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# Questions tagged [symbolic-logic]

For questions related to symbolic logic, also known as mathematical logic. Topics might range from philosophical implications of metamathematical results to technical questions.

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11 answers
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### What is the difference between Law of Excluded Middle and Principle of Bivalence?

Law of Excluded Middle: In logic, the law of excluded middle (or the principle of excluded middle) is the third of the so-called three classic laws of thought. It states that for any proposition, ...
• 987
13 votes
6 answers
3k views

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

This is an excerpt from Alfred Tarski's Introduction to Logic and the Methodology of Deductive Sciences: As variables we employ, as a rule, selected letters, e.g. in arithmetic the small letters of ...
12 votes
9 answers
13k views

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

Obviously since A → C and B → D then if A v B one of C or D must be true. 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?
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10 votes
7 answers
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### What did Russell mean when he wrote that the null-class, the class having no members, did not exist?

I am not quite sure I interpret the following sentence correctly in Bertrand Russell's paper on existential import: and among classes there is just one which does not exist, namely, the class having ...
• 8,173
9 votes
2 answers
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### What's the difference among the logical relations :=, =, and ≡?

I understand that ≡ is logical equivalence, "iff". '=' is a symbol for numerical equivalence. And ':=' is an identity claim. I often only see '=' and ':=' used with variables and names, ...
8 votes
2 answers
844 views

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

The Wikipedia article on Kripke semantics suggests that they were considered a major breakthrough in part because algebraic semantics were seen as merely "syntax in disguise". But Kripke ...
• 193
8 votes
2 answers
248 views

### How do proofs about logic fit into a logical framework?

I'm learning logic from Michael O'Leary's A First Course in Mathematical Logic and Set Theory. In chapter 1 he carefully explains the meaning of logical implication (p ⊨ q), logical inference (p ⟹ q), ...
• 361
7 votes
3 answers
695 views

### Is it true that (P∧Q≡P)⇔(Q≡⊤)?

Consider the statement (P∧Q≡P)⇔(Q≡⊤) Where P and Q are statements, and ⊤ denotes the tautology (true) statement. It seems intuitively true that the above biconditional statement is true. But I ...
• 819
7 votes
2 answers
504 views

### How did symbolic logic show that Heidegger's assertions about the nothing were illogical?

In his inaugural address at Freiburg University in 1929, Heidegger explicitly challenged the central place given to logical principles in neo-Kantianism, on the basis of a radical account of ‘the ...
• 4,259
7 votes
2 answers
971 views

### What exactly is a first-order logic?

Can someone explain in simple terms what exactly is a first-order logic? From my amateur standpoint, I think that first-order logic is a some kind of a system of symbols and general logical rules and ...
• 325
7 votes
4 answers
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### What does the truth-value of a material implication represent?

This question comes from my attempts to understand what the truth value for a material implication with a false antecedent represents. I have seen several justifications for this convention, usually ...
7 votes
3 answers
303 views

### Why is the law of the excluded middle not a exclusive disjunction?

So the law of the excluded middle, as I have read in every logic textbook that I have read, has been ( ϕ ∨ ¬ ϕ ) , but this seems somewhat unintuitive, since I was under the impression that the ...
6 votes
8 answers
3k views

### What is the explicit reasoning behind proof by contradiction?

From my understanding, proof by contradiction consists of the following steps. 1. Show that p -> q, where "->" is the conditional. 2. Show that q is false. 3. Deduce from a truth table that p must be ...
6 votes
3 answers
561 views

### Can something proved by contradiction always be proved without a proof by contradiction?

Proof by contradictions work by assuming that something is true, and then using logic (along with other assumptions which you know are true) to show that that leads to a contradiction, thus proving ...
• 423
6 votes
3 answers
3k views

### In Fitch, how does one prove "(P → Q)" from the premise "(¬P ∨ Q)"?

It's all in the question really. I am working on a proof in Fitch for a class, but I am very much stuck. I am proving the tautology that "(P → Q) ↔ (¬P ∨ Q)", and I have already finished half of it, ...
• 61
6 votes
3 answers
5k views

### Propositional Logic: How to prove the contraposition in the Fitch system?

Given that: p ⇒ q prove that: ¬q ⇒ ¬p using the Fitch system. (This being the proof of the Contraposition)
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6 votes
3 answers
188 views

### How to model "forget about" in first order logic?

The other day, my housemate said "Don't forget to not leave the spoon at the bottom of the container". I understood what he meant: "Do not leave the spoon at the bottom of the ...
5 votes
3 answers
2k views

### What's the difference between "iff" and "=df"?

Just a quick question I stumbled upon from my readings. When some philosophers write A ↔ B and others write A =df B, is there supposed to be a difference?
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5 votes
2 answers
764 views

### Are contradictory propositions in the propositional logic still contradictory in the predicate logic?

There is one seeming issue I happened upon that bothers me to no end. Take a proposition like “Snow is white”. “Snow is white” and its negation “Snow is not white” are obviously contradictory. However,...
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5 votes
2 answers
2k views

### What does the symbol ":=" mean in formal logic?

:= What does the "colon-equal symbol" mean, and how is it used?
5 votes
3 answers
2k views

### What did Bertrand Russell mean exactly when he said that *such that*, while fundamental both to formal logic and to mathematics, is "undefinable"?

Bertrand Russell in Principles of mathematics (1903) presents the notion of such that as fundamental to logic and mathematics, and states that it is “undefinable”: The Indefinables of Mathematics ...
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5 votes
1 answer
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### Why does Gensler's Star Test not work on some syllogisms? [duplicate]

All teachers are intelligent. All teachers are well-paid. From the Star Test, we can deduce that the argument must be invalid with whatever conclusion (according to the classical syllogism figures), ...
5 votes
3 answers
3k views

### What exactly are the identity rules in logic?

In first order logic, I have read that there are a couple of identity rules. If I have "a=b" does it mean that I can also write it as "b=a"? Is it true one-way or both? And if I have two ...
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5 votes
1 answer
346 views

### Origins of the syntactic form for rules of inference in modern presentations

I have been wondering where the form originates from. The turnstile ⊢ famously comes from Frege, but I haven't been able to find where the vertical notation was introduced. In the field of ...
• 153
5 votes
2 answers
603 views

### Is it possible to construct infinitely many non-equivalent formulas in predicate logic?

In the language of predicate logic with only identity and no predicates, function symbols, or constants, is it possible to construct infinitely many non-equivalent formulas?
5 votes
1 answer
307 views

### Can/Do there exist any quantifiers other than "there exists" and "for all"?

I'm curious about why there are only the two logical quantifiers there exists and for all. Intuition and human language support the idea that these quantifiers make sense, but otherwise it seems ...
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5 votes
1 answer
7k views

### What is the logical law proving "if not p then q" is equivalent to "p or q"?

I know that (¬p → q) ≡ (p v q) from comparing the truth tables. But is there a law that states this? Something like the laws of propositional logic: idempotent, associative, commutative, distributive, ...
5 votes
5 answers
362 views

### What would be an intuitive understanding of Peirce's law?

Wikipedia describes Peirce's law as In propositional calculus, Peirce's law says that ((P→Q)→P)→P. Written out, this means that P must be true if there is a proposition Q such that the truth of P ...
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4 votes
6 answers
2k views

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

I’m working on a practice question on my logic textbook. And I’m stuck at this question. This is what I have so far: 1. ~(P & Q) Assumption/ Negated Eelimination 2. P Assumption/...
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4 votes
4 answers
3k views

### What is the difference between logical consistency and logical entailment in deductive logic?

I am having a little trouble sorting out two definitions from the first chapter in my logic textbook, The Logic Book by Bergmann, Moor and Nelson. I am under the impression that a set in a sentence ...
• 41
4 votes
1 answer
2k views

### What, at present, are the major unsolved problems of logic?

In the 1900s, Hilbert published a list of 23 (later 24) unsolved problems in mathematics, which sparked increased research into each of them and the subsequent resolution of several of these problems. ...
4 votes
2 answers
642 views

### Implication Introduction formulated as a theorem?

While making a list of the rules of inference for my math students, I came across this list on Wikipedia: I noticed a pattern: for every introduction rule, there seems to be an elimination rule, and ...
• 819
4 votes
3 answers
630 views

### Symbolic Logic Proof: Leprechauns Exist?

I am reviewing a study guide for an introductory logic course (basic predicate, syllogistic etc.). The problem asks me to symbolize that "leprechauns exist" and prove that it is a logical truth and ...
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4 votes
2 answers
2k views

### What does the colon (:) mean in conjunction with material implication?

Errol E. Harris does an excellent job of explaining dialectical logic in Formal, Transcendental, and Dialectical Thinking, but in the section on formal logic, he assumes a familiarity with symbolic ...
4 votes
5 answers
985 views

### Is there a logical symbol for "why"?

Is there a formal logic symbol for "why"? For example how would you formulate "Why is 2^4 > 4^2?" Could that be formulated in pure symbols of logic if possible? Also, the phrase "what is" can it be ...
4 votes
2 answers
672 views

### Subformulas of the WFF (∀x) ((∀y) ((x ∈ y) ∨ (y ∈ x )))

Consider the well-formed formula in set theory (∀x) ((∀y) ((x ∈ y) ∨ (y ∈ x ))). I believe there are 5 subformulas: (x ∈ y) (y ∈ x) ((x ∈ y)∨(y ∈ x)) (∀y) ((x ∈ y)∨(y ∈ x)) (∀x) ((∀y) ((x ∈ y)∨(y ∈ x)...
• 819
4 votes
2 answers
199 views

### need some guidance for this easy symbolic logic question [closed]

Every dog and cat who is well trained is a good pet. (F: a is a dog; G: a is a cat; H: a is well trained; I: a is a good pet.) Here are my options: a) ∀x((Fx∨Gx)∧Ix→Hx) b) ∀x((Fx∨Gx)∧Hx→Ix) c) ∀x(...
• 163
4 votes
3 answers
416 views

### P <=> (Q v R), P, -Q⊢R Propositional Logic Question

<=> is bi-conditional, "-" is negation, "v" is disjunction. I can't figure out where to take it from line 4. Negated Q is throwing me for a loop. P <=> (Q v R), P, -Q ⊢ R P <=> (Q v R) P ...
4 votes
2 answers
3k views

### Fitch style disjunction elimination

I am having difficulty in formally proving a simple argument. Consider P(x) v Q(x) not P(x) ---------- Q(x) It is easy to see that the argument is indeed valid, but I cannot seem to prove it ...
4 votes
2 answers
166 views

### help with deductive proof

∀x (Fx ∨ x=c), ¬Fb ∧ Gb |- ¬Fa → Ga So far I don't understand how to switch variables around to prove the result. I've got a subproof set up assuming "¬Fa" in order to derive "Ga". In that proof I ...
• 153
4 votes
2 answers
5k views

### Fitch Proof - LPL Exercise 8.17

I am currently finding the third part of this exercise (Conditional 3) difficult to prove. I was sure that my proof was correct, but the Fitch program is saying otherwise. I am finding it ...
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4 votes
2 answers
240 views

### Can knowledge about argumentation be sufficient for philosophical logic without too symbolic or mathematical concepts?

The most important element for expression of truth is trough an argument, with premises and conclusion. Argumentation requires to avoid fallacies and adhere to the truth. However logic if treated as a ...
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4 votes
1 answer
148 views

### Prove ∀w(∀v((v=w∧φ(v))⇔φ(w)))

In this math question of mine, an answer pointed me to this theorem: ∀w(∀v((v=w∧φ(v))⇔φ(w))) which in turn, the answerer stated, implies another theorem: ∃v(v=t∧φ(v))⇔φ(t) which was the fact I ...
• 819
4 votes
3 answers
2k views

### Disjunctive Syllogism in a Fitch Style System

I'm trying to prove an argument of the form: B ~(C & B) Therefore: ~C. I can expand out ~(C & B) into ~C OR ~B, and with the premise B, it is clear that ~C is the case. ...
• 163
4 votes
1 answer
58 views

### Can assumption in Hilbert style proof system be contradictory?

⊢（¬A→A）→A I don't know how to solve this proof with the Axiom, Theorem and Inference rule in Hilbert-style proof system so I ask my classmate and he show me his answer. After viewing his proof, I was ...
4 votes
1 answer
189 views

### Are there famous unsolved problems in logic akin to the Millenium Prize problems?

Are there major theorems that logicians have yet to tackle? And I don't mean any problems that pertain to the philosophy of logic (i.e. logical pluralism, the nature of logical consequence, etc), but ...
4 votes
1 answer
155 views

### Is there a formalized logic for adpositional connectives?

Certain words in natural language are more amenable to logical formalization. The conjunction "and" or weak conditional "unless" are easily applied to break statements into their constituent atomic ...
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4 votes
1 answer
85 views

### Unusual change of meaning of word "any" in negative sentences form "for all" to "there exists". Predicate logic

Question. Why does the word "any" in negative sentences changes its meaning from "for all" to "there exists"? Origin of the question. I have a question about translating ...
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3 votes
4 answers
2k views

### Is Russell's Paradox a semantic paradox or a syntactic paradox?

Is Russell's Paradox a semantic paradox or a syntactic paradox? I ask because of the following: Let P be a predicate Let SEP be the property of being a set of things that satisfies P Let SP be the ...
• 423
3 votes
3 answers
284 views

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

Are these the same in predicate logic with identity: ¬ (a = b) a ≠ b I'm not quite sure whether they can be used interchangeably in proofs. Any help would be great!
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