If P Then Q Q Therefore P Example

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 true.

If p then q q therefore p example. If p then q;. And if r then s;. (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 a.

Therefore, God exists” is an example of begging the question a True. If we read it as "if P, then Q", we have that Q is "animals will eat plants" Thus with "2 My dog eat grass" the issue is that it is not a new "instance" of Q, but it is a different. One example of such an argument is If it rains, then the squirrels hide p = it rains / is raining q = the squirrels hide / are hiding ' 05Œ09, N Van Cleave 1 then Brad sings in the choir Therefore, Brad sings in the choir ' 05Œ09, N Van Cleave 33 If the Bobble head doll craze continues, then Beanie Babies.

If Q, then R 3 Therefore, notP 3 notR 4 Therefore, notP Disjunctive Syllogism (DS) Categorical Syllogism 1 P or Q 1 P or Q 1 All A's are B If P is sufficient for Q, then Q is necessary for P Some examples Being male is a necessary condition for being a bachelor. If p, then q if q, then r therefore, if p, then r "hypothetical" is just another term for conditional "syllogism" is a deductive argument made up of three statements two premises and a. Therefore if p is true.

Recall that if p is false then p > q is always true, thus the only way our implication can be false is if p is true and q is false In practice then, we assume our premise is true but our conclusion is false and use these assumptions to derive a contradiction.  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 friend. 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 true.

Examples Valid All men are mortal Socrates is a man Therefore Socrates is mortal The picture is above the desk The desk is above the floor If p then q;. 11 PROPOSITIONS 8 “if 4 = 7 then London is in Denmark” (false → false) However the following one is false “if 2 < 4 then London is in Denmark” (true → false) In might seem strange that “p → q” is considered true when p is false, regardless of the truth value of qThis will become clearer when. If, on the other hand, introduces a sufficient condition P if Q means that the truth of Q is sufficient, or enough, for P to be true as well That is, P if Q rules out just one possibility that Q is true and P is false But that is exactly what Q → P rules out So it’s obviously correct to read Q.

The writer assumes that you know when "if P, then Q" is false The writer also assumes that you know getting a contradiction from assuming "if P, then Q" is false means that "if P then Q" must be therefore be true Why waste words on explaining the obvious?. Modus Tollens (MT) o If p, then q o Not q o Therefore not p Example If you work in a bar, you’re over 19 You’re not over 19 So, you must not work in a bar Or a variation of MT p → q ~p → ~q ~q q ~p p 3 Hypothetical Syllogism (HS) o If p, then q. 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 p.

Therefore, P and Q Therefore, Q and P Hypothetical Syllogism (HS) 1 If P, then Q 2 If Q, then R 3. "If p, then q" is equivalent to "All p are q" "If p, then not q" is equivalent to "No p are q" For example, "If something is a poodle, then it is a dog" is a roundabout way of saying "All poodles are dogs" Likewise, "If something is a dog, then it isn't a cat" means the same as "No dogs are cats" EXAMPLE 2210 1. Otherwise it is true The contrapositive of a conditional statement of the form "If p then q ".

Therefore either not p or not r Simplišcation (p∧q) ∴ p p and q are true;.  If p, then q It is not the case that q It is not the case that p Example If an all powerful and all merciful God exists, then there is no evil in the world It is not the case that there is no evil in the world It is not the case that an all powerful and all merciful God exists. 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;.

If an independent premise is removed, the support that other premises supply to the conclusion is not. Example Show that (P → Q)∨ (Q→ P) is a tautology I construct the truth table for (P → Q)∨ (Q→ P) and show that the formula is always true P Q P → Q Q→ P (P → Q)∨ (Q→ P) T T T T T T F F T T F T T F T F F T T T The last column contains only T’s Therefore, the formula is a tautology Example. Is rational Then 2 = p/q for some integer p and q with q ≠ 0 and gcd(p,q) = 1 Since 2 1/2 = p/q, we know (p/q)2 = 2 (ie, square root of 2, to the power of 2) Then we get (p/q) 2 = p 2 /q 2 = 2, and p 2 = 2* q 2 So p 2 is even Let p 2= 2k for some integer k So p 2 = (2k) = 4k 2, which is equal to 2* q Dividing both sides by 2, we get 2k 2 = q 2 Since k 2.

If p then q p Therefore, q If p then q Notq Therefore, notp Exposition So, in the Form given above, the consequent is "q" For example, in the statement "if today is Tuesday, then I have logic class", "I have logic class" is the consequent To affirm the consequent is, of course, to claim that the consequent is true Thus. After awhile, it does indeed become obvious. Now let's try comparing two more complex statements to see if they are equivalent.

Logically they are different In the first (only if), there exists exactly one condition, Q, that will produce P If the antecedent Q is denied (notQ), then notP immediately follows In the second, the restriction on conditions is gone The usual rules apply, and nothing follows from denying the antecedent Q Share Improve this answer. 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!. The Bible is true because God wrote it;.

Therefore they are true conjointly Addition p ∴ (p∨q) p is true;. But if Q →R and ~R are both true, then ~Q is also true For ~Q follows from Q →R and ~R, in virtue of modus tollens So, if the premises are all true, then so is ~Q That means that all the following formulas are true – P →Q, Q →R, ~R, ~Q So, in particular, P →Q and ~Q are both true But if these are true, then so is ~P (the. \(P\) is true in the first two rows, and of those, only the first row has \(P \imp Q\) true as well And loandbehold, in this one case, \(Q\) is also true So if \(P\imp Q\) and \(P\) are both true, we see that \(Q\) must be true as well Here are a few more examples Example 316 Show that.

The "logical form" must be "if P, then Q, and Q Therefore P" In your first example, we have "1 If the animals are herbivore, then they will eat plants";. But either not q or not s;. " If not q and p implies q, then not p Example p = it is sunny, q = it is hot p → q, it is hot whenever it is sunny “Given the above, if it is not hot, it cannot be sunny” 11 Hypothe’calSyllogism " If p implies q, and q implies r, then p implies r Example p = it is sunny, q = it is hot, r = it is dry.

Mathematics, a variety of terminology is used to express p !. 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. False An argument of this form—If p, then q;.

Q) p^˘q Finally, write down a conditional statement and then negate it. If p→q is a conditional statement, then (1) q→p is called converse of p→q (2) ¬q→¬p is called contrapositive of (3) ¬p→¬q is called inverse of p→q EXAMPLE p Ram is a computer science student q Ram study DBMS p→q If Ram is a computer science student, then the will study DBMS. Common Formal Fallacies Affirming the Consequent Denying the Antecedent Illicit Hypothetical Syllogism Illicit Disjunctive Syllogism Common Formal Fallacies Affirming The Consequent If P then Q, Q Therefore P Denying the Antecedent If P then Q, P is false Therefore Q is false Affirming the Consequent If P then Q Q is true Therefore P If it rained last night then the grass.

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;. Therefore, the argument must be valid 5 Valid We can test for validity by substituting statement variables into the argument If we do the substitution, we get If p, then q Not q Therefore, not p Upon inspection, we find that this is one of the common patterns of valid inference discussed above—modus tollens.  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).

P q r q p r ∴ q aka Disjunction Elimination Corresponding Tautology ((p q) ∧ (r q) ∧ (p r )) q Example Let p be “I will study discrete math” Let q be “I will study Computer Science” Let r be “I will study databases” “If I will study discrete math, then I will study Computer Science”. For example, if the claims P and Q express the same proposition, then the argument would be trivially valid, as it would beg the question In everyday discourse, however, such cases are rare, typically only occurring when the "ifthen" premise is actually an "if and only if" claim (ie, a biconditional/equality) The following argument is not valid, but would be if the first premise was. P∨ q ∼ q p Result 26 (Transitivity) Suppose p, q and r are statement forms Then the following argument (called transitivity) is valid p → q q → r p → r Result 27 (Proof by Division into Cases) Suppose p, q and r are statement forms Then the following argument (called proofby division into cases) is valid p∨q p → r q → r.

–premises p 1, p 2, , p k –conclusion q then (p 1Ùp 2Ù Ùp k) ®q is a tautology •Ex ( (p®q)Ùp ) ®q is a tautology •Some simple valid argument forms, called rules of inference, are derived and can be used to construct complicated argument form. And if p then r;. Therefore, not q —is called modus tollens a True b False This argument form known as modus tollens is valid a True.

Therefore if p is true then q and r are true De Morgan's Theorem (1) ¬ (p ∧ q) ├. Therefore p is true Conjunction p,q ∴ (p∧q) p and q are true separately;. Therefore the disjunction (p or q) is true Composition (p → q) (p → r) ∴ (p → (q∧r)) if p then q;.

And if p then r;. To say that q is a "necessary component" of p is to mean that if one has p one must also have q, that is "if p then q" For example, "an engine is a necessary component of a functioning automobile" means that if one has a functioning car then one has an engine, rather than if one has an engine then one has a functioning car So, Morris' argument is as follows If you believe in. A valid argument form made up of three hypothetical, or conditional, statements If p, then q If q, then r Therefore, if p, then r Independent Premise A premise that does not depend on other premises to provide support to a conclusion;.

You can write p !q as ˘p_q The negation of "if p then q" is logically equivalent to "p and not q," that is, ˘(p !q) p^˘q 13 In Class Group Work First, show that p !q ˘p_q Then, show that ˘(p !.