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Results tagged with lo.logic
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user 3154
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
7
votes
How much of the current logic is about syntax?
We don't know how to abstract away from syntax in proof theory. If we say there are three main branches in proof theory:
Axiomatics seem to be necessarily syntactic: formulae are what it is about;
…
- 2,283
5
votes
What does it mean for a mathematical statement to be true?
Both the optimistic view that all true mathematical statements can be proven and its denial are respectable positions in the philosophy of mathematics, with the pessimistic view being more popular. T …
1
vote
Formulas for the liar paradox
Aladdin M. Yaqūb (1993) The Liar Speaks the Truth, OUP, formalises a very simple language for naturally expressing the liar paradox, consisting of:
First-order equational logic which may, but need n …
- 2,283
2
votes
cut elimination
To elaborate on Alexey's answer, for "usual" sequent calculi, the rules other than cut "build structure" in the proof: the left rules build up structure of the assumptions from smaller formulae, and t …
- 2,283
4
votes
Where are we working when we prove metamathematical theorems?
Real Logicians(*) use primitive recursive arithmetic as their metatheory. As a purely equational, and so unquantified, theory, it is agnostic about the distinction between classical and intuitionisti …
- 2,283
5
votes
What does it mean to 'discharge assumptions or premises'?
In the spirit of Kenny's observation, note also that we can formulate classical logic using a Peircian inference rule (equivalent to the usual theory in the presence of ex falso quodlibet) which clear …
- 2,283
2
votes
Are all mathematical theorems necessarily true?
John Goodrick writes, very much to the point: The last I heard, there was no consensus in the philosophical community as to what exactly a "possible world" is, or whether this is even a coherent notio …
- 2,283
10
votes
Accepted
What does the disjunction elimination rule say?
The first rule is not the regular disjunction elimination rule, but is known as disjunctive syllogism, and is essentially the modus tollendo ponens rule of term logic. The two rules are mutually admi …
- 2,283
5
votes
When is something too big to be a set?
I find the terminology of "too big" to be misleading. I think it comes about from thinking that the strength of a set theory comes from the generosity of its comprehension axioms, that stronger set t …
- 2,283
4
votes
Can we prove set theory is consistent?
I think you are describing a process that is a fairly accurate description of how set theorists typically think about issues of consistency, where Set1 is the informal account of the cumulative hierar …
- 2,283
9
votes
Most general formulation of Gödel's incompleteness theorems
Theories can be be represented recursion-theoretically by an encoding of the language as natural numbers (most simply, a bijective encoding, which I assume), and a Turing machine that accepts all and …
- 2,283
3
votes
Accepted
Can infinite first-order categories be specified other than as categories of models?
Sure, by direct construction. Rings, preorders, the category of paths of a given graph, etc. But that's not what you wanted to know, is it?
- 2,283
8
votes
Do you know any good introductory resource on sequent calculus?
Gentzen, 1934, 'Investigations into Logical Deduction' — This is very readable, and introduces so many ideas that later synthetic works invariably miss some. If you're serious, this, and some other p …
- 2,283
4
votes
How do they verify a verifier of formalized proofs?
The key point is the idea of the kernel of a theorem prover, as Adam mentioned. To put it another way the kernel is the smallest subset of the theorem prover's code base (and operating system and mac …
- 2,283
7
votes
Accepted
Is there a relationship between model theory and category theory?
Between model theory and category theory broadly conceived: not anything really compelling, because a category, on its own, does not stand as an interpretation for anything.
Between model theory and …
- 2,283