<< /Length 5 0 R /Filter /FlateDecode >> 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. Modus tollens ¬ q p → q … Domain: all dogs Therefore, P(his dog). Conjunction p q ∴ p ∧ q 4. Inference rules for propositional logic Two rules per binary connective: to introduce and eliminate it. (p ^q ) conjunction q) p ^q p p ! ∧ 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 � *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� This slide discusses a set of four basic rules of inference involving the quantifiers. 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. Domain: all dogs Determine the argument using P(x) x P(x). Modus ponens p p → q ∴ q p ∧ (p → q) → q 5. Solution: Determine individual propositional function P(x): x is cute. Is the argument valid? If a statement is true about every single object, then it … Most of the rules of inference will come from tautologies. (p _q ) addition) p _q p _q [(p _q )^(:p _r )] ! Inference Rules 3. The rules of inference are the essential building block in the construction of valid arguments. Simplification p ∧ q ∴ p (p ∧ q) → p 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 )] ! Does the conclusion must be true? The argument is valid: modus ponens inference rule. Since a tautology is a statement which is “always true”, it makes sense to use them in drawing conclusions. 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. Intro Elim Intro Elim Direct Proof Rule Modus Ponens Direct Proof Rule is special: not like the other rules. Propositional Logic 2. ∀ ∴ foranyarbitraryc 2. 4 0 obj }��������˾�/�?�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^[��� %PDF-1.3 Therefore, (p 2)2 = 2 > (3 2) 2 = 9 4. Inference rules for propositional logic plus additional inference rules to handle variables and quantifiers. We know that p 2 > 3 2. 6��5\�k�B��zw+Q�hՕ�o���8��x5�l�ȗH��H. predicate logic. What is wrong? x P(x) P(c) Rules of inference for quantified statement (example) State which rule of inference is applied in the following argument. This is called universal instantiation. Rules of Inference 1. Addition p ∴ p ∨ q p → (p ∨ q) 2. (q _r ) … 1. stream • Using the inference rules, construct a valid argument for the conclusion: “We will be home by sunset.” Solution: 1. All dogs are cute. If a statement is true about all objects, then it is true about any specific, given object. Like most proofs, logic proofs usually begin with premises — statements that you’re allowed to assume. Predicate Logic 4. Modus Ponens (method of affirming) premises: p, p q conclusion: q 2. 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 a rule of inference. Therefore, his dog is cute. Modus Tollens (method of denying) premises: q, p q conclusion: p 6 %��������� 1 Table of Inference Rules Number and Name of the Rule Rule of Inference Corresponding Tautology 1. 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���? 1.
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