24 votes

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 ...
Frank Hubeny's user avatar
  • 19.3k
16 votes

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 ...
sequitur's user avatar
  • 1,388
15 votes
Accepted

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) ...
Tanner Swett's user avatar
12 votes
Accepted

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 ...
Bumble's user avatar
  • 24.2k
12 votes

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 ...
Jo Wehler's user avatar
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11 votes

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 ‘...
Frank Hubeny's user avatar
  • 19.3k
9 votes

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 ...
virmaior's user avatar
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9 votes
Accepted

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", ...
Conifold's user avatar
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8 votes

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 ...
Conifold's user avatar
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8 votes
Accepted

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 ...
Noah Schweber's user avatar
8 votes

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 ...
Daniel Prendergast's user avatar
8 votes

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) ~...
Steven's user avatar
  • 81
8 votes
Accepted

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 ...
E...'s user avatar
  • 6,466
8 votes

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 ...
J D's user avatar
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7 votes

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 ...
Dolphin 613 Motorboat's user avatar
7 votes
Accepted

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 ...
Mauro ALLEGRANZA's user avatar
6 votes
Accepted

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 ...
Mauro ALLEGRANZA's user avatar
6 votes

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
virmaior's user avatar
  • 24.7k
6 votes

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 ...
Bumble's user avatar
  • 24.2k
6 votes

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 ...
Natalie Clarius's user avatar
6 votes

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 ...
transitionsynthesis's user avatar
6 votes

Is ¬(a = b) the same as (a ≠ b) in logic

In classical mathematics these are the same, and indeed more generally the "strike-through" notation is just shorthand for logical negation. In constructive mathematics, however, my understanding is ...
Noah Schweber's user avatar
6 votes
Accepted

Why aren't Kripke semantics "syntax in disguise"?

Algebraic semantics give a good organising framework for models of a logic, but they don’t give examples of models, except syntax itself. Kripke models give an easy way to construct lots of concrete ...
Peter LeFanu Lumsdaine's user avatar
6 votes

What did Russell mean when he wrote that the null-class, the class having no members, did not exist?

In the passage above Russell discusses two uses of existence: (a) is the "common sense" use: "which occurs in philosophy and in daily life is the meaning which can be predicated of an ...
Mauro ALLEGRANZA's user avatar
5 votes
Accepted

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 ...
Bumble's user avatar
  • 24.2k
5 votes
Accepted

Why do “L” and “M” name the strong and weak modal operators in modal logic?

From the Polish logician and philosopher Jan Łukasiewicz who invented the Polish notation for logic (named after his nationality) : M ϕ for możliwość : possibility L ϕ for konieczność : necessity [...
Mauro ALLEGRANZA's user avatar
5 votes

Step by step natural deduction: (T > E) ^ (A > L) /... (T v A) > (E v L)

Premise : 0) (T⇒E) ∧ (A⇒L). Use Material Implication on the premise to get : 1) (~T∨E) ∧ (~A∨L) Use Simplification to get respectively : 2) (~T∨E) and : 3) (~A∨L) Using Addition on 2) get : 4)...
Mauro ALLEGRANZA's user avatar
5 votes

What exactly is a first-order logic?

FOL is the natural logic environment to formalize mathematical theories. The basic characteristic of predicate calculus is the use of quantifiers : first-order logic is predicate calculus where ...
Mauro ALLEGRANZA's user avatar
5 votes
Accepted

How to translate "No dolphin sings unless it jumps" into predicate logic?

Because of the double negation (no + unless) I would first try to paraphrase the sentence before trying to translate it into predicate logic. All the following have the same meaning: No dolphin sings ...
E...'s user avatar
  • 6,466

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