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 the===== 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 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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If p then q q therefore p-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 pIn 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!
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An argument of this form—If p, then q;Solution 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 friendThe 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 conclusion
How 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);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" thatIf 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
Given "p implies q", there are two possibilities We could have "p", and therefore "q" (so q is possibility 1)Therefore p is true Conjunction p,q ∴ (p∧q) p and q are true separately;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) \imp
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Mathematics, a variety of terminology is used to express p !If 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 argumentTherefore, q—is called modus ponens
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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 attendanceIf 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 logic(See this post for an explanation of the conditional) Even if you have If ( P implies Q ) then ( P implies R )
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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 aTherefore if p is trueAnswer (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 s
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But either not q or not s;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 trueTherefore they are true conjointly Addition p ∴ (p∨q) p is true;
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P 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 trueP 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 falseP 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 23
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Therefore the disjunction (p or q) is true Composition (p → q) (p → r) ∴ (p → (q∧r)) if p then q;If 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 QIf 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
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If p then q;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, 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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P→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, QBecause 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!Either 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 job
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One 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?The 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;Now let's try comparing two more complex statements to see if they are equivalent
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It 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 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 answerP → 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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And if r then s;Therefore either not p or not r Simplišcation (p∧q) ∴ p p and q are true;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 ~p
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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 intoLet 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 = 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, for
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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 that If P then Q If P then R It does not at all imply that If Q then R Why?And if p then r;
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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 62So, "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 sameIf 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 worldview
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