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Can you prove the existence/subsistence of a universal set?

Consider the following statement, which is similar to Frege's axiom of Unrestricted Comprehension. ∀P2∃y[y denotes a set ∧ ∀x[x ∈ y iff P2(x,x)]] Can you use it to prove there is a universal set? ...
lee pappas's user avatar
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Is the derivation of Russell's paradox from Frege's axiom of unrestricted comprehension based on an n-ary relation error?

My question is, is the derivation of Russell's paradox from Frege's axiom of Unrestricted Comprehension based on an n-ary relation error? Here is Gottlob Frege's axiom of unrestricted comprehension. ...
lee pappas's user avatar
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4 votes
4 answers
205 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'...
lee pappas's user avatar
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2 votes
0 answers
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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, ...
AUTIST INC's user avatar
1 vote
0 answers
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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 ...
lee pappas's user avatar
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3 votes
2 answers
158 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 ...
AUTIST INC's user avatar
1 vote
4 answers
201 views

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 ...
lee pappas's user avatar
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215 views

Is equality necessarily transitive?

Consider the equation b2 - 1 =0 If b is a variable, it's neither true nor false. So let the symbol b be a constant, thus the equation denotes a proposition. The symbol b is a referrer, and the ...
lee pappas's user avatar
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-2 votes
2 answers
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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 ...
lee pappas's user avatar
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1 vote
2 answers
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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] [...
lee pappas's user avatar
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3 votes
0 answers
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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: ...
lee pappas's user avatar
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5 votes
4 answers
677 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 ...
lee pappas's user avatar
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1 vote
1 answer
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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 ...
lee pappas's user avatar
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6 votes
2 answers
209 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 ...
lee pappas's user avatar
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-1 votes
1 answer
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Can Frege's axiom of unrestricted comprehension be slightly modified to avoid the Russell paradox?

Russell showed that Frege's axiom 5, the principle of unrestricted comprehension is false, by considering the set of all and only sets that aren't elements of themselves. It's my suspicion that that ...
lee pappas's user avatar
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2 votes
0 answers
109 views

Does First order Logic require the subsistence/existence of a universal set, i.e. a set of all things? [closed]

In Irving M. Copi's 'Symbolic Logic' he postulates an unlimited set of individual constants. Each constant denotes an individual, so it seems he accounted for a set of all things U. First order logic ...
lee pappas's user avatar
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30 votes
5 answers
19k views

How is first-order logic complete but not decidable?

Why doesn't completeness imply decidability for first-order logic? First order logic is complete, which means (I think) given a set of sentences A and a sentence B, then either B or ~B can be arrived ...
Taylor Hornby's user avatar
35 votes
3 answers
5k views

Is first-order logic 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 ...
Mono's user avatar
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