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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Is There a Maximally Consistent Set That is Not $\omega$-complete? [on hold]

I was going back through Mates 2nd edition Elementary Logic and am having a bit of trouble on one of the exercises; it says, "Find a counter example to the following assertion: for any set of ...
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43 views

Proof by natural deduction advice

Any advice on shortening my right to left proof will be appreciated.
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Proof by natural deduction help

This is where I’ve gotten so far. I’ve proven it from left to right but I’m getting some trouble proving it from right to left. I’m trying to reach the conclusion by double negation.
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How to prove ~(~A v ~B) |- A & B by natural deduction? [closed]

Help with proof please. I can't seem to see a proper way to go about this specific proof.
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51 views

How to derive P > (Q > R) from (P > Q) > R in Fitch?

I am having a little bit of difficulty coming up with a Fitch-style natural deduction proof. Presumably, I need to use a few conditional introduction rules, but I am not sure what I can get out of ...
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Would it create difficulties if a proposed model theory had two or more distinct representations for the same predicate? [closed]

If there would be difficulties using such a model theory, then what is an example of a difficulty that would arise? The reason that I ask is that if no difficulties arise, or if the difficulties can ...
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In classical model theory, how does one represent the one-place predicate R(x) where for all x, R(x) iff (x∉x)? [closed]

If we try to represent the predicate by means of the set of values that satisfy it, then of course we run into Russell's paradox. Now, in ZF, we could simply use the whole domain of the theory, but ...
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2answers
79 views

Fitch Proof - Logic LPL 6.31

I am trying to complete the following proof in Fitch but am completely clueless on how to approach it. Any help would be appreciated! Thanks
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1answer
38 views

Why doesn't one assert in metamathematics that a sentence S is a logical consequence of the conjunction of a set of sentences?

In other words, why isn't there -- at least in standard textbook presentations of things like the deduction theorem and the compactness theorem -- a conjunction connective that is applied to sets of ...
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Logic Question in a fitch style system - disjunction elimination

I am having difficulty in formally proving two arguments. Firstly, given the premises A ∨ (B∧C) ¬B∨¬C∨ D derive A v D. I can see why it has to be either A or D, because if it's B and C, ...
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What is the difference between a function and a relation in first-order logic?

Consider this definition of first-order (or Herbrand) logic syntax. Here is the vocabulary: Definition (Vocabulary): A vocabulary V consists of: A set of relation constants {r1, ...,...
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67 views

What's the difference between a second-order relation and a relation between objects?

I was reading an article in philosophy and found this: Some philosophers have denied that there is such a relation as identity. Thus Ludwig Wittgenstein writes (Tractatus 5.5301): "That ...
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How should one symbolize “and then” in logic?

Graham Priest observed that not all uses of "and" in English commute: (page 15) ...according to the truth table for &, 'a and b' always has the same value as 'b and a', namely, they are both ...
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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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Logic terminology: does “conditional” etc refer to the operator or the WFF?

To be precise.... Do the terms we use to talk about the truth-functional operators (conditional, negation, conjunction, disjunction, biconditional) refer to the operator in isolation, or the WFF that ...
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310 views

What is model theory?

I have never been able to understand any need or even any benefit of model theory. Both Rudolf Caranp and Richard Montague showed how to encode semantics directly in the syntax. Can you help me ...
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1answer
31 views

What are the legal quantifier negation rules?

Is using Quantifier Negation to flip two quantifiers at once legal in symbolic logic? Example: ~∀x∀y(Hhx & Hhy) ƎxƎy~(Hhx & Hhy) 1 QN Or do I need to do this in 2 steps? Example: ~∀x∀...
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Are verb tenses actually irrelevant in logic?

In my Introduction to Logic course, we learned that verb tenses are irrelevant when symbolizing and deducing arguments. However, it seems to me that the verb tenses could sometimes choose whether or ...
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65 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, ...
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3answers
91 views

A deontic premise that leads to a necessity from a permission

I wanted to devise some rules for myself, then formulate those rules using formal symbolic logic, and one of the rules that I have set for myself is : "Do not say what is unnecessary", in other words :...
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2answers
120 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 ...
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2answers
73 views

How does one tell if logical expressions are equivalent?

How do I check if these expressions are equivalent? ∀a,b [P(a) ∧ ¬R(a) ∧ S(b)] → G(a,b) ∀a [(P(a) ∧ ¬R(a)) → (∀b [S(b) → G(a,b)])]
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Decidability of 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?
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How to solve this natural deduction problem?

This one is driving me crazy. I don't understand most keys for de morgan, modus ponens, etc, so please abbreviate if possible? EX: DM, MP, SIMP, HS, Conj, Imp (material Implication). Thank you anybody ...
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Honestly have no Idea how to prove A v ¬¬B from A v B (LPL Q. 6.18)

Premise A v B Goal A v ¬¬B Please help. It seems so self evident but I don't know how to get there.
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How to define a new logical language

There are a number of logical languages defined that are so called flavors of a certain logic (for example, flavors of First Order Logic). Such new logical languages sometimes extend or restrict the ...
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367 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 ...
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59 views

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

i have a silly logic question again. How would you translate the following sentence into predicate logic? No dolphin sings unless it jumps. My guess is that it is an E-sentence of the form "no A ...
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How to translate “Only dogs and dolphins jump if petted” into predicate logic?

The Logic2010 software provides exercises for the symbolization of English sentences. But i'm stuck with one symbolization, which closely resembles another one I solved correctly. The first one ...
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1answer
62 views

What is the meaning of free variables?

In some books I see something like: phi(x1,...,xn) with free variables among x1,...,xn. An example of this is on 2nd paragraph, page 25 of the book Knowledge in Action. I have included a snapshot of ...
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1answer
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Logical, semantic and self-referential paradoxes: The Truth teller and the Liar (draft) can an expert on the matter give feedback?

Title: Logical semantic and self-referential paradoxes: The Truthteller and the Liar (draft, informal) (major) assumption: A statement is either true or not true (law of excluded middle, classical ...
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Fitch Proof - Arrow's logic of preferences

I've been stumped on this one question in particular for several days now and I'm hoping to get some help on where I'm going wrong. Given the following premises: ∀x∀y(StrongPref(x,y)→ ¬StrongPref(y,...
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265 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 ...
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1answer
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Do computer languages instantiate only predicate calculus?

All the computer languages I'm familiar with, be they imperative or declarative have the same core mechanics (arithmetic and logic).They have the same loops, conditionals etc. Whatever the language it ...
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A question about possibility

If A, then B ~ A So, possible that B Valid or not? My take: Not valid. Reason: Valid means if all the premises are true, the conclusion must be true That means adding new information should not ...
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1answer
79 views

Proof Tree to Fitch Proof

I was wondering if anyone could help me on a proof I've been working on: I was able to check that it is valid with a proof tree generator (prooftools): However, I still haven't figured out the proof....
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62 views

Fitch Question Please Help Me [closed]

I'm having trouble understanding writing out a proof. The proof I'm trying to work with is : [![enter image description here][1]][1] How do I reach this goal? Which rules do I use and with which ...
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1answer
71 views

Fitch Questions Please Help Me

I'm having trouble understanding writing out a proof. The proof I'm trying to work with is : How do I reach this goal? Which rules do I use and with which support steps to each rule (proofs to prove ...
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3answers
800 views

How is Wittgenstein’s “notorious paragraph” about the Gödel's Theorem not obviously correct?

Timm Lampert quotes from Wittgenstein's "notorious paragraph" (§8 of Remarks on the Foundations of Mathematics, Appendix 3) in http://wab.uib.no/agora/tools/alws/collection-6-issue-1-article-6....
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Why do we need model theory to express semantics?

https://en.wikipedia.org/wiki/Model_theory Why can't semantics be directly expressed in the formal language? This is the key part of model theory that I don't understand: https://www.lesswrong.com/...
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74 views

Hard Predicate Proof Help

I have been working on this proof for over a week now, and I can't seem to figure it out: Pd ⟷ (Hj & Mj), Gsd, ∀x∀y∃z(((Gxy & (Py ➝ Pz)) & Rxyz) ➝ Gxz), Pe ⟷ ∀x(Hx ➝ Mx), Rsde |- Gse I ...
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152 views

Can classical logic have deduction with infinite steps

I've been reading the Stanford Encyclopedia of Philosophy article on classical logic, and I've been confused about Theorem 9, and the preceding statement. They mention how (*), the clause which ...
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Do you know of any mathematical theorem whose proof relies on the use of the principle of explosion (ECQ)?

Ex contradictione (sequitur) quodlibet (ECQ) is almost universally recognised in mathematical logic as a valid inference. In symbolic logic, this inference is usually expressed in the following way: ...
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131 views

Checking the validity of the logical conclusion gleaned from a heated conversation

I have two friends - call them John and Jane. I was recently privy to an argument concerning a book between John and Jane that went like this: John: This book did not make a single coherent, ...
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108 views

Fitch-style natural deduction

How to prove the following questions? (a) p from assumption ¬(p → q) (b) ¬¬p → p from no assumptions.
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trouble with rules of inference practice problems [closed]

Prove the following symbolized arguments applying the appropriate rules of inference: 1) P ∨ Q = M ⊃ ¬ Q M =conjunction Therefore P 2) (P V Q) ∧ ¬ Q P ⊃ R =...
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107 views

How should I use the propositional logic rules for → and ↔?

My question is how should I use the propositional logic rules for → and ↔ (although other rules may be required) to prove the following: A → B, B → C ⊢ (AvB) → C A ↔ B ⊢ ¬A ↔ ¬B Please use the ...
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2answers
175 views

Known self-evident unproven logical truths

Is there any authoritative source for all known self-evident logical truths that most specialists would agree are true although they can't be proven? There are many different axiomatic systems, and ...
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55 views

Does 'until' imply a conditional with a negative consequent?

Suppose a father tells his kid that he can play video games whenever he wants. Then, one day, when the kid fell sick, the father told him that he can play video games until he recovers. Does this '...
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95 views

Symbolic logic and rules of inference: two questions

Question one: (C>D) & (D>B) (B>D) & (E>C) (D>C) BvE ∴ DvB ? ? ? ? DvB I'm fairly sure this questions has constructive dilemma at the end, but after four hours of working on these two ...