# 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.

306 questions
Filter by
Sorted by
Tagged with
20 votes
11 answers
20k views

### 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
19 votes
7 answers
6k views

### Selection of logical connectives {¬,∨,∧,⇒,⇔} in set theory?

Nearly every treatment of set theory, whether Paul Halmos' Naive Set Theory, Herbert Enderton's Elements of Set Theory, Patrick Suppes' Axiomatic Set Theory, etc. introduce a common set of logical ...
• 799
12 votes
9 answers
12k 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?
• 137
9 votes
7 answers
3k views

### 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 ...
• 6,241
9 votes
2 answers
9k views

### 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
245 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), ...
• 351
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 ...
• 799
7 votes
2 answers
793 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 ...
• 183
7 votes
2 answers
488 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 ...
• 3,787
7 votes
2 answers
817 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
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
2k 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)
• 101
6 votes
3 answers
185 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 ...
6 votes
4 answers
1k views

### 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 ...
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?
• 495
5 votes
2 answers
733 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,...
• 63
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
1 answer
896 views

### 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 ...
• 527
5 votes
1 answer
336 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 ...
• 151
5 votes
2 answers
539 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
235 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 ...
• 351
5 votes
1 answer
4k 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
2 answers
196 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 ...
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/...
• 43
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
3 answers
491 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 ...
• 403
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
3 answers
626 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 ...
• 163
4 votes
2 answers
1k 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
930 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
644 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)...
• 799
4 votes
2 answers
180 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
403 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
160 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
5 answers
322 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 ...
• 19.2k
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 ...
• 41
4 votes
1 answer
540 views

### What it the relationship between Type theory and logic?

I am aware that a similar question was asked about the type theory in the principia, but I'm more interested in what the relationship between, say Martin-Lof Type theory and intuitionistic logic is.
• 432
4 votes
2 answers
237 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 ...
• 153
4 votes
1 answer
146 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 ...
• 799
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
168 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
152 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 ...
• 2,426
4 votes
0 answers
53 views

### Does quantifier dependence involve putting ∃ before ∀ (or vice versa)?

I don't know why I'm having such trouble getting the gist of the SEP article on independence-friendly logic, but I am. I also remain perplexed about a comment I received on the MathOverflow about the ...
• 12.2k
3 votes
3 answers
256 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!
• 31
3 votes
3 answers
461 views

### Does the existential quantifier express existence?

Does the existential quantifier express existence? The existential quantifier is a symbol of symbolic logic which expresses that the statements within its scope are true for at least one ...
3 votes
2 answers
347 views

### Is there a uniform way of differentiating sufficient and necessary conditions?

I am struggling to formulate symbolic conditional logic rules from basic sentences (studying for the LSAT). It seems that subtle differences in syntax are throwing me off. Is the conditional ...
3 votes
3 answers
4k views

### Prove P v ~P using most basic rules?

Is there a way to prove P v ~P in basic inference rules? I can't think of where to start because nothing applies to this. I was thinking about usinig Conditional proof, but I don't know what should I ...
• 155