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Questions tagged [formal-logic]

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Is there an interesting relationship between formula formation rules and the rules of sequent calculus?

I am happily proceeding to chapter 4 of Ebbinghaus et al.’s Mathematical Logic and able to ask a new range of clarifying questions on first-order logic. A first-order theory, regardless of its ...
Julius Hamilton's user avatar
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An Analogy Between The Goal of the Tractatus and Formal Axiomatic Systems

After struggling with a few sections of the Tractatus, as well as the explanations of said sections is Monk's How to Read Wittgenstein and Glock's A Wittgenstein's Dictionary, I've come to a certain ...
Sam's user avatar
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Groupings of set theoretic axioms, or even “algebras of axioms”

I want to understand how to group the axioms of a set theory to study the effect that each axiom has in relation to the others. Here’s what I mean: First of all, assume “a set theory” is not a well-...
Julius Hamilton's user avatar
3 votes
1 answer
71 views

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

In all of the established propositional logics that I’m aware of, a propositional atom is treated as a meta-variable. In certain first-order proof systems, this does not hold for those same logics ...
PW_246's user avatar
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3 votes
4 answers
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What are some logically equivalent formulations of “uniqueness”?

A monoid is a mathematical structure with an associative law of composition and an identity element. It can be proven that if an element of a monoid has an inverse, then the inverse is unique: Assume ...
Julius Hamilton's user avatar
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3 answers
217 views

The smallest possible formal definition of FOL

I find the common presentation of first order logic somewhat confusing. I feel that I often don’t understand why we need the exact terms and concepts we do. My current recapitulation of “standard FOL” ...
Julius Hamilton's user avatar
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Is B(p) V B(~p) an instance of LEM in doxastic logic?

So in classical logic either p is T or p is F. But is it same in doxastic logic, ie, is B(p) V B(~p) an instance of LEM? And the second issue, is it equivalent to B(p) V ~B(p)?
Vihan 's user avatar
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Do intutionists think the law of the excluded middle is universally, metaphysically false?

The law of the excluded middle (LEM) is that every well-formed formula of a sound logical system is either true or false. In systems that do not reject the law of the excluded middle, there can be ...
Julius Hamilton's user avatar
6 votes
3 answers
329 views

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

So the idea is to model reasoning as a preference relation over sets of propositions. Given sets of propositions S1 and S2, we might have the relation S1 < S2, which we can read as "S2 is ...
causative's user avatar
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2 answers
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Do you require a more expressive logic to describe a less expressive one?

If we consider this sentence: ¬(P → Q) ⊢ ¬Q as a purely symbolic calculus, I would like to explore some kind of “reverse mathematics” where the question is, which axioms are needed in order for that ...
Julius Hamilton's user avatar
2 votes
1 answer
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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

I think my main issue is understanding if I am translating the negation of the statement correctly. I feel as if it is ultimately not logical when negated. Can someone please translate the first line ...
mechesad24's user avatar
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1 answer
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What are the possible ways to symbolically represent entities, within formal logic?

What are the different solutions proposed in the academic literature to represents symbolically individual entities within formal logic expressions? One solution I am aware of is to use Latin letters. ...
Speakpigeon's user avatar
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6 votes
2 answers
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How does Gödel’s encoding of mathematical statements into natural numbers enable self-referential propositions?

As part of his proof for the first incompleteness theorem, Gödel encoded mathematical expressions into unique numbers. These were used to construct statements exhibiting self-referentiality, such as ...
edelex's user avatar
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2 answers
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Treating truth as a predicate

It is interesting to me that in some conventions of logic I have seen (generally, common ones), the form of logical language is designed to make “truth” implicit. For example, merely to write: P(x) is ...
Julius Hamilton's user avatar
3 votes
8 answers
744 views

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 sounds conceptually intuitive. Logic commonly has a syntax and a semantics. The syntax is a set of symbols with formation rules for creating new expressions from currently ...
Julius Hamilton's user avatar
0 votes
1 answer
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Deductive systems which deduce the structure of logics

“Logics” are frequently defined as having both a “syntax” and a “semantics”. For example, first-order logic is a deductive system or formal language with an alphabet (collection of symbols) and ...
Julius Hamilton's user avatar
4 votes
4 answers
423 views

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

I have been thinking about some pretty speculative material that has been hard to wrap my mind around, but I feel like there is more to discover in this train of thought. I can't remember exactly how ...
Julius Hamilton's user avatar
35 votes
6 answers
10k views

Three statements that contradict each other

In formal logic, it seems that a contradiction only arises between two statements. Is it possible to have a set of three statements that together are a contradiction, but where any two of the ...
Marc Bacvanski's user avatar