TR1413: Discrete Mathematics For Computer Science Lecture 4: System L

Introduction One of the famous propositional formal system was developed by Lukasiewicz, known as System L.

Alphabets in System L Consists of –The infinite set of propositional variables {p 1,p 2,…,p n } –The set of punctuation symbols {‘,’,’(‘,’)’} –The set of logical operators { ㄱ,}

wffs in System L A propositional statement P is a wff in System L if it conforms to one of the following conditions: –P is a propositional variables; or –P is of the form ( ㄱ Q) where Q is a wff; or –P is of the form (Q  R) where Q and R are wffs.

Axioms in System L Three axioms: –L1: (U  (V  U)); –L2: ((U  (V  W))  ((U  V)  (U  W))) –L3: ((( ㄱ U) ( ㄱ V))  (V  U)) where U, V and W are wffs.

The Inference Rule of System L System L has only one rule of inference, known as modus ponens (MP). The rule: –If V and (V  W) are wffs of L which are members of a proof sequence the the wff W can be added to the proof sequence.

Proof in System L A proof is defined to be a sequence of wffs of L such that each wff is either –An instance of an axiom of L; or –Derivable from two earlier wffs in the sequence using the rule MP. The two earlier wffs must be of the form P and (P  Q). Every steps in a proof must be justified clearly.

Examples of proof in L

Deduction in System L A deduction from the set of wffs T in System L is defined to be a sequence of wffs such that each wff is either –An instance of one of the axioms of L, or –One of the hypotheses, or –Is derivable from two earlier wffs in the sequence using the rule MP. If such a sequence exists, the the sequence of wffs leading to a given wff P is said to be a “deduction of P from T in L”.

Examples of deduction in L

Derived Rules of deduction The Deduction Theorem The Inverse Deduction Theorem Hypothetical syllogism

The Deduction Theorem Can be stated as: –If T  {U} ᅡ V where U and V are wffs, and T is a set of wffs, then T ᅡ (UV) Meaning: –If V is deducible from a set of hypotheses which include U, then UV is deducible from the set with U removed.

The Inverse Deduction Theorem Can be stated as –If the deduction T ᅡ (UV) where U and V are wffs, and T is a set of wffs, can be established, then the result T  {U} ᅡ V holds.

Hypothetical Syllogism Can be stated as: –(UV), (VW) ᅡ (UW)

Examples

Notation In the textbook, rules are written by using a horizontal bar. For example, Hypothetical Syllogism is written as (UV), (VW) (UW)