# 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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### 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 ...
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 ...
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 ...
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) ∧ ...
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/...
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 ...
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 ...
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 ...
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!
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 ...
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, ...
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 &...