Questions tagged [first-order-logic]
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18
questions
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2
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63
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Question regarding 0-ary relations
EDIT-
Definition. A denotes a unary relation iff
∀x[if x∈ A then ∃y[x= (y)]]
Using this definition, since individuals don't contain elements, any individual is a unary relation. Since the empty set ...
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-4
votes
2
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124
views
Is equality necessarily transitive? [duplicate]
I want to introduce three definitions into the philosophy of logic for the purpose of improving first order logic.
Consider the following three definitions.
Definitions
C is an arbitrary constant iff ∀...
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11
votes
2
answers
1k
views
Why is completeness (as in Gödel completeness theorem) a desirable feature?
When justifying the dominance of first-order theory, an argument that is often made is that it is complete (as shown by Gödel).
This means that a theory formulated in first-order logic has a model if ...
- 227
4
votes
4
answers
237
views
How do you prove mathematical induction without the notion of a set?
EDIT - Peano's axioms for N can't be used to answer this question, because they assume induction. So what axioms can be used? I am thinking the following:
P1. x ∈ N iff x=1 ∨ ∃y (x=y' ∧ y ∈ N)
P2. 0'...
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2
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0
answers
59
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Translating a part of the Lowenheim-Skolem Theorem into first order logic
The part of the Lowenheim-Skolem theorem that I want to translate into first order logic is the following: For every signature A, every infinite A-structure B, and every infinite cardinal number C, ...
1
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0
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40
views
Can the modal logic S5 be reduced to Rosser's system for a first order function calculus?
From the SEP
In propositional logic, a valuation of the atomic sentences (or row of a truth table) assigns a truth value
(
T
or
F
)
to each propositional variable
p
. Then the truth values of the ...
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3
votes
2
answers
165
views
On Relations Versus Relational Properties
According to the Stanford Encyclopedia of Philosophy, the following holds: Relations and relational properties can be distinguished. A relation is borne from one thing to another thing. A relational ...
1
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4
answers
216
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Looking for a formal proof that x=x isn't a contingency
EDIT - My original question was answered, but not to my satisfaction. What I really want is a formal proof α = α isn't a contingency, using the modal logic version of Hao Wang's axiom of Identity. I ...
- 1,450
-2
votes
2
answers
100
views
What is the proper form of universal instantiation?
Definitions
C is a specific constant iff ∃! x [x=C]
C is a general constant iff ∀x [x=C]
C is an arbitrary constant iff ∀x [x=C] ∨ ∃! x [x=C]
Consider the commonly accepted form of the rule of ...
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0
votes
3
answers
933
views
Is Frege's axiom of unrestricted comprehension actually true after all?
Consider the following demonstration whose first line is the assumption called the axiom of unrestricted comprehension.
∀F∃y ∀x[x ∈ y iff F(x)] [OSC1]
∀F∃y [α ∈ y iff F(α)] [UI]
∃y [α ∈ y iff α ∉ α] [...
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2
votes
0
answers
140
views
Can the entirety of first order logic be reduced to the propositional calculus?
I've been wondering, whether or not first order logic can be reduced to the propositional calculus.
Rosser's system RS_1, described by Irving M. Copi in 'Symbolic Logic', has 5 axioms or postulates:
...
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5
votes
4
answers
682
views
Must a domain of discourse always be specified in universally quantified statements?
Some logic texts formulate universally quantified statements without specifying a domain of discourse D.
For example
For any x: x isn't alive.
They take it for granted that x ∈ U, where it's true that
...
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1
vote
1
answer
85
views
Can modal logic be used to define the notion of an “arbitrary constant” in FOL?
I was wondering if first-order logic can be reduced to propositional calculus if we eliminate quantification.
For example, instead of saying “for all x in a domain D, P(x)”, we could state “P(x)” for ...
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6
votes
2
answers
220
views
What is the difference between a model and an interpretation in logic?
On page 319 of Irving M. Copi's 'Symbolic Logic', he states, "if we want our logical system to be applicable to any possible universe, regardless of the exact number of individuals it contains ...
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-1
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1
answer
121
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Can Frege's axiom of unrestricted comprehension be slightly modified to avoid the Russell paradox?
EDIT - The universal quantification of F should be at the far left, so the axiom I'm proposing is
PRINCIPLE OF RESTRICTED COMPREHENSION
∀F∃y [y is a set & ∀x[ not(F(x) ↔ x ∉ x ) → (x ∈ y ↔ F(x))]]
...
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