Tool
Quick reference
Everything from 2.1 and 2.2 you’re likely to look up, on one page.
English to SL
| English | Paraphrase | SL |
|---|---|---|
| not p; p isn’t so; un-, in-, non- | it is not the case that p | ∼P |
| p and q; p but q; although p, q; p yet q; p however / moreover / nevertheless q; both p and q | p and q | P & Q |
| p or q; either p or q; p unless q; at least one of p, q | p or q | P ∨ Q |
| p or q but not both | (p or q) and it is not the case that (p and q) | (P ∨ Q) & ∼(P & Q) |
| if p then q; q if p; q provided that p; q given p; assuming p, q; should p, q | if p then q | P ⊃ Q |
| p only if q; only if q, p | if p then q | P ⊃ Q |
| p if and only if q; p if but only if q; p just in case q (when it states an exact condition) | p if and only if q | P ≡ Q |
Common traps
- “Only if”
- Introduces the consequent. “A only if B” is A ⊃ B, never B ⊃ A.
- “Unless”
- Becomes “or”. “A unless B” is A ∨ B.
- Someone / no one
- “Someone isn’t here” is ∼(everyone is here). “No one is here” is ∼(someone is here).
- Not both
- “Not both A and B” is ∼(A & B), which differs from ∼A & ∼B (neither).
- Inclusive “or”
- A ∨ B is true when both are true. Add ∼(A & B) to exclude that.
- Just in case
- A biconditional only when it states an exact condition, not when it explains a precaution.
Syntax
- Sentence letters
- A–Z, with or without positive-integer subscripts.
- Connectives
- ∼ tilde (unary); & ampersand, ∨ wedge, ⊃ horseshoe, ≡ triple bar (binary).
- Metavariables
- Boldface P, Q, R, S range over sentences of SL; p, q, r, s over English sentences. Not part of SL.
- Definition
- (1) Every sentence letter is a sentence. (2) ∼P. (3–6) (P & Q), (P ∨ Q), (P ⊃ Q), (P ≡ Q). (7) Nothing else.
- Main connective
- The connective whose scope is the whole sentence. It sets the type: negation, conjunction, disjunction, material conditional, material biconditional.
- Components
- The sentence itself, its immediate components, their components, and so on down to sentence letters.
- Conventions
- Outer parentheses may be dropped (never after a ∼). Square brackets may replace parentheses.
Typing on this site
| Symbol | Type |
|---|---|
| ∼ | ~ or - |
| & | & or ^ |
| ∨ | v or | |
| ⊃ | > |
| ≡ | = |
Lowercase letters become capitals as you type, so a gives A. Use Shift+V for the sentence letter V, since lowercase v is the wedge.