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59 votes
Accepted

Three statements that contradict each other

To answer specifically the question in your last sentence, the answer is yes. The set {P, P→Q, and ¬Q} is one such set. Any two formulas from the set do not contradict, but all together they do. {P, ...
Adam Sharpe's user avatar
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35 votes

Three statements that contradict each other

Sure! This sandwich has ham This sandwich has butter This sandwich does not have ham OR it does not have butter Or less yummingly, consider (P, Q, not P OR not Q). The nice property of our sandwich-...
Julien Lopez's user avatar
12 votes

Do intutionists think the law of the excluded middle is universally, metaphysically false?

Intuitionists reject the LEM, but they don’t reject proof of negation. That is, if A can be demonstrated to imply an absurdity, then A is false to an intuitionist. See here https://ncatlab.org/nlab/...
PW_246's user avatar
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10 votes

Three statements that contradict each other

It depends on the precise definition of Contradiction. If we define it "syntactically" as a pair made of a statement and its negation, the answer is obvious. But we may define it "semantically",...
Mauro ALLEGRANZA's user avatar
9 votes

Three statements that contradict each other

This is a multi sentence version of the Liar's paradox: i) Statement iii) is false ii) Statement i) is false iii) Statement ii) is false Any two of these sentences do not contradict each other ...
Matt's user avatar
  • 207
5 votes

How can logical soundness be determined, if it is the rules of the logic itself which dictate what is true and false?

The idea of soundness has two different flavours, and perhaps you are confusing them. A logical argument can be sound if the logical is correctly applied and the premises are true. For example: All ...
Marco Ocram's user avatar
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5 votes
Accepted

Is there any formal logic system that considers tautologies to not be well-formed?

I can't find the direct example of what I thought I'd seen about A → A being disallowed in some logic system. Still, in the SEP article on relevance logic,S they do quickly bring up a subinstance of ...
Kristian Berry's user avatar
5 votes

Do intutionists think the law of the excluded middle is universally, metaphysically false?

In a nutshell, the answer to the title question is: no. See: Luitzen Egbertus Jan Brouwer, Unreliability of the logical principles (new English transl. of the original 1908): Now the principium ...
Mauro ALLEGRANZA's user avatar
4 votes

What are some logically equivalent formulations of “uniqueness”?

To me, an intuitive way to express uniqueness would be something like: Francis is Pope and anybody who is not Francis is not Pope. But, "Anybody who is not Francis is not Pope," is the ...
Bumble's user avatar
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4 votes

What are some logically equivalent formulations of “uniqueness”?

You are reading the definition in a way that is not quite right. There is no logical term for "other". A better reading is "x is P and anything that is P is x". This seems to ...
David Gudeman's user avatar
4 votes

Can reasoning be modeled as a preference relation over sets of propositions?

Instead of Can reasoning be modeled as a preference relation over sets of propositions? Are there any aspects of reasoning that can't be captured by a system like this? I'll focus on "Is there ...
ac15's user avatar
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4 votes

How does Gödel’s encoding of mathematical statements into natural numbers enable self-referential propositions?

As I have recently learned, the Gödel sentence was not originally "directly" self-referential. But first, then, here's what Gödel says about the matter in his introduction to his famous ...
Kristian Berry's user avatar
3 votes

Is there any formal logic system that considers tautologies to not be well-formed?

There's a lot here, but I'm going to give a response a shot. I'll latch on to this, and respond: guess I was wondering if the properties of the integers are kind of random, in a way. Random? No. But ...
J D's user avatar
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2 votes

How can logical soundness be determined, if it is the rules of the logic itself which dictate what is true and false?

You are probing the limits of logic, and of certainty. Both are suspect per contemporary thinking in both logic and empiricism. The Duhem-Quine Thesis is crucial in answering your question. https://...
Dcleve's user avatar
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2 votes
Accepted

Treating truth as a predicate

Your question is quite wide-ranging. To some extent, the issue of whether truth is a predicate leads back to the philosophical question, What is truth? And this in turn informs the issue of how to ...
Bumble's user avatar
  • 26k
2 votes

What are the possible ways to symbolically represent entities, within formal logic?

The sense of idea of ἄτομον, translated into Latin as individuum, that is, what we get by individuation is so primordial for us that it is uniformly an invariable constituent of thought and language (...
Tankut Beygu's user avatar
  • 2,255
2 votes

The smallest possible formal definition of FOL

With the premise that I do not have Ebbinghaus, so I cannot check what the book exactly is saying or quote from it: First, you have an “alphabet” - a collection of symbols. I do not know if this ...
Julio Di Egidio - inactive's user avatar
2 votes

Do intutionists think the law of the excluded middle is universally, metaphysically false?

In systems that do not reject the law of the excluded middle, there can be undecidable propositions, which means that the syntax is incomplete. This is based on a very bad misconception. The ...
user21820's user avatar
  • 466
1 vote

On the difference between a meta-variable and a propositional atom

Hmm, I see where you're coming from, but I think you might be overthinking this a bit. It seems you're kinda hung up on the whole meta-variable thing. Look, in propositional logic, we're dealing with ...
Epic Five's user avatar
1 vote

What are some logically equivalent formulations of “uniqueness”?

Uniqueness means different things for different mathematical objects, all related to how we view equality between objects. This is observed in your post, where you notice that two sets being equal, ...
Michael Carey's user avatar
1 vote

What are some logically equivalent formulations of “uniqueness”?

As far as I have seen, the ubiquitous way of proving “x has property P uniquely” in mathematics is by showing “if there was some other y which also had property P, it would follow necessarily that y ...
lee pappas's user avatar
1 vote

Can reasoning be modeled as a preference relation over sets of propositions?

I personally think all of the formal logic will fall into place easily if you can spend some time thinking about what you mean by "preference" and produce a more precise criteria. I prefer ...
Julius Hamilton's user avatar
1 vote
Accepted

Do you require a more expressive logic to describe a less expressive one?

So, I've been thinking about this question, and I think I'll clarify how I see it. You ask: Do you require a more expressive logic to describe a less expressive one? Yes. But I want to widen the ...
J D's user avatar
  • 27.6k
1 vote

Do you require a more expressive logic to describe a less expressive one?

consider this sentence: ¬(P → Q) ⊢ ¬Q . . . which axioms are needed in order for that string of symbols to be “true”? I initially proposed P ⊢ ¬(P → Q) → ¬Q, but this is wrong. Sorry! Edited answer: ...
Speakpigeon's user avatar
  • 7,992
1 vote

I am stuck on this homework question (Formal Logic class w/ Fitch) my proofs are messed up in the end. I need to start over, but that is what i have

Cogito ... (A = B) ∨(B = C) ∨¬(A = C) Where do you see this kinda statement? Re cogito .... There are at most 2 (whatever). ∀x∀y∀z((Gx ⋀ Gy ⋀ Gz) ⊃ (x = y) ⋁ (y = z) ⋁ (x = z)) If you want to say at ...
Hudjefa's user avatar
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1 vote

How does Gödel’s encoding of mathematical statements into natural numbers enable self-referential propositions?

You ask: How does Gödel’s encoding of mathematical statements into natural numbers enable self-referential propositions? The short answer is, it doesn't. A longer answer is to say, strictly speaking ...
Bumble's user avatar
  • 26k
1 vote
Accepted

How can logical soundness be determined, if it is the rules of the logic itself which dictate what is true and false?

The updated version of your question indicates that you are interested in the property of soundness as it applies to a formal system of logic. Soundness in this sense is a relation between a formal ...
Bumble's user avatar
  • 26k
1 vote

How can logical soundness be determined, if it is the rules of the logic itself which dictate what is true and false?

Meaning and Context When I am ten, eleven, and twelve years old, my reading comprehension teacher, Ms. Wexler, tells us to decode the meaning of words, phrases, and sentences in context. The thing ...
SystemTheory's user avatar
  • 1,909
1 vote

How can logical soundness be determined, if it is the rules of the logic itself which dictate what is true and false?

Let's untangle 'logic' and 'soundness' first: Logic works with "true" and "false" tags, and rules to manipulate them. Additionally, it's used for communication and checking of ...
sanjarcode's user avatar

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