# natural deduction rules

So the major premise is 'All A is C'. Natural Deduction uses two kinds of rules: Rules of inference and rules of replacement. A Natural Deduction proof in PC is a sequence of wffs beginning with one or more wffs as premises; fresh premises may be added at any point in the course of a proof. Simplification (Simp.) Table 2. If the sun is shining, then Mary is at the beach. )(p∨q) ⇔ (q∨p)p∧q ⇔ q∧p As above, a truth table would show that the wff representing MT, namely [(p ⇒ q)∧ ~q] ⇒ ~p, New wffs are generated by applying "rules" to any wff or a group of wffs that have already occurred in the sequence. But this rule as stated (from p & q to infer p) seems to permit only the left conjunct to be inferred. For propositional logic and natural deduction, this means that all tautologies must have natural deduction proofs. 8. Any tautologous conditionals and biconditionals could serve as valid rules of inference; but these are infinite in number. Then the instance of MP is: The following three inference schemes are among the ones we will use: The logical validity of these inference schemes can be verified by truth tables or truth-value analysis, but thi… So we turn the situation around and say that all mathematics proofs are instances of Natural Deduction. s, on the branch) NATURAL DEDUCTION RULES FOR IDENTITY AND FUNCTION SYMBOLS - Function symbols: Treat all constant terms alike in @ E appl~h3 * and 3 I. ADD~V V I and 3 E oxy io names. )[p∧(q∨r)] ⇔ [(p∧q) ∨ (p∧r)] Prove Constructive Dilemma Thus it is similar to the procedure of deriving theorems in mathematics. Conjunction (Conj.) Last updated - January 20, 2007. p _______⊢ p ∨ q, Table 2. Equiv. The symbol A ⇒ B is called a conditional, A is the antecedent (premise), and B is the consequent (conclusion). Association (Assoc. For instance, let p stand for 'the sun is shining' and q for 'Mary is at the beach'. I myself needed to study it before the exam, but couldn’t ﬁnd anything useful The reason for keeping MT in a list is that it is convenient in that it would often save a few steps in a proof. The former apply only to entire lines of proof, while the latter apply to components within a line as well as to the whole line. Transposition (Trans. This natural process is mimicked by the "Natural" Deduction Method of Propositional Logic (also called Propositional Calculus, abbreviated PC). p q______⊢ p∧q 4 License. Introduction rules introduce the use of a logical operator … p _______⊢ p ∨ q In other words, in any proof, there is a finite set of hypotheses { B, C, … } and a conclusion A, and what the proof shows is that A follows from B, C, …. The proof rules we have given above are in fact sound and complete for propositional logic: every theorem is … 1.2 Why do I write this Some reasons: • There’s a big gap in the search “natural deduction” at Google. )(p ⇔ q) ⇔ [(p ⇒ q)∧(q ⇒ p)] (2)~B[Premise] Conversely, a deductive system is called soundif all theorems are true. )(p ⇒ q) ⇔ (~p ∨ q) Therefore it is raining. The symbol A ⇒ B is called a conditional, A is the antecedent (premise), and B is the consequent (conclusion). Therefore Mary is at the beach. (p ⇔ q) ⇔ [(p∧q) ∨ (~p∧~q)] Rules of Natural Deduction Rules of Implication 1. The other premise (minor premise) contains the term that is the subject of the conclusion. Also known as "chain reasoning", a hypothetical syllogism is not limited to two premises. True! 2. Disjunctive Syllogism (DS): From p ∨ q and ~p to infer q. If John moves to Alaska, then he will freeze this winter; Therefore Mary is at the beach. Constructive Dilemma (P Q) & (R If the sun is shining, then Mary is at the beach.Mary is not at the beach. For a discussion that is more detailed than is warranted in the above short introduction, please visit A Primer for Logic and Proof. Natural Deduction uses two kinds of rules: Rules of inference and rules of replacement. (10)B ∨ D[Commutation]. This natural process is mimicked by the "Natural" Deduction Method of Propositional Logic (also called Propositional Calculus, abbreviated PC). but if he moves to Miami, then he will burn up next summer. When using a printed copy of the document, these hyperlinks So, if you know that p is true, you can add any other statement whatsoever to it by means of the disjunctive logical operator and the resulting compound statement will be true. Therefore Mary is at the beach. To see it, let us rewrite 'if p ⇒ q, and p, then q'. Hypothetical Syllogism (HS): From p É q and q É r to infer p É r. All but two (Addition and Simplication) rules in Table 1 are Syllogisms. 2 Why do I write this; 1. 2. Law of Exportation[(p∧q) ⇒ r] ⇔ [(p ⇒ (q ⇒ r)]. So the major premise is 'All A is C'. Example 3. Transposition (Trans. Copyright Ó 2001-2007 MathPath Rules of Natural Deduction Rules of Implication 1. Destructive Dilemma (DD)(p ⇒ q) ∧(r ⇒ s)~q ∨ ~s _________⊢ ~p ∨ ~r New wffs are generated by applying "rules" to any wff or a group of wffs that have already occurred in the sequence. It corresponds to a Proof Line beginning with the word therefore. is a tautology. (We have seen the convention that False can imply either True or False. Notice in this example that some steps involved the use of more than one rule. Material Implication (M. horns'. New wffs are generated by applying "rules" to any wff or a group of wffs that have already occurred in the sequence. Look at its truth table. Normal human reasoning is generally a train of thought moving linearly from the premises to the conclusion. Inference Rules               Daniel Clemente Laboreo. p∧q _________⊢ p It becomes ((p ⇒ q) ∧ p) ⇒ q. 3 Functioning; 3. Here is a popular set, known as Copi's set of rules . A ∨ C _________⊢ B ∨ D, Proof: Constructive Dilemma (CD): From (p ⇒ q) and (r ⇒ s) and p ∨ r (6)~D ⇒ A [5, Transposition, DN] In natural deduction, every proof is a proof from hypotheses. The theory of proof is not difficult. De Morgan's Theorem (DM)~(p∧q) ⇔ (~p ∨ ~q) Normal human reasoning is generally a train of thought moving linearly from the premises to the conclusion. Substitution of a tautolgy in a single wff changes the wff but does not change its truth value; this is what a rule of replacement does. What is difficult is coming up with the chain of steps to reach the conclusion. Most of the deduction rules come in one of two flavors, introduction or elimination. 2 Basic concepts. 7. (4)~A ⇒ C [2, Material Implication] 3 Natural deduction. TRUTH TREE RULES FOR IDENTITY AND FUNCTION SYMBOIS Rule #: Close any branch on which s # s appears. Imp. The sun is shining. Addition (Add.) The inference rules in Table 1 operate at once on one or more than one of the previous wffs in the deduction sequence and produces a new wff. The predicate of the conclusion in our example is 'is C'. So why is this a tautology? [p ∨ (q∧r)] ⇔ [(p ∨ q)∧(p ∨ r)] It consists in constructing proofs that certain premises logically imply a certain conclusion by using previously accepted simple inference schemes or equivalence schemes. 1. )(p ⇔ q) ⇔ [(p ⇒ q)∧(q ⇒ p)] natural deduction, this means that all tautologies must have natural deduction proofs. A small number of tautologies are chosen that are able to transform other wffs. Association (Assoc. 4 Notation. And so on! Replacement Rules               [p ∨ (q∧r)] ⇔ [(p ∨ q)∧(p ∨ r)] p∧q _________⊢ p If A ⇒ B is true, and ~B true, then ~A is true. We use ¬e because it eliminates a negation. We use the fact that ~(p∧~p) being a tautology in PC, its negation (p∧~p) is a contradiction. ˚ ¬˚ Œ ¬e L The proof rule could be called Œi. Colloquially, we call a 'dilemma' a predicament where we have to choose between alternatives neither of which yields a pleasant outcome. A small number of tautologies are chosen that are able to transform other wffs. This method in PC is what is used in mathematics proofs. However, note the 'commutation' rule in Table 2: p∧q ⇔ q∧p. Both the premises and the conclusion may contain meta-variables representing arbitrary propositions. All but two (Addition and Simplication) rules in Table 1 are Syllogisms. Disjunctive Syllogism (DS)p ∨ q~p_________⊢ q Hypothetical Syllogism (HS)p ⇒ qq ⇒ r_________⊢ p ⇒ r If p ⇒ q is true, and p is true, then q is true. A way of writing this is: Simplification (Simp.) Conversely, a deductive system is called sound if all theorems are true. The sun is not shining. (7)A ⇒ B [1,Simpl.] Constructive Dilemma (CD)(p ⇒ q) ∧(r ⇒ s)p ∨ r _________⊢ q ∨ s Inference Rules               (5)~A ⇒ D [3,4, HS] Addition (Add): From p to infer p Ú q. 3 Whom is it addressed to; 1. So, we have derived MT from two inference rules (Material Implication and DS) in the list. Proof: Let us look at the rules of Table 1. Table 1. Now let us consider an 'instance' of this wff by substituting declarative sentences to stand for p and q. Constructive Dilemma (CD)(p ⇒ q) ∧(r ⇒ s)p ∨ r _________⊢ q ∨ s Replacement Rules               The inference rules in Table 1 operate at once on one or more than one of the previous wffs in the deduction sequence and produces a new wff.