# PDF Natural Deduction

It embodies proofs by contradiction. It says that if by assuming that P is false we can derive a contradiction, then P must be true. The assumption x is discharged in the application of this rule. This rule is present in classical logic but not in intuitionistic constructive logic. In intuitionistic logic, a proposition is not considered true simply because its negation is false.

We will take it as an axiom in our system. The Latin name for this rule is tertium non datur , but we will call it magic. A proof of proposition P in natural deduction starts from axioms and assumptions and derives P with all assumptions discharged. Every step in the proof is an instance of an inference rule with metavariables substituted consistently with expressions of the appropriate syntactic class.

The proof tree for this example has the following form, with the proved proposition at the root and axioms and assumptions at the leaves. A proposition that has a complete proof in a deductive system is called a theorem of that system. A measure of a deductive system's power is whether it is powerful enough to prove all true statements. A deductive system is said to be complete if all true statements are theorems have proofs in the system. For propositional logic and natural deduction, this means that all tautologies must have natural deduction proofs.

Conversely, a deductive system is called sound if all theorems are true. The proof rules we have given above are in fact sound and complete for propositional logic: every theorem is a tautology, and every tautology is a theorem. Finding a proof for a given tautology can be difficult. But once the proof is found, checking that it is indeed a proof is completely mechanical, requiring no intelligence or insight whatsoever. Such a solution simplifies a formulation of rules and eliminates the risk of a clash of variables while applying the rules. When we provide ND rules for more standard approaches with just individual variables which may have free or bound occurrences, we must be careful to define precisely the operation of proper substitution of a term for all free occurrences of a variable.

The first application of introduces a parameter in place of. In line 3 and 7 the assumptions for the applications of in line 5 and 10 respectively are introduced, each time with a new eigenparameter in place of. Note that both applications of are correct since neither nor are present in the formulas ending suitable subproofs.

Also the application of in line 6 is correct since is not present in line 1. The fact that is a proof construction rule is obscured here since there is no need to introduce a subproof by means of a new assumption. We just require that in order to apply there be no occurrence of an involved parameter here in active assumptions.

Although the idea is simple its correct implementation leads to troubles. Carefull formulations of such a rule as in Quine are correct but hard to follow; simple formulations as in several editions of Copi make the system unsound. For a detailed analysis of the relations between Gentzen-style and Quine-style quantifier rules one should consult Fine , Hazen and Pelletier All these problems with providing correct and simple rules for quantifiers led some authors to doubt if it is really possible see Anellis ND systems were also offered for many important non-classical logics.

## Natural Deduction

It appeared that for many non-classical logics one can obtain a satisfying result by putting restrictions on the rule of repetition in the case of some subproofs. Let us take as an example the ND formalization of well known propositional modal logic T; for simplicity we restrict considerations to rules for necessity.

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With the situation is more complicated since it is based on the following principle:. If in such modal subproof we deduce , it can be closed and can be put into the outer subproof. In line 4 a modal subproof was initiated which is shown by putting a sole in place of the assumption.

## Propositional Logic and Natural Deduction

Lines 5 and 6 result from the application of modal repetition. Such an approach may be easily extended to other modal logics by modifying conditions of modal repetition; for example, for S4 it is enough to admit that formulas with no deletion also may be repeated; for S5, formulas with negated are also allowed. This modus of formalizing logics in ND was also applied for other non-classical logics including conditional logics Thomason , temporal logics Indrzejczak and relevant logics Anderson and Belnap In the latter the technique of restricted repetition is not enough however and even not required for some logics of this kind.

Far more important is the technique of labeling all formulas with sets of numbers annotating active assumptions which is necessary for keeping track of relevance conditions. Vigano provides a good survey of this approach. When constructing proofs one can easily make some inferences which are unnecessary for obtaining a goal.

### Bulletin of Symbolic Logic

Gentzen was interested not only in providing an adequate system of ND but also in showing that everything which may be proved in such a system may be proved in the most straightforward way. In particular, such unnecessary moves are performed if one first applies some introduction rule for logical constant and then uses the conclusion of this rule application as a premise for the application of the elimination rule for. In such cases the final conclusion is either already present in the proof as one of the premises of respective introduction rule or may be directly deduced from premises of the application of introduction rule.

For example, if one is deducing on the basis of and then by is deducing from this implication and , then it is simpler to deduce directly from ; the existence of such a proof is guaranteed because it is a subproof introducing. Let us call a maximal formula any formula which is at the same time the conclusion of an introduction rule and the main premise of an elimination rule.

A proof is called normal iff no maximal formula is present in it. Roughly speaking we can obtain such a proof if first we apply elimination rules to our assumptions premises and then introduction rules to obtain the conclusion. Such proofs are analytic in the sense of having the subformula property : all formulas occurring in such a proof are subformulas or negations of subformulas of the conclusion or premises undischarged assumptions.

Example Probelms in Natural Deduction

Although the idea of a normal proof is rather simple to grasp it is not so simple to show that everything provable in ND system may have a normal proof. In fact for many ND systems especially for many non-classical logics such a result does not hold. First he introduced an auxiliary technical system of sequent calculus and proved for it both in the classical and intuitionistic cases the famous Cut-Elimination Theorem. Then he showed that this result implies the existence of a normal proof for every thesis and valid argument provable in his ND systems.

The first published versions of proofs of Normalization theorems appeared in the s due to Raggio and Prawitz who proved this result also for ND systems for some non-classical logics.

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• For a detailed account of these problems see Troelstra and Schwichtenberg or Negri and von Plato One thing should be noticed with respect to proofs in normal form. Although normal proofs are in a sense the most direct proofs, this does not mean that they are the most economical. In fact, non-normal proofs often may be shorter and easier to understand than normal ones. Perhaps it is simpler to understand if we recall that normalization in ND is the counterpart of cut-elimination in sequent calculi. Applications of cuts in proofs correspond to applications of previously proved things as lemmas and may drastically shorten proofs.

What is important in normal proofs is that, due to their conceptual simplicity, they provide a proof theoretical justification of deduction and a new way of understanding the meaning of logical constants. He also wanted to realise a deeper philosophical intuition concerning the meaning of logical constants. It is claimed that if a set of rules is intuitive and sufficient for adequate characterisation of a constant, then it in fact expresses our way of understanding this constant.

In this particular case the meaning of logical constants is characterised by their use via rules in proof construction. There is also a strong connection with anti-realistic position in the philosophy of meaning where it is claimed that the notion of truth may be successfully replaced with the notion of a proof Dummett However, inferentialism is not particularly connected with ND nor with the specific shapes of rules as giving rise to the meaning of logical constants. Leaving aside the far-reaching program of inferentialism, one can quite reasonably ask whether the characteristic rules of logical constants may be treated as definitions.

He himself preferred introduction rules as a kind of definition of a constant. Hence one can directly obtain on the basis of these proofs with no application of. As these sufficient conditions for deductions of premises are characterised by introduction rules, we can easily see that the inversion principle is strongly connected with the possibility of proving normalization theorems; it justifies making reduction steps for maximal formulas in normalization procedures. Not all authors dealing with proof-theoretic semantics followed Gentzen in his particular solutions.

Popper was the first who tried to construct deductive systems in which all rules for a constant were treated together as its definition. There are also approaches such as Dummett , chapter 13, and Prawitz in which elimination rules are treated as the most fundamental. No matter which kind of rules should be taken as basic for characterization of logical constants, it is obvious that not any set of rules may be treated as a candidate for definition. Prior paid attention to this fact by means of his famous example. One can easily show that any formula is deducible from any formula after adding such rules to ND system.

Schroeder-Heister provides one of the recent solutions to this problem whereas Schroeder-Heister offers extensive discussion of other approaches. Andrzej Indrzejczak Email: indrzej filozof. Volume Email: a.