Prepositional Logic: Semantics and Syntax

Artificial languages: used for logic because natural languages are not precise enough and have double meanings.

Formal language: represents similarities among arguments of a natural language, such as sentential logic, or propositional logic. A more sophisticated language is predicate logic.

Sentence letters: represents complete preposition in natural languages.

Parenthesis: (         )

• Parenthesis may be dropped for convenience so long as there is no ambiguity resulted
• Strongest bond to Weakest bond: ~, & and v, -->, <-->

ex.
~P & Q --> R is read as ((~P & Q) --> R)
P --> Q <--> R is read as ((P --> Q) <--> R)
P v Q & R is not allowed, as it is ambiguous between (P v (Q & R)) and ((P v Q) & R)
P --> Q --> R is not allowed, as it is ambiguous between (P --> (Q --> R)) and ((P --> Q) --> R)

Connectives:

• Ampersand & both are true
• Wedge v or
• Arrow --> if Φ, then Ψ
• Double arrow <--> if and only if
• Tilde ~ not

Well-formed Formulas WFF (Pronounced Wuff) is any expression that accords to these 7 rules:

• Atomic sentence: a sentence letter standing alone is a WFF. Ex. A, B, D12
• Negation: If Φ is a WFF, then ~Φ is a WFF. This rule is recursive.
• Binary connectives:&, v, -->, <--> Unary connective: ~

If Φ(Phi) and Ψ(Psi) are WFFs, then...

Conjunction (Φ & Ψ) Φ and Ψ are conjuncts

ex. Φ & Ψ
Φ and Ψ
Both Φ and Ψ
Φ, although Ψ
Though Φ, Ψ

Disjunction (Φ v Ψ) Φ and Ψ are disjuncts

ex. Φ v Ψ
Φ or Ψ
Either Φ or Ψ
Φ unless Ψ (~Φ --> Ψ)

Conditional (Φ --> Ψ) Φ is antecedent. Ψ is the consequent

ex. Φ --> Ψ
If Φ, Ψ
Φ, only if Ψ
Φ provided that Ψ
Φ on the condition that Ψ
Φ is a necessary for Ψ
Whenever Φ, Ψ
Given that Φ, Ψ
In case Φ, Ψ

Biconditional (Φ <--> Ψ)

ex. Φ <--> Ψ
Φ if and only if Ψ
Φ is equivalent to Ψ
Φ is necessary and sufficient for Ψ
Φ just in case Ψ