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But either not q or not s;Lincoln is dead Therefore, Lincoln was shot The FORM IS If Lincoln was shot, then Lincoln is dead Lincoln is dead Therefore, Lincoln was shot 1 IF P , THEN Q Q Therefore P NEXT We go from FORM back to ARGUMENT IF P , THEN Q Q Therefore P IF Ed passes Phil 101, then Ed has perfect attendance===== If P, then Q If Q, then R Therefore, if P, then R ===== P → Q Q → R P → R 1 Disjunctive Syllogism (DS) Either Ralph walked the dog or he stayed home Ralph did not walk the dog Therefore, he stayed home ============================== Either P or Q Not P Therefore, Q ============================== P v Q ~P Q 1

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If p then q q therefore p-If you have p ∨ ¬ p then you also have p ∨ ¬ p ⇒ q for all q and, consequently, (p ⇒ q) ∨ (¬ p ⇒ q) Having this law available actually has subtle and interesting consequences in formal logicLet us see that for our first two inferences If P then Q, P, therefore Q If P then Q, Q, therefore P P Q ((P → Q) ∧ P) → Q T T T T T T T T F F F T T F F T T F F T T F F T F F T F 2 3 1 = 4 1 = P Q ((P → Q) ∧ Q) → P T T T T T T T T F F F F T T F T T T T F F F F T F F T F 2 3 1 = 4 1 =

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Look at the fourth (or sixth) row In this case, \((P \imp R) \vee (Q \imp R)\) is true, but \((P \vee Q) \imp R\) is false Therefore the statements are not logically equivalent While we don't have logical equivalence, it is the case that whenever \((P \vee Q) \impTherefore either not p or not r Simplišcation (p∧q) ∴ p p and q are true;If P, then Q Therefore, if not P, then not Q which may also be phrased as → (P implies Q) → (therefore, notP implies notQ) Arguments of this form are invalid Informally, this means that arguments of this form do not give good reason to establish their conclusions, even if

In this case, the truth values for ~(p∧q) and ~p∨~q are exactly the same, so we can conclude that the two statements are equivalent ~(p∧q)≡~p∨~qSo, if we ever encounter ~(p∧q), we can replace it with ~p∨~q without changing the logical meaning of the statement!P = "" Q = "" R = "Calvin Butterball has purple socks" I want to determine the truth value of Since I was given specific truth values for P, Q, and R, I set up a truth table with a single row using the given values for P, Q, and R Therefore, the statement is trueIt is convenient to read → sentences in English using if then That is, we read P → Q ( "P arrow Q") as if P, then Q But there are many other ways in English of saying the same thing, and hence many other ways of reading → sentences in English Q if P P only if Q Q provided that P Q in case P Provided P, Q In the event that P, Q

Therefore, q—is called modus ponensTherefore p is true Conjunction p,q ∴ (p∧q) p and q are true separately;The channel that brings you the best shit shows in FE o7🤡 #feclownshow 🤡 Air Date

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Otherwise it is true Contrapositive The contrapositive of a conditional statement of the form "If p then q" is "If ~q then ~p" Symbolically, the contrapositive of p q is ~q ~pIf P is false, then ¬P is true P∧ Qshould be truewhen both P and Qare true, and falseotherwise P Q P∧ Q T T T T F F F T F F F F P∨Qis trueif either P is trueor Qis true(or both — remember that we're using "or" in the inclusive P Q R P → Q ¬R (P → Q) → ¬R T If p, then q Not q Therefore, not p If you don't care, fuck off The Rubber Duck method of debugging We called it the Rubber Duck method of debugging It goes like this 1) Beg, borrow, steal, buy, fabricate or otherwise obtain a rubber duck (bathtub variety) 2) Place rubber duck on desk and inform it you are just going to go over some

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Q if p , then q q , ifp p , only if q p implies q p is sufcient for q q is necessary for p q follows from p c Xin He (University at Buffalo) CSE 191 Discrete Structures 15 / 37 Terminology for implication Example Proposition pP→Q means If P then Q ~R means NotR P ∧ Q means P and Q P ∨ Q means P or Q An argument is valid if the following conditional holds If all the premises are true, the conclusion must be true Some valid argument forms (1) 1 P 2 P→Q C Therefore, QMathematics, a variety of terminology is used to express p !

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So, when P is indeed true, so is Q The combinationof P is true with Q is false DOES NOT OCCUR Since this is the only time"if P then Q" is false, we know that "if P then Q" is true The sentence "If (if P, then Q) and (if Q, then R), then (if P, then R)" captures theEither p or q p Therefore, q Disjunctive Syllogism Premises Conclusion Either p or q Not p Therefore, q Deductive Arguments Exercise 3 I If it's raining, then the roads are wet It's raining Therefore, the roads are wet a valid b affirming the antecedent C sound 2 If you get a degree in business, then you are certain to get a jobAnd if r then s;

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Noticethat this leaves with some area of the Qcircle that is not also in the Parea That is because "if P then Q" does not mean that there cannot be instanceswhere Q is true, but P is false An important thing to notice, however,is that if you say thatTherefore the disjunction (p or q) is true Composition (p → q) (p → r) ∴ (p → (q∧r)) if p then q;Therefore they are true conjointly Addition p ∴ (p∨q) p is true;

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And if p then r;P q p → q ∼ q ∼ p T T T F F T F F T F F T T F T → F F T T T In this case there is only one critical row to consider, and its truth value it true Hence this is a valid argument Result 22 (Generalization) Suppose p and q are statement forms Then the following arguments (called generalization) are valid p p∨q q p∨ q Result 23If p, then q q Therefore, p b If p, then q If q, then r Therefore, if p, then r c Either p or q Not p Therefore, q d If p, then q p Therefore, q The first step in investigating possible implicit premises is to a Search for a credible premise that would make the argument as strong as possible b Rewrite the argument

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If p then q Notp Therefore, notq If p then q p Therefore, q If p then q Notq Therefore, notp Example I want to list seventeen summary statements which, if true, provide abundant reason why the reader should reject evolution and accept special creation as his basic worldviewTherefore, P and Q Therefore, Q and P Hypothetical Syllogism (HS) 1 If P, then Q 2 If Q, then R 3Solution We want to use the p and q given above as replacements for the p and q in the following argument form (such use is called a replacement instance) If p then q not q Therefore not p Hence we have for (a) If my car is still in the shop then I have to get a ride with a friend I don't have to get a ride with a friend

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 If P then Q P Therefore Q Here, the letters P and Q are sentence letters They are used to translate or represent statements By replacing P and Q with appropriate sentences, we can generate the original valid arguments This shows that the two arguments have a common form It is also in virtue of this form that the arguments are valid, forHow to think about "P ⊃ Q" in plain EnglishIn propositional logic, P ⊃ Q is what is called a material implicationIt doesn't mean that P and Q mean the same thing (they might not have the same truth value);P then q" or "p implies q", represented "p → q" is called a conditional proposition For instance "if John is from Chicago then John is from Illinois" The proposition p is called hypothesis or antecedent, and the proposition q is the conclusion or consequent Note that p → q is true always except when p is true and q is false

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Answer (1 of 6) In " Is p => q, 'E ~ q' Ergo p ( => q is ) a fallacy " It depends on how you define '=>' If you are using it as the operator," b follows a" ' a => b ', a weak form of cogency, then of course your conclusion is invalid Because while it may be a necessary condition for q to sThe conditional of q by p is "If p then q" or "p implies q" and is denoted by p q It is false when p is true and q is false;P → q = (~p ∨ q) In the Principia Mathematica, the "=" denotes "is defined to mean" Using this denotation, the above expression can be read "p implies q is defined to mean that either p is false or q is true" The following truth table shows the logical equivalence of "If p

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The part of a conditional statement (If p, then q) introduced by the word then Deductive Argument An argument intended to provide logically conclusive support for its conclusionIf P, then Q Q is false Therefore, P is false In logical operator notation where represents the logical assertion Or in settheoretic form ∴ ("P is a subset of Q x is not in Q Therefore, x is not in P") The argument has two premises The first premise is the conditional "ifthen" statement, namely that P implies Q QUESTION 1 Modus tollens has this argument pattern The correct option for this question is If p, then q Not q Therefore, not p It is because in Modus Tollens p and q are prepositions Modus Tollens states that if p implies q, and q is false, View the full answer

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If p then q;An argument of this form—If p, then q;If p then q p Therefore, q If p then q Notq Therefore, notp Exposition The consequent of a conditional statement is the part that usually follows "then" The part that usually follows "if" is called the "antecedent" I write "usually" here because there are many different ways to make a conditional statement, but we needn't go into

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As far as I understand, If p then Q means " if P is true, Q has to be true Any other case, I don't know " So, from what I understand, the first 2 rows of the truth table state that " If P is true and Q is true, the outcome is correct and If P is true and Q is false,All that it is, is a claim that if P is true, then Q is also true — without making any more claims than this An alternative way of considering P ⊃ Q is as a "constraint" thatSo, "if P, then P" is also always true and hence a tautology Second, consider any sentences, P and Q, each of which is true or false and neither of which is both true and false Consider the sentence, ``(P and Not(P)) or Q'' This means exactly the same

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Therefore if p is trueP if p, then q not q therefore, not p in plain English, this says let's suppose that p is true if p is true, then q must be true but there's no way that q can be true (or q being true is quite absurd) so it must not be the case that p is trueNow let's try comparing two more complex statements to see if they are equivalent

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(pVq) V (~p^q) → q p q ~p p V q ~p ^ q (p V q) V (~p ^ q) (p V q) V (~p ^ q) → q T T F T F T T T F F T F T F F T T T T T T F F T F F F T Problem 18 (15 points) Write each of the following three statements in the symbolic form and determine which pairs are logically equivalent aOne way to write the conditional is "if p, then q" Thus, if you know p, then the logical conclusion is q Consider this as you review the following truth table Why is this true? Invalid argument forms Consider the following argument form p q Therefore r If we let p be 'It is raining in the southeast', let q be 'increased rain usually helps crops produce a higher crop yield' and r be 'crops in California will produce more' then the resulting argument is not valid (check to make sure you see a possible way to have all true premises and a false conclusion)

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Given "p implies q", there are two possibilities We could have "p", and therefore "q" (so q is possibility 1)(See this post for an explanation of the conditional) Even if you have If ( P implies Q ) then ( P implies R )If p then q p Therefore q eg All humans are mortal Socrates is a human Therefore, Socrates is mortal 2 Modus Tollens ("Method of denying") If p then q ~q Therefore ~p eg All humans are mortal Zeus is not moral Therefore, Zeus is not a human • Exercise Section 23, #26, 27, p 62

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 If P then Q If P then R It does not at all imply that If Q then R Why?Because if P is false, the first two would be (vacuously) true, but it might be that also Q is true and R is false, which would make the last one false!

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