Example if p then q p therefore , q modus ponens ( affirming the consequent ) what argument form is this if its raining , the park is closed the park is not closed therefore, it is not raining example if p, then q not q therefore, not pThis is just the truth table for \(P \imp Q\text{,}\) but what matters here is that all the lines in the deduction rule have their own column in the truth table Remember that an argument is valid provided the conclusion must be true given that the premises are true The premises in this case are \(P \imp Q\) and \(P\text{}\) Both arguments are of course valid What is common between them is that they have the same structure or form 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
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If p then q q therefore p is an example of what kind of common argument
If p then q q therefore p is an example of what kind of common argument-If I have a college degree, then I am not lazy (p →~ q) I don't have a college degree )(~ p Therefore, I am lazy q Hypothesis )((p →~ q)∧~ p Conclusion q Argument in symbolic form (( p →~ q)∧~ p) →q To test to see if the argument is valid, we take the argument in symbolic form and construct a truth tableIt has the form p>q, and q You can't apply this law;
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Rule of inference Modus ponens Definition It has the form If p, then q p ∴ q The term modus ponens in Latin means "method of affirming" p q p → q p q T T T T T T F F T F T T F F F T F Example If the sum of the digits of 371,487 is divisible by 3, then 371,487 is divisible by 3 The fallacy of affirming the consequent is any argument of the following form If p, then q;Formal logic Formal logic The propositional calculus The simplest and most basic branch of logic is the propositional calculus, hereafter called PC, so named because it deals only with complete, unanalyzed propositions and certain combinations into which they enter Various notations for PC are used in the literature In that used here the symbols employed in PC first comprise variables
In fact, there is no law that you can apply, soFamous If P, Then Q Logical Argument Example Premise #1 If (P) this rock hits that window, then (Q) that window will break Thus, in the precise specifications of the P/Conditions/Causes and therefore the causes of the Q/Consequence(s)/Effect(s) of an If P, Then Q logical argument we find the basis for true knowledge, the accurateWays in which the propositions may fail to be equivalent Here is another example Example 232 Show (p!q) is equivalent to p^q Solution 1 Build a truth table containing each of the statements p q q p!q (p!q) p^q T T F T F F T F T F T T F T F T F F F F T T F F Since the truth values for (p!q) and p^qare exactly the same for all possible
Consider the argument form If P then Q Not Q Therefore, Not P Is this form modus ponens, affirming the consequent or something else?If p then q;Otherwise it is true The contrapositive of a conditional statement of the form "If p then q " is "If ~ q then ~ p "
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Therefore p is true Conjunction p,q ∴ (p∧q) p and q are true separately;For example, consider this which includes a fallacy of affirming the consequent "If it is raining the ground will be wet The ground is wet If p then q q Therefore p This kind if argument is also called a CON This is common to get someone to fall for scheme to get moneyBut either not q or not s;
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The argument can be translated into statement variables as follows p Therefore, p or q If the premise is true, then the truth of p necessitates the truth of the conclusion II Soundness 1 Sound The argument is valid and all the premises are true The argument is valid because, when we take all of the premises to be true, the conclusion must be trueExpert Answer Any argument that is in the form of "If" will be valid, and any argument that affirms the consequent will be invalid A valid argumentform If p, then q p Therefore, qAnargument formis said view the full answerP or Q 2 Not Q Therefore, 3 P Constructive dilemma 1 P or Q 2 If P then R 3 If Q then S Therefore, 4 R or S Basic difference between disjunctive and conjunctive statements it is easier for a disjunctive statement to be true than for a conjunctive statement
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The inference from the premises to the conclusion is invalid, because it could be that the premises are true and the conclusion is false For example, if p is false and q is true, then the premises are true and the conclusion is falseQ 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 p•Given statement p,q, • "~p" ("not p", "It is not the case that p") is called negation of p • "p ∧ q" ("p and q") is conjunction of p and q • "p ∨ q" ("p or q" ) is disjunction of p and q •"p⊕q" (p exclusive or q) • "p→q" (if p then q) is conditional • "p↔q" (p if and only if q) is biconditional
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I'm trying to prove the hypothetical sylloligism using boolean algebra We already have a solution using propositional logic, which relies on proof by contradiction $(p \implies q) \wedge (q \impResult 21 (Modus Ponens and Modus Tollens) Suppose p and q are statement forms Then the following are valid arguments (i) The argument called modus ponens defined as p → q p q (ii) The argument called modus tollens defined as p → q ∼ q ∼ p Proof We shall show that modus tollens is valid p q p → q ∼ q ∼ p T T T F F T F F T FAn argument with this form—"If p, then q If q, then r Therefore, if p, then r"—is known as Question 6 options a) Hypothetical syllogism b) Modus tollens
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A formal consequence must be true in all cases, however this is an incomplete definition of formal consequence, since even the argument "P is Q's brother's son, therefore P is Q's nephew" is valid in all cases, but is not a formal argument A priori property of logical consequenceSome Common Valid Argument Forms Modus Ponens (MP) 1 If P, then Q 2 P 3 Therefore, Q Multiple Modus Ponens (MMP) 1 P 2 If P, then Q 3 If Q, then R 4 ThereforeTwo compound propositions, p and q, are logically equivalent if p ↔ q is a tautology !
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1 If P then Q 2 P 3 Therefore, Q Valid (Modus Ponens) Notice that this argument is still valid even though (as far as we know) all the premises (and the conclusion) are, in fact, false F 1 If P then Q 2 Q 3 Therefore, P Invalid This is the same invalid form as argument B Notice that all the premises and the conclusion are in fact trueThe Bible is true because God wrote it;Therefore they are true conjointly Addition p ∴ (p∨q) p is true;
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What are truth values of !A statement is true when the world is the way the statement says the world is An argument is a sequence of statements, the last of which (the conclusion) is supposed to follow from the others (the premises) A valid argument is one with the following property IF all of its premises are true, then its conclusion must also be true (said another way an argument is valid when it isTherefore Q (You can negate the P or the Q interchangeably) False Dilemma – (Fallacy) Ignoring or disregarding other options and possibilities Conditional Material Implication Implication The relation that holds between the antecedent and the consequent of a conditional If P then Q (P is the antecedent and Q is the consequence)
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Let p and q be statement variables which apply to the following definitions 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; Here is an example "If P then Q And if R then also Q But either P or R So in any event, Q" Dilemmas are perfectly respectable forms of argument In an argument of this sort, P and R are called "the horns" of the dilemma If you want to reject a dilemma, thenCMSC 3 Section 01 Homework1 Solution CMSC 3 Section 01 Homework1 Solution 1 Exercise Set 11 Problem 15 Write truth table for the statement forms (5 points) ~(p ^ q) V (p V q)
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Let R(x,y) denote x beats y in Rock/Paper/ If P, then Q Not Q Therefore, Not P Disjunctive Syllogism Either P or Q Not P Therefore, Q & Introduction P Q One example of a kind of inductive argument that is quite common is called an Argument by Analogy An argument by analogy occurs whenever one makes a comparison of two or more things and concludes, because of the similarity ofSection24 Logical Arguments Definition 241 An argument is a sequence of statements ( premises) that ends with a conclusion A valid argument is one where the conclusion follows from the truth of the premises For the sequence of premises p1,p2,,pn p 1, p 2, , p n and conclusion q, q, an argument is valid if
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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" thatTherefore either not p or not r Simplišcation (p∧q) ∴ p p and q are true;If p then q p Therefore, q Modus Tollens A validating form of argument from propositional logic If p then q Notq Therefore, notp N Narrow Scope A term has narrow scope when it modifies the smallest part of a sentence that is grammatically possible
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When we assign values to x and y, then P has a truth value Example !Mathematics, a variety of terminology is used to express p !And if r then s;
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Therefore the disjunction (p or q) is true Composition (p → q) (p → r) ∴ (p → (q∧r)) if p then q;Implications are similar to the conditional statements we looked at earlier;Therefore, God exists" is an example of begging the question a True
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The first step in determining whether an argument isdeductive or inductive is to find the argument's conclusion and thenits premisesTherefore if p is true then q and r are true ∴ (¬p∨¬q)Argumentforms and common flaws Here are some examples of valid forms of argument Modus ponens The general form of a modus ponens argument is given in (1) Two examples follow (2) If P then Q P Q (2) If you videotape Gilligan's Island reruns, then you are in big trouble You videotape Gilligan's Island reruns You are in big trouble
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False The key to identifying an argument in context is to first identify the conclusion, then look for the premises a True b False This classic argument "The Bible says that God exists;7 Modus tollens premises, conclusion Common form (modus tollens) If p, then q (premise) Not q (premise) Therefore not p (conclusion) A premise is a statement presumedtrue for purposes of an argumentIn " 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
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The law of detachment has the form p>q, and p, therefore q That is, if you know that q happens whenever p happens, and you also know that p did happen, then q must happen The example about icy roads and driving conditions is NOT like that;2 It is false that some nonA are not nonB Therefore explain!, all nonA are nonBTherefore explain!, all B are ATherefore explain!, it's false that no B are ATherefore explain!, some B are AP → q is typically written as "if p then q," or "p therefore q" The difference between implications and conditionals is that conditionals we discussed earlier suggest an action—if the condition is true, then we take some action as a result
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Symbolized by p q, it is an ifthen statement in which p is a hypothesis and q is a conclusion The logical connector in a conditional statement is denoted by the symbol The conditional is defined to be true unless a true hypothesis leads to a false conclusion A truth table for p q is shown below(1) P v Q, (2) not P, therefore (3) Q Affirming the Consequent (1) P > Q, (2) Q, therefore (3) P Denying the Antecedent (1) P > Q, (2) not P, therefore (3) not Q Antecedent The P part in "If P, then Q" Consequent The Q part in "If P, then Q" Conditional If P, then Q Hypothetical If P, then Q Valid argument IF all of the premises ofAnd if p then r;
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Let Q(x,y) denote "x=y3" !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);
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