# 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 the Law of Identity a Logical Absolute? Can it be doubted, violated, or disproven (i.e., proven false)?

**The First Law of Thought: The Law of Identity** Something is what it is [L_1], and it is not what it is not [LI_2]: -| yields the logical conjunction: [LI_1 & LI_2], & = “and”; Therefore, ...
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### Fitch proof help no premise [closed]

There is no premise just a conclusion of ¬C on fitch. I am very stuck on this problem.
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### How can this shorten ( ~p v r ) \$\and\$ (~q v r) to ((~p \$\and\$ ~q) v r)?

I am just confused how this be possible, what rule is been used?
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### How to draw a truth table on Stack Exchange? [migrated]

I need to draw/input tables, such as truth tables in order to ask and answer questions about logic. Can someone please instruct me on how to make a truth table on this site. My goal is to make a truth ...
90 views

### Equivalence Thesis

What is, if any, the canonical justification accepted in mathematical logic for the Equivalence Thesis, asserting (1) that indicative conditionals are truth-functional logical expressions and (2) that ...
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### Why is Turing claiming that a complete and computable axiomatization of arithmetic would imply the decidability of first-order logic?

So I'm reading the famous paper of Turing "On Computable Numbers, with an Application to the Entscheidungsproblem". At the beginning of his proof of the undecidability of first-order logic (FOL), he ...
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### The form of elementary propositions in TLF

In Tractatus Wittgenstein states that: 4.22 An elementary proposition consists of names. It is a nexus, a concatenation, of names. Suppose now that L is a first order language. As far as I ...
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### fitch proof. P v Q Q→ ¬ R ¬ P ¬ R → ¬ S GOAL: ¬ S

Need help exercise using the FITCH program format. I'm stuck on where to start. The following 4 steps must be used to prove the goal. P v Q Q→ ¬ R ¬ P ¬ R → ¬ S GOAL: ¬ S Now I know: ¬ P and P v Q ...
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### Solving a proof with Fitch

I'm working on an assignment and I'm stuck on this proof. I feel like I'm on the right track but I can't find the way to prove the goal. B ^ D (B^¬A) → ¬C B → ¬A (D^E)→ (A v C) GOAL: ¬E
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### 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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### What is (∃!e) in Davidson 1967?

Davidson proposes a causal law for singular event causation. Here is one example of a backward-looking part of such a law (changed slightly for clarity and to avoid copyright infringement): (e)(u)((...
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### Alphabetic Substitution, Barcan, and Strict Implication

Context: I'm stuck on Axiom 8 from the introduction to Barcan 1946, "A Functional Calculus of First Order Based on Strict Implication." My instinct is that I'm missing a basic, perhaps obvious concept-...
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### Is Classical Logic the proper model of the deductive logic of human reasoning?

Which mathematicians and philosophers unambiguously claimed that Classical Logic was the proper model of the deductive logic of human reasoning, and when did they say it? The expression "Classical ...
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### 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 ...
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### How to prove (A v ¬ B), (¬ A v C), (¬ C → B) therefore (¬ D v C)

My idea is to use disjunction elimination on (¬ A v C)to obtain C, and then use disjunction introduction to obtain (¬ D v C), but I'm having a hard time obtaining C.
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### Complete a formal proof of ~(~A&~B) from A in as few lines as possible

Prove ~(~A&~B) from A in as few lines as possible. ~ = negation & = conjunction v = disjunction | = line in a subproof Here's what I have: A - Premise |~A - Assume |~B ...
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### How do I prove :((A ⊃ B) ⊃ C) ⊃ (B ⊃ C)?

How do I prove, :((A ⊃ B) ⊃ C) ⊃ (B ⊃ C), using symbolic logic derivations where ⊃ represents a conditional i.e. A ⊃ B = A implies B? The first line of my derivations is the assumption, (A ⊃ B) ⊃ C)....
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### In Quine's ontology, why does a 'recognition' of something lead to ontological commitment while a 'feeling' does not?

We are discussing Quine's On What There Is in a metaphysics class I am in. I felt like I understood what he meant, that if something has to be predicated for in a sentence, we are ontologically ...
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### 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), ...
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### Is there a semantically complete system of direct-method deductive logic?

Does anybody know of a system of direct-method deductive (propositional) logic, in other words, a system that does not require (or even incorporate) conditional (and indirect) proof method(s) and that ...
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### 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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### How can one prove ∃𝑥(𝐹(𝑥)→𝐺(𝑥)) given the premise ∃𝑥𝐹(𝑥)→∃𝑥𝐺(𝑥)? [closed]

Premise: ∃𝑥𝐹(𝑥)→∃𝑥𝐺(𝑥) Desired Conclusion: ∃𝑥(𝐹(𝑥)→𝐺(𝑥))
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### Question about v-variants of variable assignments

In Full Predicate Calculus, some variable assignment q satisfies a disjunction under interpretation U if q satisfies one or both disjuncts under U, it satisfies a conjunction under interpretation U if ...
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### Truth Value of Definite Descriptions

I'm currently studying definite descriptions in logic. My textbook postulates Bertrand Russell's view of definite descriptions, but I'm curious about other views as well (in the context of classical 2-...
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### Fitch Proof Exercise 6.20

I am working on a proof and am stuck on a step. I am not sure why I cannot assume the negation of B. Is it not allowed or am I missing something? Thank you]1
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### How can I prove the law of excluded third (p ∨ ¬p)) using Fitch?

Good day. I do not quite understand how I can get ~~p after the 11th line. According to the proof of the law itself (and all reasonable logic) I should get it, and then simplify the expression - but ...
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### Proof by natural deduction advice

Any advice on shortening my right to left proof will be appreciated.
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### Help with natural deduction by introduction and elimnation rules

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 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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### 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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### 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, ..., rn}, each with ...
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### 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 ...
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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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### 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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### 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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### 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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### 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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### 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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### 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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### 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?
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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 ...