# Tag Info

### How to prove (A v B), (A → C), (B → D) therefore (C v D)

Here is part of the question: My only idea is v must be introduced, but how would I use subproofs to show one of A/\C or B/\D is never false if A v B? It might be best to think of using ...
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### Why does the Principle of Explosion not make Mathematical Logic inconsistent?

Explosion is a property of logical consequence relations, and thus of logics, that is not trivial: Some logics have it, some don't. So there is simply no sense asking Why can't we simply get rid ...
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### What did Russell mean when he wrote that the null-class, the class having no members, did not exist?

Let me start by slightly rephrasing what Russell wrote, since Russell is using the word "exists" in an unusual and confusing way. With my changes in bold, here is what Russell wrote: (b) ...
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### Are contradictory propositions in the propositional logic still contradictory in the predicate logic?

Something that is a contradiction in the propositional logic remains a contradiction in predicate logic. The problem with your examples is that they are not particularly clear as to whether you are ...
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### What's the difference between "iff" and "=df"?

A ↔ B means that both propositions are equivalent: "x implies y" (A) is equivalent to "non-y implies non-x" (B). Equivalences have to be demonstrated. While A =df B means that the ...
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### How is Wittgenstein’s “notorious paragraph” about the Gödel's Theorem not obviously correct?

Timm Lampert, cited by the OP, quotes Wittgenstein (§8 of Remarks on the Foundations of Mathematics, Appendix 3): ‘True in Russell’s system’ means, as was said: proved in Russell’s system; and ‘...
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### What is the explicit reasoning behind proof by contradiction?

There seem to be several overlapping concerns in your issue with proof by contradiction. You have an objection to the truth table for material implication: I've never been satisfied with the ...
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### Why does Gensler's Star Test not work on some syllogisms?

Gensler's star test is a simplified method for determining the validity of a syllogism proposed in 1973. According to the test, one stars (asterisks) the first (capital) letter after "All", ...
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### What does Tarski mean when he says "variables do not posses any meaning by themselves"?

What does he mean when he says variables do not have a meaning by themselves? Does he mean variables are completely devoid of meaning? I suppose he thinks of something along the lines of "while ...
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### At what point in the history of mathematics, and why, did mathematicians come to say "A implies B" to mean "not A or B"?

The word 'implies' is definitely ambiguous. Saying that "it just means if A is true, then B is true" does not explain anything. John Corcoran wrote a paper ("The Meanings of Implication&...
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### What, at present, are the major unsolved problems of logic?

One should keep in mind that the meaning of "logic" changed over the last century, and is now more confined to formal logic, although it is broader than deductive or mathematical logic in the narrow ...
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### Is it possible to construct infinitely many non-equivalent formulas in predicate logic?

Below I've looked at sentences, as opposed to mere formulas, since the resulting situation is more interesting. If we allow formulas with free variables, then as Toothpick Anemone essentially observes ...
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### Why does the Principle of Explosion not make Mathematical Logic inconsistent?

Rather than give you the correct technical answer (edit: okay I ended up going into some technical details, oops), which has been provided, I think I'll try and diagnose where you're intuitions have ...

### How to prove (A v B), (A → C), (B → D) therefore (C v D)

You can use proof by contradiction: p1: A v B p2: A -> C p3: B -> D assume ~(C v D) ~C & ~D (from 1, De Morgan's law) ~C (from 2, conjunction elimination) ~D (from 2, conjunction elimination) ~...
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### What does the colon (:) mean in conjunction with material implication?

It's a terrible and outdated system of notation where dots (.) and colons (:) function both as conjunctions and as parentheses. When a . or : occurs between expressions, it denotes conjunction. For ...
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### What does the symbol ":=" mean in formal logic?

Answer According to Wikipedia's article on logic symbols, := is used for definition. The truth of a proposition can be determined through empirical or rational means, but sometimes it is assigned ...
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It is a set-theoretic paradox. Syntactic paradox would imply it follows from certain syntactical rules. Assuming you treat the axioms of naïve set theory as syntactic, then Russell's paradox is a ...
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The original distiction between logical and epistemological contradictions was introduced by Ramsey (1926) (but it had already been hinted at by Peano in 1906): "While logical contradictions ...

### Does the existential quantifier express existence?

Yes, the existential quantifier expresses existence. If you assert that Some pegasus are flying then you do assert that pegasuses exist, at least by the classical logical treatment of the ...
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### What is model theory?

Model theory, as it is today understood, is a formal way to study how bits of language manage to represent the world. The fundamental idea of model theory is that you have a structure that assigns ...
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### How did symbolic logic show that Heidegger's assertions about the nothing were illogical?

I guess it refers to this passage later in the article/review you were quoting from: the debate between Heidegger and Carnap -- Shirley's next topic -- precisely turns on whether Heidegger's account ...
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### Origins of the syntactic form for rules of inference in modern presentations

As per comments above, the modern use is due to Hilbert's school. Gerhard Gentzen, Über die Existenz unabhängiger Axiomensysteme zu unendlichen Satzsystemen (1932) derived it from Paul Hertz, Über ...

### What does Tarski mean when he says "variables do not posses any meaning by themselves"?

Tarski means exactly what he says: it is the fundamental idea that enables someone to "get" algebra. I still don't understand how you teach this idea. I have done some tutoring, and students ...
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### Implication Introduction formulated as a theorem?

The two different symbols on the page you link to are indeed different. The first is the turnstile symbol Ⱶ which may be read as 'proves', while the arrow → is material implication. These are very ...
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### Problems with Existential Instantiation

The EI rule formalizes the fact that if we know that ∃xP(x), we are licensed to give to "that P" a name. But we have to avoid that the said name is not already "in use" because, if so, it may denote ...

### How to show that (P & Q) v (~P v ~ Q) is a theorem in SD

another approach giving the minimum number of steps (though not a formal proof): 1. (P & Q) v ~(P & Q) law of excluded middle 2. (P & Q) v (~P v ~Q) DeM 1
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### Known self-evident unproven logical truths

It is contentious even to suppose that logic is concerned with being 'self-evident' at all. The old-fashioned idea that logic represents the immutable laws of thought that hold everywhere for all ...
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