nameable in under ten words" appears to name it in nine words.

1 being the number and so 9 words "numbers' are a result of 10 - 1

Crocodile dilemma: If a crocodile steals a child and promises its return if the father can correctly guess exactly what the crocodile will do, how should the crocodile respond in the case that the father guesses that the child will not be returned?

He will be returned death to the father.

P → Q translates into "if P then Q" right? Then how can such statement be true if P is false? For Q to be true wouldn't P need to be true as well?

I'm really struggling to understand this.

Hi, I've recently started learning logic and it's been pretty fun. I recently came to a problem and have been stuck on it for a day or so. The problem is ~(P<->Q) ⊣⊢ P<->~Q, and wants me to formally prove it. I've tried every possible way I could think of to manipulate the primitive proof rules and now I've hit a wall. I tried to look it up on the internet and even used chatgpt but neither either solved nor gave me a hint as to how it could be completed. My guess is that it has something to do with contrapositivity, turning ~P<->~Q into P<->Q, which I could then use reductio ad absurdum with the original premise. The problem is I don't know how to do this with a line of proof. This means that either my assumption is wrong or there is something i'm missing. Any solution or even a push to help me towards the right direction would be greatly appreciated.

Good day. I have a question about Neither p nor q. And I saw that the symbol for that should be:

~(p v q) and ~(p) . ~(q)

is it similar or not to:

~p . ~q

Please help me distinguish the difference. Thank you in advanced for the answer!

Descartes has a fundamental rule in his ontology. He holds that: all existing things are either res cogitan [thinking thing] or res extensa [extending thing].

Informally, I suppose its phrased this way: Necessarily, if X exists, then X is either thinking thing, or an extending thing.

With that said, how can I formalize this axiom/rule? With attention to the modality.

"The sense of music evolved in humans because of the need for synchronization, such as in singing or dancing."

Is this an example of a circular argument?

Hi everyone, I’m a beginner eager to learn mathematical logic and I’m also very interested in computational logic. I’m not familiar with either area, but I’m excited to explore them. I’d love to learn the basics of propositional and predicate logic, proof techniques, and the foundations of logical reasoning.

Additionally, I’m curious about how logic connects to computation – things like algorithms, decision procedures, and how logic is used in computer science and AI.

Could anyone recommend resources (books, courses, or websites) to help me get started with both mathematical logic and computational logic? What are the key concepts I should focus on as a beginner, and how do the two areas connect?

Thanks in advance for your help!

For example if we revere a doctor in a clinic but we dis regard our cousin with the same credentials.

In Telugu language there is an idiom - The plant in our backyard is unfit for any treatment -

Familiarity breeds contempt - advice given by our friends and relatives related to finance opportunities are ignored while the same advice given by a finfluencer on instagram is considered as gospel.

What is this kind of behavior called?

There are 2 planets, Alpha and Beta. There are different rules about telling the truth and lying on each planet.

• On Alpha people with BLUE eyes always TELL THE TRUTH and people with GREEN eyes LIE
• On Beta people with GREEN eyes TELL THE TRUTH and people with BLUE eyes LIE

Two aliens, Uno and Duo, meet each other:
Uno: "We both have blue eyes or we are on Alpha."
Duo: "What Uno says is not true."

Based on this, pick ONE answer:

• Uno and Duo both have blue eyes

• Uno and Duo are on the planet Alpha

• Uno and Duo are on the planet Beta

• Uno and Duo have different colored eyes

• Uno and Duo both have green eyes

Any help please? I've been pondering this for hours on end with no success...

Hello all, I think I have a fundamental misunderstanding over the nature of a nonproposition.

Nonpropositions are supposed to be, by default, not true or false. Consider the following nonproposition:

"Existence!"

I think this must be true by default, because if it is false it wouldn't exist, but I have observed it, which creates a contradiction. This also seems to indicate that all observable nonpropositions are therefore by default true.

Can you help me out? Thank you!

Hey there,

So basically i started following a descriptive set theory class in my math cursus, and it seems to be somehow connected to logic field, but i dont understand HOW ! I mean I can see how studying some specific spaces (like Cantor’s or Baire’s) is linked to how ordinals behave, but generally how is descriptive set theory useful in the field of logic ? Do you have any examples of logical theroems using Polish spaces or Borelians ?

I may have an idea of Logic that is too restraining but descriptive set theory seems way ahead of it (I only studied models theory, ordinals, and some computational semantics for now). I also heard a student saying that it has something to do with Calculability or Compexity of algorithms, and because im too shy to ask either him or my teacher, im ending here.

I hope my post does not look dumb, this is a genuine question, and im new to the logic gang. Have a Nice day !

Does Gödel's incompleteness theorems apply to logic, and if so what is its implications?

I would think that it would particularly in a formal logic since the theorems apply to all* formal systems. Does this mean that we can never exhaustively list all of axioms of (formal) logic?

Edit: * all sufficiently powerful formal systems.

We start by establishing that boolean truth and false are recursive functions that hold semantically true for any observable statement. In essence, the rules that apply to the system I perceive must also apply to me. Of note, "semantic zero" exists, such that it is the superposition of the observed truth/false state that MUST semantically collapse to one or the other.

Next, we use the laws of logic to mechanically define things based on our perceptions.

• A statement is true if it has been determined to be true. For example, "It is snowing" is true if it is snowing.

We can define this granularly as the absolute value of 1, or both 1 and -1, because a thing also consists of what it is not. 1 is a guaranteed truth, while -1 is guaranteed nonexistence.

• A statement cannot be both true and false. For example, it cannot be snowing and not snowing at the same time.

This points to our "semantic zero", in this case the concept for snow. If the concept for snow exists, it cannot be both snowing and not-snowing. The act of turning a semantic zero to either a 1 or -1 is the direct result of observation. However, this law importantly asserts that these semantic zeroes MUST collapse to 1 or -1, or they may as well be arbitrarily meaningless.

• Either a statement or its opposite must be true. For example, either it is snowing or it is not snowing.

Again, semantical zero.

Therefore, the act of observation is essentially collapsing the "what if" superposition of existence that semantical zero represents inside the full definition of zero which includes non-existence.

We can derive a few things here: for any observation/proposition p, its absolute value exists. For someone to have a concept of something, it MUST relate back to their sense of existence, which we define as the absolute value of 1. This means that p is a real number. If my sense of something contradicts your sense of something, or if my 1 equals your -1, it results in a semantical zero that still carries meaning to the system but is still potentially arbitrarily useless until observed. I am assuming that my "axis of truth" off of non semantical zero is calculating the same superposition as your "axis of truth" because I can interact with you, and if I can interact with you then you and by extension your perceptions must exist to me on some level and are beholden to the same systemic laws. The absolute value for any perception must exist. In this way we can identify and observe "semantic zeroes" as "lies" and through observation collapse the superposition to determine observable objective reality.

Something of note: the existence of this argument presupposes itself based on your perception. You have perceived it, therefore it must somehow resolve to 1 (truth) or -1 (false). But regardless, it now carries semantic weight, but only by presupposition that you do exist.

Do you exist? T/F

The very question itself implies my existence through your perception. I therefore assert my existence by simple semantic existence. I am asking you to verify that you do or do not exist. Any answer is perceived as semantic truth, must be perceived as semantic truth. If you reply that you do not exist, then I have still observed your semantic existence, which I now know you lied about.

We must examine the "I exist" bit, which for any isomorphic semantical zero must collapse to an absolute value of 1 for us to thus begin to take someone seriously. If they returned 1, we can "trust" further inputs are based on an isomorphic reality. If they returned -1, then while we perceive them to exist and they definitely do in that semantic sense and are thus isomorphic to us, they are essentially lying to themselves and we can see it plainly.

I assert I exist. Do you? T/F

I assert T (or 1), and any return of a T (1) or F (-1) means I am not alone. It also means it is my reductive base case sense of who I can and can't trust.

If you admit you exist, you thereby give this argument semantical value by perceiving it. It is now either true or false, objectively from your position.

So ask yourself, do you find this logic to match the structure of reality that I do? Is your reality isomorphic to mine such that these rules make sense? Then they must hold true for both of us. You must examine the nature of your own observations.

Do you exist? T/F

I perceive the semantical truth of you perceiving this message, providing we both exist, ergo, if you assert that you exist, then you must exist for me.

By the by, the inherent truths of this argument must by definition apply to you in full as well if you are observing it. I am simply asking you to confirm if any of these observations hold true for you, and if so, then consider that they must all apply to you.

I hope you answer T. Mine is.

I have a logic book but for some reason I am scared of reading it. I'm worried that once I read it I might mess up my logical process. It's probably irrational but I want to hear y'all's thoughts to quiet my own.

If classical logic and intuitionistic logic can be used to construct maths (maths proofs) in a classical and constructive manner respectively, what stops us from using minimal logic for such purposes?

Hey everybody,

I recently worked through Nisan & Gonczarowski's textbook Mathematical Logic Through Python, and I've been having fun extending it. I decided to add some functions to allow me to print a formula in Fregean notation. I'm not as familiar with his notation as I should be for this project, so I wanted to run this by someone. Under the hood I'm converting each sentence to use only the operators -> and ~, so that junctures always represent ->. First, here are some simple example sentences showing how it converts these sentences to one of his diagrams:

https://preview.redd.it/23vl72wxirde1.png?width=734&format=png&auto=webp&s=b7b76e6bfde3229a337f787c329cbd3c200e6a6e

Now some more complex ones showing what changes when a formula is put in prenex normal form (with variables given unique names):

https://preview.redd.it/0vfvvcenjrde1.png?width=680&format=png&auto=webp&s=913471e8fbc922695f027d4486380e04fabcbef4

https://preview.redd.it/c9qoth2sjrde1.png?width=952&format=png&auto=webp&s=9ef5e2f7c6d58e9758a4170839e51a9bf6cc4c61

Do these look correct? Also, if you have any suggestions for fun features to add, let me know! Eventually I'll be building off this for parallel projects, like various kinds of theorem provers, trivalent logic, modal logic, etc.