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Questions tagged [formal-theory]

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Why is there no “Mario Bunge” entry in the SEP? Where could I find a valuable presentation of Mario Bunge's philosophy?

I only have a rudimentary knowledge of Mario Bunge's philosophy. But what I've read in the Ontology part of his Treatise of basic philosophy seems much promising. I'm astonished that Bunge is not ...
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2answers
79 views

Formal proof errors

I am trying to prove A ^ B from the premises shown in the screenshot. As you can see in the screenshot I am struggling with the second sub-sub proof. Do you have recommendations for how to continue/...
2
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1answer
108 views

Why is first-order logic defined as a collection of formal systems?

I think I understand what a formal system is and what formal languages are. But I have trouble grasping why first-order logic is referred to as a collection of formal systems whereas propositional ...
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1answer
117 views

What does Tarski mean when he says that truth is a property of sentences?

A fundamental statement of Tarski's Theory of Truth is that truth is a property of sentences. What does this statement mean? What kind of Truth is it referring to? What is the formal definition of '...
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4answers
197 views

Is the phenomenom of “subjective consciousness” or “qualia” formally captured by any state-of-the-art Theoretical Model in Physics?

Is the phenomenon of "subjective consciousness" or "qualia" formally captured or defined by any state-of-the-art Theoretical Model in Physics? If so, can you share a brief summary of such ...
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4answers
2k views

Difference between logic and mathematics

I’ve read the article in the SEP about the philosophy of mathematics. I believe I follow most of it. However, I am a bit puzzled by something that may be due to some basic misunderstanding on my part....
2
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1answer
100 views

Formal epistemology starting point

After an introductory course in logic (propositional and predicate calculus) and the article on the SEP, where and how should one start studying formal epistemology?
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1answer
478 views

Is it valid to prove the axioms of a system from themselves? How does it square with Gödel's incompleteness?

I recently asked whether the axioms are tautologies, and got comments that seemed to me highly suspicious. Namely, that you can always prove an axiom from itself, that you can trivially say A ...
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0answers
97 views

What is the difference between Herbrand Logic and Relational Logic or Predicate Logic?

I am learning a course from Stanford University, and it introduces the notion of Herbrand Logic. However in Wikipedia I cannot find a definition specifically for "Herbrand Logic", only for Herbrand ...
6
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3answers
775 views

Can you list examples of problems that can not be solved within a formal system but human beings have solved through construction or creativity?

This kind of problem is mentioned in a book I have read, but the book did not give a concrete example. If any such problem existed, this might help me understand human creativity. I think it would ...
3
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2answers
312 views

Formal proof : predicate logic

This is what I need to prove formally: 1.∃x Cube(x) ∧ Small(d) . . . . Goal :∃x (Cube(x) ∧ Small(d)) I have already tried different ways, but I still can't prove the goal. 1. ∃x Cube(x) ∧ Small(d) ...
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4answers
378 views

In what sense is set theory the “meta theory” of analysis?

Here, Terence Tao said: I deliberately chose not to be excessively formal with regards to the set-theoretic foundations of mathematics here, which I am regarding as part of the metatheory of ...
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4answers
331 views

Is there (or does something exist that is close to) a theory of arguments?

I'm looking for any extensive work on a framework for "arguments", that works something along these lines: When two parties are debating, they are making assertions on a particular domain, D. Those ...
0
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1answer
96 views

Derivation of the law of the conditional using the disjunct form

How can we prove that P ⊃ Q is logicaly equivalent to ¬P ∨ Q using the laws of derivation? Thank you very much in advance!
9
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5answers
2k views

What are computable numbers, and what is their philosophical significance?

What are Computable Numbers? Is computability (or non-computability) some sort of technology-dependent characteristic of numbers (via e.g. Turing Machines)? What are the philosophical implications or ...
6
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4answers
800 views

Is there a way to avoid Gödel's incompleteness affecting mathematics as a whole?

I have been thinking about Gödel's incompleteness theorems and their ramifications for the whole of mathematics. In this question I assume some fixed formal system F expressive enough for the ...
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3answers
248 views

What is(are) the importance(s) of formal reasoning

In Mathematics, we as an undergraduate are exposed for the first time (at least for me it was the case) to 'rigor'. For example, in Real Analysis classes we often use logical quantifiers in our ...
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3answers
653 views

What are the “undefinable numbers” in real analysis and philosophy?

What if any important results in real analysis make use of the notion of an "undefinable" real number? (Whatever "important" may mean to the reader.) Or is it used more in the philosophy of ...
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4answers
152 views

Does Philosophy have a Good Answer to Personal Inadequacy?

Some of the answers and comments to this question got me thinking about the problem of personal inadequacy. By this I mean, people seem to regularly hold values/ideals/morals beyond anything they ...
9
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1answer
329 views

Where did Gödel write that first-order logic is the “true” logic?

In "On How Logic Became First-Order" Matti Eklund writes (p. 2/148): It appears to be widely held today that arguments from Skolem and Kurt Gödel, both alleged proponents of the thesis that ...
6
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1answer
399 views

Was there a Kantian influence on Hilbert's formalist programme?

In this paper by Cassou-Nogues which is on an aspect of the mathematical philosophy of Cavailles he quotes the mathematician Hilbert (a colloborator of Einstein in Gottingen) ...We find ourselves ...
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2answers
149 views

Prove a logical formula is equivalent to the contradiction if and only if the set it describes is empty

Let ψ be a well-formed-formula (wff). Prove that (ψ ≡ ⊥) ⇔ {x:ψ(x)}=Ø that is, the formula ψ is a contradiction if and only if the set it describes has no members. Note This question is not about ...
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3answers
1k views

Are there rules for dealing with self-reference “paradoxes” in logic?

My favorite paradox that leads to an endless regress, and also leads to a question: The sentence after this is true. The sentence before this is false. When contradictions appear in proofs, ...
2
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3answers
679 views

What is the minimum number of axioms required for a system of axioms?

What is the minimum number of axioms you need, apart from definitions and usage of the notation, such that you have a system that does not contradict itself? I would just think that the answer is ...
2
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1answer
81 views

Transitive Incompleteness of Logical WFF's Due to Godel's Incompleteness Theorem

If a set of theorems, or wff's, are used in conjunction with one another, does this have an impact on their completeness in terms of soundness? For example, I have five theorems of logic, or well-...
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2answers
606 views

How does abstraction/generalization in mathematics fit into inductive reasoning?

I have a question about the nature of generalization and abstraction. Human reasoning is commonly split up into two categories: deductive and inductive reasoning. Are all instances of generalization ...
3
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2answers
919 views

Definition of concept

What definition does contemporary analytic philosophy give of 'concept'? And what is the difference between a concept of something and a conception of something? Then what's the difference between ...
7
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4answers
286 views

Logic as a (bad) model language

Can someone name the most well known philosophers to explicitly put forward an idea along the lines that formal systems can only be used descriptively, not prescriptively - that they're just a model, ...
2
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1answer
267 views

What are the ramifications of the limitations of ZFC set theory?

In the Wikipedia article on Zermelo-Fraenkel set theory says that the theory sets out to formalize a notion of sets such that "all entities in the universe of discourse are such sets." It goes on to ...
9
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2answers
356 views

How does “higher-order logic” differ from “normal” (first order?) predicate logic?

How does “higher-order logic” differ from “normal” predicate logic? I assume the latter is consistently called ”first order logic”. So where are the differences between these? What kinds of statements ...
32
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3answers
3k views

Is First Order Logic (FOL) the only fundamental logic?

I'm far from being an expert in the field of mathematical logic, but I've been reading about the academic work invested in the foundations of mathematics, both in a historical and objetive sense; and ...
7
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3answers
271 views

Is there a formal presentation of Maturana's “Autopoiesis”

I have been reading through "Autopoiesis and Cognition" by Humberto R. Maturana and Francisco J. Varela. One of their goals in defining autopoiesis and the supporting concepts of simple and composite ...
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10answers
2k views

Is it possible to scientifically determine good and evil?

Sam Harris has argued on many occasions - the earliest of which I'm aware of being in his book, The End of Faith, as well as later on in The Moral Landscape - that it is (at least theoretically) ...
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4answers
11k views

What are the philosophical implications of Gödel's First Incompleteness Theorem?

Gödel's First Incompleteness Theorem states Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any ...