Formal Logic Proof Methods Direct Proof / Natural Deduction Conditional Proof (Implication Introduction) Reductio ad Absurdum Resolution Refutation (Negation as Failure) Tableaux Induction

References 1.Direct Proof / Natural Deduction 2.Conditional Proof (Implication Introduction) 3.Reductio ad Absurdum 4.Resolution Refutation (Negation as Failure) 5.Tableaux 6.Induction For 1 see Frost (handout) or any other source. For 2 and 3, see Konyndyk (handout). For 4, see Frost (handout). Other sources as well. For 5, see Frost (handout) For 6, see Hughes&Creswell.

Direct Proof 1.Axioms and Inference Rules of the Logic System (e.g. PC, FOPL) are given. 2.Find a derivation from the set of universally valid formulas of the given logic, its axioms, with the conclusion  (the formula to be proven), by repeatedly applying inference rules and thus generating formulae. 3.A chain of inferences, leading from the axioms (or a subset thereof) to the conclusion  is a proof of . 4.If nothing else is used than the axioms and inference rules of the given logic system, the derived formulae are theorems of the logic system. They are universally valid formulae of this logic system. 5.If proper axioms are used (i.e. a theory), the derived formula is a theorem of this theory.

Natural Deduction 1.Use empty set of logical axioms and special sets of inference rules. 2.Inference rules are divided into Exploitation Rules and Introduction Rules. 3.To prove a formula, we make "assumption" (suppose P), and use inference rules, to derive formulae. 4.The assumption is the condition part in an implication formula, as a result of this process. 5.Reduce this formula until you derive the formula to be proven. There are different variations of this type of proof method.

Conditional Proof (Implication Introduction) 1.State what you want to proof. 2.Make assumption  (antecedent). This is start of (a part of) the proof. Each assumption has a scope. Scopes can be nested (like programs). 3.Derive formulae  (consequent), within the scope of an assumption, based on the assumption (and earlier consequents), through applying inference rules. 4.Terminate the assumption by adding a new line (outside the scope), stating the implication  (implication introduction). This discharges the assumption. You can also terminate an assumption by leaving its scope without conclusion. 5.If you can finally introduce an implication reflecting the original formula, you are done.

Reductio ad Absurdum (Negation Introduction) Reductio ad Absurdum is a modification of the Conditional Proof (Implication Introduction) method. Add: If you find two formulae  and  within the scope of an assumption , you can terminate the assumption with the conclusion .

Resolution Refutation (Negation as Failure) 1.Transform formula into conjunctive normal form (CNF) = a conjunction of disjunctions of literals. (p 1 ...  p n ) ...  (q 1 ...  q m ) A literal is an atomic formula or its negation. Steps: eliminate implications  reduce the scope of negation  apply distributivity rules to collect disjuncts

Resolution Refutation (Negation as Failure) 2.Clause form A disjunction of literals written as a set. p  q   r  {p, q,  r} Such a set {p, q,  r} is called a clause. 3.Horn Clause If a clause contains at most one positive literal, it is called a Horn clause (  PROLOG). 4.Clause Set A conjunction of clauses written as a set. {{p, q,  r}, {p, q, r}}  ((p  q   r)  (p  q  r)) Such a set is called a clause set (or Horn clause set).

Resolution Refutation (Negation as Failure) Principle of Resolution Refutation: Given a formula (in conjunctive NF / as clause set) ((p  q   r)  (  p   q  r)) Obviously, such a formula is not satisfiable, if each literal in one clause has a corresponding negated literal in the other clause (why?). Let's do this: If two clauses have the same atomic formula, one clause as positive literal, the other one as negative literal, then throw the literals of the two clauses together into one clause, crossing out positive literals and their negations, e.g. p and  p. The resulting clause is called the Resolvent (of the two clauses).

The Resolution Rule Resolution Rule: If two clauses contain the same atomic formula, one clause as a positive literal, the other clause as a negative literal, then throw the literals of both clauses together into one clause, crossing out positive literals and their negations, e.g. p and  p, one each from each clause. The resulting clause is called the Resolvent (of the two clauses). Duplicates have to be removed, e.g. p, p. Continue this process until you either cannot build a new resolvent, or until you resolve the empty clause. If you derive the empty clause, the negated assumption has been proven to be contradictory with the other (true) formulae. Thus, the (negated) assumption is false, and the (positive) assumption must therefore be true. This means that the to be proven formula β can be derived from the given set of formulae .

Resolution Refutation (Negation as Failure) Principle of Resolution Refutation (contd.): If we have several clauses in a clause set, like: ((p  q   r)  (  p   q)  (  q  r)) we can resolve incrementally, in each single step building the resolvent of two clauses. If we get at some point in time an empty clause, we are done (why?).

FOPL and Proof Theory A logic system is decidable, if it is always possible to determine whether a theory is consistent or not. A logic system is semi-decidable, if it is always possible to determine inconsistency but not always possible to determine consistency. !Full FOPL is semi-decidable. !Function-free FOPL is decidable.

Tableau Proof Methods see Priest, section 1.5, page 8ff, and Frost, section 4.2.5, page 186ff. The basis of the Tableau Proof method is to find out whether a formula (to be proven)  leads to a contradiction with a true formula (like a logical axiom or proper axiom) . We construct a tree of formulae starting with the initial formulae (the true formula) and the negated conclusion, i.e. the formulae to be proven in negated form. Successively split up the formulae (according to reverse inference rules) in order to derive simpler formulae. Similar to resolution, branches of the tree which contain a formula and its negation are closed. They are "cut out". If every branch of a tableau closes, the tableau is compete and the proof is done.

Tableau Proof see Priest, section 1.5, page 8ff Steps to set up the Tableau (as tree): 1.Write assumption and negated conclusion under each other. Formulae underneath each other are conjunctive. 2.Split formulae (if necessary take out implications). 3.Conjunctively joint sub-formulae go into the same branch of the tree, under each other. 4.Disjunctively joined sub-formulae go into separate sub- branches of the tree. 5.Compare atomic propositions in the branches. 6.If a branch contains a proposition and its negation, it will be closed. Write x on its leaves. 7.If all branches are closed, the tableau is complete. Done.

Tableau Proof - Rules (Priest)

Tableau Proof (example 1)

Tableau Proof (example 2)

Tableau Proof with Inverse Sequent Rules (Frost) see Frost, section 4.2.5, page 186ff. Steps to set up the Tableau (as table): 1.Write true formula  and to be proven conclusion  on two sides of a column. (  left and  right) 2.Apply inverse sequent logic rules (  inverse inference rules). 1)E.g. substitute P  Q by P and Q in same column. 2)E.g. substitute R  S on the right side by adding R to the left side and S to the right side. 3.If the same atomic formula appears on both sides, the tableau closes. Done.

Sequent Logic and Sequent Proofs see Frost, section 4.2.3, page 183ff. Sequent:A 1,..., A m  C 1,..., C n A 1,..., A m, C 1,..., C n arbitrary formulae A 1,..., A m  C 1,..., C n corresponds to A 1 ...  A m  C 1 ...  C n  inference in sequent calculus; not the same as  nor as |- Inference Rules in Sequent Logic:  Antecedent Introduction Rules  Consequent Introduction Rules  Modification Rules

Sequent Logic and Sequent Proofs see Frost, section 4.2.3, page 183ff. Inference Rules in Sequent Logic:  Antecedent Introduction Rules T  U, ab, W  x (A  ) a  b, T, W  U, x a, T  Ub, T  U (A  ) a  b, T  Ua  b, T  U a, T  U b, T  U (A  ) a  b, T  U T  U, a (A  )  a, T  U

Sequent Logic and Sequent Proofs see Frost, section 4.2.3, page 183ff. Inference Rules in Sequent Logic:  Consequent Introduction Rules a, T  U, b (C  ) T  U, a  b T  U, aT  U, b (C  ) T  U, a  b T  U, a T  U, b (C  ) T  U, a  bT  U, a  b a, T  U (C  ) T  U,  a

Sequent Logic and Sequent Proofs see Frost, section 4.2.3, page 183ff. Inference Rules in Sequent Logic:  Modification Rules T  UT  U Thinning / Weakening a, T  U T  U, a a, a, T  UT  U, a, a Contraction a, T  UT  U, a T, a, b, U  WT  U, a, b, W Interchange T, b, a, U  W T  U, b, a, W T  U, aa, W  x Cut T, W  U, x

Resolution Theorem Proving in FOPL

see Frost, section and 4.4, page 227ff. Principle of Resolution Refutation in FOPL: 1.Bring formula into prenex form, i.e. move all quantifiers to the left end (apply equivalences). Change variable- names if necessary (rectification). 2.In negatively quantified formulae, replace their bound variables with Skolem-functions (or -constants). 3.Introduce Skolem-Constants to replace existentially quantified variables. 4.Remaining formula in Skolem NF contains only (implicitly) all quantified variables.

Prenex Form - Equivalences

Skolemization

Steps for Resolution in FOPL

Resolution - Example