a rule of inference. If a statement is true about every single object, then it … Domain: all dogs Therefore, P(his dog). (p ^q ) conjunction q) p ^q p p ! Conjunction p q ∴ p ∧ q 4. Therefore, his dog is cute. Intro Elim Intro Elim Direct Proof Rule Modus Ponens Direct Proof Rule is special: not like the other rules. Inference rules for propositional logic Two rules per binary connective: to introduce and eliminate it. Most of the rules of inference will come from tautologies. Inference rules for propositional logic plus additional inference rules to handle variables and quantifiers. Rules of Inference 1. 1. � *ΗaڶL{ؿ���pĚ�c���L�ż��Hc�'L���,�oq[JP��9Gᄳ5�B���G��g��Q�掠ZsI�>����4����s���Fҭ-c�aw�VIW]7{BD&3��\Y�75�"x1�[('FNL1K�Y�4s���P�A��E�I���+Ba��MĬJM�xMŇ�GH_NF��j�`N� Predicate Logic 4. The argument is valid: modus ponens inference rule. This slide discusses a set of four basic rules of inference involving the quantifiers. Inference Rules 3. Inference rules 1 The following rules make it possible to derive next steps of a proof based on the previous steps or premises and axioms: Rule of inference autologyT Name p ^q (p ^q ) !p simpli cation) p p [(p )^(q )] ! Like most proofs, logic proofs usually begin with premises — statements that you’re allowed to assume. %��������� Is the argument valid? We know that p 2 > 3 2. 6��5\�k�B��zw+Q�hՕ�o���8��x5�l�ȗH��H. Rules of Inference for Propositional Logic Determine whether the argument is valid and whether the conclusion must be true If p 2 > 3 2 then (p 2)2 > (3 2) 2. Modus ponens p p → q ∴ q p ∧ (p → q) → q 5. << /Length 5 0 R /Filter /FlateDecode >> Modus Ponens (method of affirming) premises: p, p q conclusion: q 2. %PDF-1.3 1. (p _q ) addition) p _q p _q [(p _q )^(:p _r )] ! If a statement is true about all objects, then it is true about any specific, given object. Since a tautology is a statement which is “always true”, it makes sense to use them in drawing conclusions. Addition p ∴ p ∨ q p → (p ∨ q) 2. Does the conclusion must be true? x�I�%G���+j�z�r���BL���ba=�Ѝq����;�C�7��� ��U����ɈȬ��������Ͻ����w�������{�?�ߝ�{��Q�>eƌ�G�!i��~�S�hѾ���I�뵌?|ͩ �Q\�7+n������ M�勒Jr�����O�H-�o���? Domain: all dogs Determine the argument using P(x) x P(x). Propositional Logic 2. Solution: Determine individual propositional function P(x): x is cute. (q _r ) … x P(x) P(c) 4 0 obj ∧ A;B ∴ A∧B ∧ A∧B ∴ A,B ∨ A ∴ A∨B,B∨A ∨ A∨B;¬A ∴ B A B ∴ A → B A;A → B ∴ B 13 This is called universal instantiation. stream n��EL�'Y6~cn�Û!�����:$�m|?�.q��L߃���j/2�|����C ��2'�g�#mK��]�&�PF-+l��ƙ�^�Kۿʄ�����{���/m���Ը{wxwF��;��kN���B�P#�?r;��36C�Q���t-�����t���Qk��*"�Q�HJ�߹�$��5�Yg+;�*�I��� ��b�?Ru�)���:z�Ý�+���R�5 Therefore, (p 2)2 = 2 > (3 2) 2 = 9 4. Rules of inference for quantified statement (example) State which rule of inference is applied in the following argument. Choose propositional variables: p: “It is sunny this afternoon.” q: “It is colder than yesterday.” r: “We will go swimming.” s : “We will take a canoe trip.” t : “We will be home by sunset.” 2. Simplification p ∧ q ∴ p (p ∧ q) → p 3. Modus Tollens (method of denying) premises: q, p q conclusion: p 6 What is wrong? • Using the inference rules, construct a valid argument for the conclusion: “We will be home by sunset.” Solution: 1. The rules of inference are the essential building block in the construction of valid arguments. ∀ ∴ foranyarbitraryc 2. predicate logic. }��������˾�/�?�w�~� y��x��^m��{��o�ۅ���?��wz������;��}�v7���2�?O���~����{��e�Fo��ǖF����|�Q�oz���W{;�)v(�������s�2�� �����xb��}3�7�,����ow-�ٔ�;}a+�}�o�3j*�����A��I�G��遹,�2�P��OQ�l�Z3�-�7���S�t�tś�,rA>�F�x~) i��p��6~c�����Kb'�_��X��z^[��� 1 Table of Inference Rules Number and Name of the Rule Rule of Inference Corresponding Tautology 1. Ò1PÀ±îrrÍ&{ÍÙ�(RÁö/£.�KD ÓƒÀWŞ)Å$6a(6ùD½¼fıGĞɸ‘�C}šü3(Ò!Ï.QId)/Xn£¸ÁR¡ä”�åñÛÚ|”À cn4ì˜lû:pëزÀ‡è¡ÓOÇÍòàT ×Å!Ì÷h¡G‘ÆĞ�,Q\ö¶Ğ%ÈC"”ª‚ºC_óÜŠ¡Ôú22œØ>Bæ�¨qB1RE¯â7°ÿ¦n‚sk!ÚĞ*Ñ#_PÍ¢ÿ7„Õß»’•_DdCåLö7+ø[ÇÉš°ûPU«‘¾Ì½EürñVsö‰0Üöaõ}qãu#ê¿Ï�~+X/Fí¼fää~`üÒÛZ+ÀÒ2ZpYEx_a4€ô P`½•�ÓVÛZï/ ®cI’ï)²Š\Ï£tEÜK¾hÀ•)´4�‘ë]5#‹Š¡YÇ�3;+Šªƒ„û«“Œ"d¸ğ×YÁâ–fd¹ÎaïŠ]‹Ëºàî–ö b¸Jg]YÃîäîÊ/KówuÁ¦;YWô2ù#ß4 6V!�‹wˆäO©ÔÁŠ(ÛSzgƒ³òê8j[%%Ó»©ûòJ½*­`èˆO`ŒïLÒly†gš?ñéşÆ}(9zX›ë2oçJ#È£ª81q�fğ[t0?Ë. All dogs are cute. Modus tollens ¬ q p → q …