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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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Philosophers or philosophical traditions that reject symbolic reasoning

I'm most familiar with philosophy in the context of discussing various flavors of logic, such as independence-friendly logic, various extensions of first-order logic with plurals, relevant logic, and ...
Greg Nisbet's user avatar
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20 votes
11 answers
21k 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, ...
Tames's user avatar
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7 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 ...
IgnorantCuriosity's user avatar
3 votes
1 answer
137 views

What are the arguments of philosophers against the reasoning which justifies the horseshoe from truth-functionality?

There is a reasoning in mathematical logic which is meant to prove that the horseshoe is the only logical operation which fits our notion of conditional. The reasoning starts from the idea that the ...
Speakpigeon's user avatar
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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?
sumsum2's user avatar
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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, ...
Zenreon's user avatar
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3 votes
2 answers
881 views

In modern logic, why does "All S is P" contradict "Some S is not P"?

In modern logic, the existential import is removed from universal statements. So All S is P may still be true if there is no S at all. Contradictory statements must have opposite truth values. Why ...
KohLP's user avatar
  • 91
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, ...
RECURSIVE FARTS's user avatar
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 ...
arileah's user avatar
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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. ...
BeingOfNothingness's user avatar
4 votes
2 answers
168 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 ...
John's user avatar
  • 153
3 votes
2 answers
348 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 ...
QuizzicalTest's user avatar
3 votes
3 answers
3k views

Prove A ∨ D from A ∨ (B ∧ C) and (¬ B ∨ ¬ C) ∨ D ( LPL Q6.26) without using --> or material implication

This is a repeated question: Language Logic and Proof Q. 6.26 Using the natural deduction rules, give a formal proof of A ∨ D from the premises A ∨ (B ∧ C) (¬ B ∨ ¬ C) ∨ D ...
Billy Bob's user avatar
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3 votes
5 answers
292 views

At what point in the history of mathematics, and why, did mathematicians come to say "A implies B" to mean "not A or B"?

Here is what one respondent to my previous question says: A big part of the problem here lies with interpreting the word ‘implies’, which is ambiguous in English. Unfortunately, mathematicians get ...
Speakpigeon's user avatar
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3 votes
2 answers
1k views

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

I'm having trouble proving the following using natural deduction: (T > E) ^ (A > L) /... (T v A) > (E v L) I checked the answer but I didn't quite understanding the reason why the proof progressed ...
anonymous's user avatar
2 votes
2 answers
101 views

How does 'use-mention' apply to formulas?

When we use 'terms' such as words it is generally clear however, if we have a formula: And I write: 'x+1=2 is true for x=1' is this 'using' or 'mentioning'? If a formula contains variables, it has no ...
Confused's user avatar
  • 1,191
2 votes
3 answers
5k views

Prove (¬P ∨ Q) ↔ (P → Q)

How can one use a standard logic proof to prove this without using any premises? I've tried doing subproofs and splitting up ¬P and Q to try to get to P → Q but I'm very stuck!
Ethan Seiler's user avatar
2 votes
2 answers
2k views

Verum, Falsum, Atoms

I have been somewhat confused about the definition of atoms, or atomic formulae. Some sources say that verum (⊤) and falsum (⊥) are atoms, some not. Is there any consensus within the community or is ...
Somniare's user avatar
1 vote
3 answers
770 views

What are the rules for discharging a premise in a Zero-Premise Deduction?

If I have the problem (A → B) v (B → C), is there a way to prove this from no premises without first using Material Implication to convert the statement into ¬(A → B) → (B &...
PWoeks's user avatar
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1 vote
1 answer
103 views

Wetzel's 'occurences'

I was reading this often quoted article by Linda Wetzel (1993) where she discusses the 'occurence' of expressions in others and Quine's issues with the idea, she describes an expression as a sequence ...
Confused's user avatar
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1 vote
2 answers
519 views

Can inductive arguments be made in first order logic and, if not, why not?

After reading a question by rus9384 Why is faulty generalization called an informal fallacy? I wondered whether induction can be part of any argument in first order logic (FOL). rus9384 symbolized ...
Frank Hubeny's user avatar
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1 vote
0 answers
167 views

Semantic expressiveness of modal logic

I am wondering how much of the semantic of basic philosophical questions can be expressed by formal arguments in modal logic. Here is one argument I formalised myself: P1 ◇ ∀a, ∃x // GNB(x, a) ∧ C(a)...
Speakpigeon's user avatar
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1 vote
2 answers
2k views

How to prove ~ (~P & ~Q) : P ∨ Q by natural deduction

Here's another of Tomassi's exercises I can't solve (Logic, page 106): ~ (~P & ~Q) : P ∨ Q I have to use natural deduction and the only rules I know are: assumptions, modus ponendo ponens, ...
Diego Ruiz Haro's user avatar
1 vote
2 answers
6k views

How to get proof using proof editor and checker

How can I use Natural deduction proof editor and checker or The Logic Daemon to derive the given conclusion from the given premise: (∃x) ( Fx ∙ (y) (Fy → y = x) ) / (∃x) (y) (Fy ≡ y = x) It tells me ...
wa7d's user avatar
  • 289
1 vote
3 answers
264 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 ...
Moritz Loritz's user avatar
0 votes
1 answer
64 views

Question about proving a set that is quantificationally inconsistent in PD+ (Finished the proof but want it to be checked)

Does ∃x(Nx & ~Nx) contradiction itself? Is there an error in my proof? Thank you
Stanley's user avatar
0 votes
2 answers
256 views

Is the material implication the correct model of conditional reasoning in mathematics?

Question: Do you believe that the material implication correctly models the kind of conditional reasoning necessary in mathematics to prove a theorem? Example: If x > y and y > 0, then x > ...
Speakpigeon's user avatar
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-3 votes
8 answers
2k views

Why does the Principle of Explosion not make Mathematical Logic inconsistent? [closed]

Step Proposition Derivation 1 ------P --------- Assumption 2 ---- ¬P --------- Assumption 3 ----- P ∨ Q ----- Disjunction introduction (1) 4 ----- Q --------- Disjunctive syllogism (3,2) https:...
polcott's user avatar
  • 477
-3 votes
2 answers
775 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 ...
polcott's user avatar
  • 477