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Questions tagged [mathematical-logic]

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How do skeptics explain axioms not being arbitrary?

I get infinite regress but surely the axioms of ZFC or arithmetic were not so much chosen as discovered and intuited and thought about. They certainly didn't just grab whatever was around them and say ...
Ehudjd Ejeijr's user avatar
-1 votes
1 answer
34 views

Can two formulas be “valid under the same interpretations”?

Ebbinghaus 2021: A formula φ is valid (written “⊨ φ”) iff ∅ ⊨ φ. Thus, a formula is valid if and only if it holds under all interpretations. Let’s break this down: First of all, it is stated in the ...
Julius Hamilton's user avatar
4 votes
2 answers
222 views

The validity of the reasoning in Halting Theorem

Here is an example of the Halting Theorem from Wikipedia (Halting Problem) Christopher Strachey outlined a proof by contradiction that the halting problem is not solvable.The proof proceeds as ...
kouty's user avatar
  • 343
-1 votes
2 answers
71 views

What difference between the truth of a conditional* and its logical validity?

I am confused . . . Here is a remark on the "classical analysis" of the implication: On the classical analysis, logical implication is the same, not as the truth of a conditional statement, ...
Speakpigeon's user avatar
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1 vote
0 answers
32 views

Is there a limited number of 'pragmatic' logic rules?

What you have cited is a pragmatic limit, as you have not seen logic systems with more than 8 or so precepts. IF there were such a limit to precept quantity, then YES there would be a limit to the ...
Sayaman's user avatar
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1 vote
1 answer
288 views

What is the relationship between the logical operators in a logic, and how can that relationship be generalized?

One common type of logic, first-order logic, is commonly presented as having a certain collection of well-known logical operators, including: AND OR IMPLIES NOT etc. Interestingly, some of the ...
Julius Hamilton's user avatar
0 votes
1 answer
40 views

Extending the use-mention distinction to account for variables and predicates

When we talk about the use-mention distinction, often the following is said: To use an expression means to refer to its meaning, to mention an expression means to refer to the expression itself. I ...
Harshit Rajput's user avatar
1 vote
0 answers
79 views

What would be the algebraic generalization of the concept of “soundness” in mathematical logic?

Is there a corresponding/generalized concept of “soundness” as we abstract logical structures into algebraic ones? In the manner of algebraic logic: en.m.wikipedia.org/wiki/Algebraic_logic Let us say ...
Julius Hamilton's user avatar
1 vote
0 answers
53 views

Is there a formal logic that distinguishes between a priori and a posteriori truths?

Briefly, In a previous post, I explored the question of if logical systems have any way of distinguishing between true statements that are obvious or tautological, and thereby not ‘meaningful’, versus ...
Julius Hamilton's user avatar
3 votes
3 answers
453 views

Is it a problem for arithmetic or our representation (or both) that there is incompleteness?

Is this a settled (as much as it can be) philosophical area? I feel like I understand that there will always be incompleteness for a finite set of axioms trying to capture all of arithmetic. But I ...
J Kusin's user avatar
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4 votes
1 answer
104 views

Demonstrate that a term cannot be well-typed?

This problem is coming from Exercise 3.3 in Bacon's A Philosophical Introduction to Higher-order Logics. I am trying to do my due-diligence here and not skip problems, but this one stuck out to me. ...
C D's user avatar
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2 votes
1 answer
205 views

What is the difference between Aristotelian logic and mathematical logic?

I don't have much prior knowledge on this topic, but I am looking for a fundamental difference/difference between Aristotelian logic and mathematical logic. The difference, which I read previously ...
زكريا حسناوي's user avatar
17 votes
21 answers
3k views

What is a natural number?

It’s been on my mind lately. I do maths and work with them daily, but I’m not entirely sure of what they really are. I understand they are symbols at a surface level, but there is obviously more to it....
Fraser Pye's user avatar
1 vote
2 answers
112 views

Do philosophers do research in mathematical logic at the same level as mathematicians? [closed]

I am neither a philosopher nor a mathematician but I am assuming that philosophers don't know a lot of math. So, does that keep them from doing similar research in mathematical logic as mathematicians ...
user56417's user avatar
7 votes
5 answers
2k views

Is mathematics analytic or synthetic?

This question is related to another question I posted but I think it requires its own treatment since of the already wide scope of the other question i.e. Is the classical theory of concepts ...
user21312's user avatar
  • 139
3 votes
1 answer
160 views

Is the classical theory of concepts compatible with logical positivism's view on analyticity of mathematics?

Doing some work on theory of mathematical concepts and need a good framework that suits my own views. Is the classical theory of concepts, which seems to no to suffer very much when considered in ...
user21312's user avatar
  • 139
2 votes
2 answers
135 views

Can someone translate this into quantified modal logic?

So an attempt to translate- its not possible for two necessary beings to exist- in quantified modal logic. Is it correct? ¬◇∃x∃y[[□Nx ∧ □Ny] ∧ x ≠ y]
Vihan 's user avatar
  • 121
0 votes
1 answer
80 views

A technical question about the limitation of z of "jointing together" or "zus(x,y)" in Gödel Arithmetization

I am recently reading Professor Carnap's Logical Syntax of Language. In p.61 D18.1., the limitation of z is not greater than: pot [prim (sum[lng(x), lng(y)]), sum(x,y)]. Remarks: z is the series-...
Rational Reconstruction's user avatar
3 votes
4 answers
240 views

Prerequistes for mathematical logic

I have a working knowledge of calculus and linear algebra. But when I pick up books on mathematical logic (for example the ones listed in the logic study guide by Peter Smith), they often use ...
user56417's user avatar
5 votes
1 answer
78 views

Questions Regarding Tarski's Semantical Formalization of the Colloquial Usage of Truth

My question is in regard to a problem (albeit a simple one) that I ran into reading Tarski's paper "Concept of Truth in Formalized Languages". On page 159 Tarski states: (5) for all p, ‘p' ...
Max Maxman's user avatar
0 votes
0 answers
106 views

How does Barthes' ideas of Metalanguage differ from simple propositional logic?

On the wikipedia for post structuralism, In Elements of Semiology (1967), Barthes advances the concept of the metalanguage, a systematized way of talking about concepts like meaning and grammar ...
Babu's user avatar
  • 2,188
1 vote
0 answers
59 views

How does understanding of fragments differ from understanding of the whole?

Consider a person reading a mathematical proof, then each syllogism from it's antecedent maybe understood by that person, yet they may find it difficult to understand the whole proof. At times however,...
Babu's user avatar
  • 2,188
0 votes
0 answers
44 views

Are brains geometrically equivalent to three-dimensional Venn diagrams?

I had a coworker who was kind of obsessed with Christopher Langan's supposed "theory of everything," and one article of evidence he introduced was his thought that the way our eyes are ...
Kristian Berry's user avatar
2 votes
1 answer
167 views

Does every mathematical question have an unambiguous answer?

Does every mathematical question have an unambiguous answer? For example, suppose I were to assert "In the decimal expansion of pi, does there occur in at least one location a billion 1's in a ...
user107952's user avatar
  • 7,170
6 votes
1 answer
171 views

Is the overlap between Yoneda's lemma and Peirce's pragmatic maxim known?

In philosophy, particularly in ethics, the pragmatic maxim (WP) states that an object may be considered solely in terms of its effects on the surrounding context. In the special case of ethics, the ...
Corbin's user avatar
  • 1,546
-6 votes
3 answers
335 views

Can we logically derive a value for 0÷0? [closed]

I have a "proof" that 0÷0 = 2: 0÷0 = (100 - 100) ÷ (100 - 100)   = (10⋅10 - 10⋅10) ÷ (10⋅10 - 10⋅10)   = (10² - 10²) ÷ 10(10 - 10)   = (10 + 10)(10 - 10) ÷ 10(10 - 10)   = (10 + 10) ÷ 10   =...
Wenura's user avatar
  • 137
0 votes
1 answer
184 views

Philosophy books for mathematicians

Are there any books on philosophy that make relatively heavy use of math? I'm not looking for anything on formal epistemology, logic, or philosophy of math. Two examples of books that fall in the ...
user236343's user avatar
3 votes
3 answers
398 views

Does logic give us a single definitive and universal answer for comparing the odds of unlikely events?

As an amateur who has interest in logic and mathematics I've been reading about the concept of different probability perceptions. I'd like to have your opinions over the subject below. When it comes ...
Geerts's user avatar
  • 31
2 votes
2 answers
168 views

Per Mathematical Structuralism, can a pure mathematical theory have semantics that is not closed on isomorphism?

This question is the philosophical side of a question that I've recently posted to MathOverflow. Here, I'm specifically asking about the output of Mathematical Structuralism on that question that I'll ...
Zuhair's user avatar
  • 387
0 votes
0 answers
33 views

is there such a thing as non-constructive computational “proof”?

For the problems which can’t modified into a constructive proof, is there some useful notion of proving them to some computational approximation? I’m referencing: “Interpretations come at a cost: for ...
J Kusin's user avatar
  • 2,794
0 votes
1 answer
60 views

Why is conjunction interposed with intersection instead of union?

My philosophical background going into set theory was heavily laden with Kantian and neo-Kantian elements, so one of my essential premises was read off the following passage from the first Critique [...
Kristian Berry's user avatar
0 votes
5 answers
495 views

Why do many philosophers state their arguments without using mathematics or formal language?

I am an amateur lover of philosophy and a researcher in physics and computer science. When reading a book of philosophy, I always find it frustrating that philosophers are so polysemous and ambiguous ...
Light Yagmi's user avatar
8 votes
2 answers
844 views

Why aren't Kripke semantics "syntax in disguise"?

The Wikipedia article on Kripke semantics suggests that they were considered a major breakthrough in part because algebraic semantics were seen as merely "syntax in disguise". But Kripke ...
jdonland's user avatar
  • 193