Questions tagged [first-order-logic]
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For questions about formal deduction of first-order logic formula or metamathematical properties of first-order logic.
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Understanding the use of the Axiom of Choice in David Marker's proof of Compactness
I am trying to understand the use of the Axiom of Choice in the proof of Lemma 2.1.8 (page 37 below) in David Marker's Model Theory: An Introduction Pages 35 and 36 can be found here. Lemma 2.1.8 is ... logic first-order-logic predicate-logic model-theory- 3,118
Can we reduce the number of occurrences of the metavariable in Peano Arithmetic's axiom schema of induction?
Is there an axiomatization of Peano arithmetic where the axiom schema contains fewer than 4 distinct occurrences of the metavariable? Ideally, as few as possible. The immediate motivation is to come ... elementary-number-theory logic first-order-logic beth-tableau- 6,946
Beginner question about first-order logic
In first order logic, is it correct to say that: $\vdash\forall x(A\rightarrow B)\rightarrow\forall xB$ if and only if $\vdash\forall x(\neg A\rightarrow B)$ The right-to-left is easy to argue for. I ... first-order-logic- 21
First-order logic: where's the the flaw in this argument?
There must be a flaw in the following argument, but I don't see it at the moment. Who can point it out? In first-order logic, suppose that a structure $\mathfrak{U}$ is a model of the formula ($x = 3$... logic first-order-logic natural-deduction- 121
Requirements for being a model of ZF
It is a follow up to this other question Is it true that "Axiom of Choice is true or Axiom of Choice is false"? In order to proof that something is a model of ZF, what should I do? I need to ... first-order-logic- 184
How to apply axiom of choice on the universe of sets? [duplicate]
If we have $\forall x(x\in X\to\exists y(y\in Y\land\varphi(x,y)))$, then we can define a function $f$ from $X$ to $\mathcal P(Y)\setminus\{\emptyset\}$ such that $f(x)=\{y\in Y|\varphi(x,y)\}$, and ... first-order-logic axiom-of-choice- 31
Is there work on free logic in algebraic logic?
Has there been work on algebraic versions of free logic? I've been playing with mechanizing various forms of algebraic logic. I think maybe free logic could simplify some things. Trivially you can add ... logic reference-request first-order-logic- 1,133
This conditional statement is evaluated as true (by the program Tarski's world) however it seems false could you help detect my mistake?
So the way I read the statement is: there is an object x such that if it's a cube then it is between a and b. However, there is a cube but it is not between a and b, making the statement false. The ... first-order-logic predicate-logic- 101
Measure of similarity between boolean formulas?
Apologies, if this is a dumb question... Is there a measure of structural similarity between Boolean formulas A and B, such that if we know some information about A's behavior under a set of inputs $... logic propositional-calculus first-order-logic- 35
Soundness and Completeness for a single Model Only?
Question modified to hopefully answer the questions (I'm a physicist to all might not be mathematically watertight) In Enderton "A Mathematical Introduction to Logic", logical Implication is ... logic first-order-logic model-theory nonstandard-models- 555
Downsides of defining $\models$ to be false when there's a mismatch in signature between the structure and formula.
The way that $M \models \varphi$ is usually defined, we know implicitly that for some $L$, $M$ is an $L$-structure and $\varphi$ is an $L$-sentence. I'm curious what the downsides are to defining $M \... logic first-order-logic model-theory- 6,946
Making sense of: "there is an uncountable set" is a logical consequence of $\mathrm{ZFC}$, yet $\mathrm{ZFC}$ is satisfiable in a countable domain.
As I learn the Compactness and Löwenheim-Skolem theorems of first-order logic and I begin to have a better understanding of what they really mean, something has me baffled. As I have learned it, the ... logic first-order-logic predicate-logic model-theory- 3,118
Witnesses in the Henkin construction and the canonical model
See the images below, from Model theory: An Introduction, by David Marker: In the proof of Lemma 2.1.7 the canonical model is defined using equivalence classes of constant symbols of $\mathcal{L}$, ... logic first-order-logic predicate-logic model-theory- 3,118
Link between Tarski Truth, Negation, Consistency and Deductions
In Enderton "A Mathematical Introduction to Logic" negation in a given structure M is defined as: $$(M \models \neg A) \; \Longleftrightarrow \; [\text{ not true } (M \models A)] \tag{1}$$. ... logic first-order-logic model-theory forcing- 555
Identifying fragments of First Order Logic, where the number of models increases strictly monotonically with the domain size?
Given a function-free first order logic language with only universal quantification. The problem of model counting aims at finding the number of models that an FOL formula of the form $\forall {\bf{x}}... combinatorics first-order-logic- 1,515
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