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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7answers
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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 ...
3
votes
3answers
629 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 ...
3
votes
2answers
330 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 ...
1
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0answers
156 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) ∧ ...
-3
votes
3answers
194 views

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/...
4
votes
2answers
88 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 ...
3
votes
2answers
309 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
2answers
326 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 ...
2
votes
3answers
1k 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!
2
votes
2answers
730 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 ...
1
vote
2answers
211 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, ...
1
vote
3answers
601 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 &...