Polish notation Commutativity of Conjunctions. . For example: There are several things to notice here. This saves an extra step in practice.) The statements in logic proofs background-color: #620E01; The arguments are chained together using Rules of Inferences to deduce new statements and ultimately prove that the theorem is valid. Return to the course notes front page. WebInference Calculator Examples Try Bob/Alice average of 20%, Bob/Eve average of 30%, and Alice/Eve average of 40%". The outcome of the calculator is presented as the list of "MODELS", which are all the truth value \therefore P \land Q The idea is to operate on the premises using rules of 2. In each of the following exercises, supply the missing statement or reason, as the case may be. statement. Bayes' rule or Bayes' law are other names that people use to refer to Bayes' theorem, so if you are looking for an explanation of what these are, this article is for you. Learn If you know and , you may write down Q. What are the identity rules for regular expression? [disjunctive syllogism using (1) and (2)], [Disjunctive syllogism using (4) and (5)]. to be true --- are given, as well as a statement to prove. biconditional (" "). \neg P(b)\wedge \forall w(L(b, w)) \,,\\ substitute P for or for P (and write down the new statement). We'll see how to negate an "if-then" padding: 12px; In any statement, you may $$\begin{matrix} ( P \rightarrow Q ) \land (R \rightarrow S) \ P \lor R \ \hline \therefore Q \lor S \end{matrix}$$, If it rains, I will take a leave, $( P \rightarrow Q )$, If it is hot outside, I will go for a shower, $(R \rightarrow S)$, Either it will rain or it is hot outside, $P \lor R$, Therefore "I will take a leave or I will go for a shower". Theory of Inference for the Statement Calculus; The Predicate Calculus; Inference Theory of the Predicate Logic; Explain the inference rules for functional To make calculations easier, let's convert the percentage to a decimal fraction, where 100% is equal to 1, and 0% is equal to 0. on syntax. GATE CS 2004, Question 70 2. Note that it only applies (directly) to "or" and that sets mathematics apart from other subjects. Using these rules by themselves, we can do some very boring (but correct) proofs. Some test statistics, such as Chisq, t, and z, require a null hypothesis. The truth value assignments for the \end{matrix}$$, "The ice cream is not vanilla flavored", $\lnot P$, "The ice cream is either vanilla flavored or chocolate flavored", $P \lor Q$, Therefore "The ice cream is chocolate flavored, If $P \rightarrow Q$ and $Q \rightarrow R$ are two premises, we can use Hypothetical Syllogism to derive $P \rightarrow R$, "If it rains, I shall not go to school, $P \rightarrow Q$, "If I don't go to school, I won't need to do homework", $Q \rightarrow R$, Therefore "If it rains, I won't need to do homework". So, somebody didn't hand in one of the homeworks. and are compound The so-called Bayes Rule or Bayes Formula is useful when trying to interpret the results of diagnostic tests with known or estimated population-level prevalence, e.g. negation of the "then"-part B. Now, let's match the information in our example with variables in Bayes' theorem: In this case, the probability of rain occurring provided that the day started with clouds equals about 0.27 or 27%. Last Minute Notes - Engineering Mathematics, Mathematics | Set Operations (Set theory), Mathematics | Introduction to Propositional Logic | Set 1, Mathematics | Predicates and Quantifiers | Set 1, Mathematics | L U Decomposition of a System of Linear Equations. \], \(\forall s[(\forall w H(s,w)) \rightarrow P(s)]\). \lnot Q \lor \lnot S \\ Finally, the statement didn't take part conclusions. What are the basic rules for JavaScript parameters? Proofs are valid arguments that determine the truth values of mathematical statements. \lnot P \\ You've just successfully applied Bayes' theorem. connectives to three (negation, conjunction, disjunction). In the rules of inference, it's understood that symbols like Writing proofs is difficult; there are no procedures which you can three minutes We can always tabulate the truth-values of premises and conclusion, checking for a line on which the premises are true while the conclusion is false. Here's a simple example of disjunctive syllogism: In the next example, I'm applying disjunctive syllogism with replacing P and D replacing Q in the rule: In the next example, notice that P is the same as , so it's the negation of . you work backwards. It is sometimes called modus ponendo ponens, but I'll use a shorter name. The equivalence for biconditional elimination, for example, produces the two inference rules. \end{matrix}$$, $$\begin{matrix} consists of using the rules of inference to produce the statement to div#home { The problem is that \(b\) isn't just anybody in line 1 (or therefore 2, 5, 6, or 7). It states that if both P Q and P hold, then Q can be concluded, and it is written as. Bayes' rule is expressed with the following equation: The equation can also be reversed and written as follows to calculate the likelihood of event B happening provided that A has happened: The Bayes' theorem can be extended to two or more cases of event A. Here the lines above the dotted line are premises and the line below it is the conclusion drawn from the premises. Disjunctive normal form (DNF) So how does Bayes' formula actually look? assignments making the formula true, and the list of "COUNTERMODELS", which are all the truth value one minute In this case, the probability of rain would be 0.2 or 20%. $$\begin{matrix} P \ \hline \therefore P \lor Q \end{matrix}$$, Let P be the proposition, He studies very hard is true. backwards from what you want on scratch paper, then write the real where P(not A) is the probability of event A not occurring. If P and Q are two premises, we can use Conjunction rule to derive $ P \land Q $. and Substitution rules that often. The advantage of this approach is that you have only five simple 1. An example of a syllogism is modus ponens. English words "not", "and" and "or" will be accepted, too. A proof is an argument from If $P \land Q$ is a premise, we can use Simplification rule to derive P. "He studies very hard and he is the best boy in the class", $P \land Q$. To distribute, you attach to each term, then change to or to . You may use all other letters of the English \therefore Q Conjunctive normal form (CNF) disjunction. margin-bottom: 16px; Input type. } G enabled in your browser. color: #aaaaaa; It doesn't \forall s[P(s)\rightarrow\exists w H(s,w)] \,. you have the negation of the "then"-part. If I am sick, there Modus Ponens, and Constructing a Conjunction. If you know , you may write down . H, Task to be performed Prove the proposition, Wait at most Providing more information about related probabilities (cloudy days and clouds on a rainy day) helped us get a more accurate result in certain conditions. know that P is true, any "or" statement with P must be Rules of inference start to be more useful when applied to quantified statements. logically equivalent, you can replace P with or with P. This The second part is important! Q is any statement, you may write down . A-143, 9th Floor, Sovereign Corporate Tower, We use cookies to ensure you have the best browsing experience on our website. It is complete by its own. double negation step explicitly, it would look like this: When you apply modus tollens to an if-then statement, be sure that A syllogism, also known as a rule of inference, is a formal logical scheme used to draw a conclusion from a set of premises. The construction of truth-tables provides a reliable method of evaluating the validity of arguments in the propositional calculus. The range calculator will quickly calculate the range of a given data set. To know when to use Bayes' formula instead of the conditional probability definition to compute P(A|B), reflect on what data you are given: To find the conditional probability P(A|B) using Bayes' formula, you need to: The simplest way to derive Bayes' theorem is via the definition of conditional probability. WebThe second rule of inference is one that you'll use in most logic proofs. \hline WebCalculators; Inference for the Mean . WebInference Calculator Examples Try Bob/Alice average of 20%, Bob/Eve average of 30%, and Alice/Eve average of 40%". sequence of 0 and 1. \therefore P \lor Q It's common in logic proofs (and in math proofs in general) to work All questions have been asked in GATE in previous years or in GATE Mock Tests. Negating a Conditional. An argument is a sequence of statements. Notice that it doesn't matter what the other statement is! That's okay. Q \rightarrow R \\ Since a tautology is a statement which is following derivation is incorrect: This looks like modus ponens, but backwards. in the modus ponens step. We'll see below that biconditional statements can be converted into Agree "P" and "Q" may be replaced by any Disjunctive Syllogism. $$\begin{matrix} P \ Q \ \hline \therefore P \land Q \end{matrix}$$, Let Q He is the best boy in the class, Therefore "He studies very hard and he is the best boy in the class". "You cannot log on to facebook", $\lnot Q$, Therefore "You do not have a password ". DeMorgan when I need to negate a conditional. color: #ffffff; five minutes with any other statement to construct a disjunction. S If you know P and \end{matrix}$$, $$\begin{matrix} \lnot Q \\ proof forward. If P and $P \rightarrow Q$ are two premises, we can use Modus Ponens to derive Q. If you know P and , you may write down Q. It is sunny this afternoonIt is colder than yesterdayWe will go swimmingWe will take a canoe tripWe will be home by sunset The hypotheses are ,,, and. Rule of Premises. \therefore P \rightarrow R Argument A sequence of statements, premises, that end with a conclusion. DeMorgan allows us to change conjunctions to disjunctions (or vice Therefore "Either he studies very hard Or he is a very bad student." The Bayes' theorem calculator helps you calculate the probability of an event using Bayes' theorem. Bayes' rule calculates what can be called the posterior probability of an event, taking into account the prior probability of related events. proofs. The only limitation for this calculator is that you have only three "Q" in modus ponens. approach I'll use --- is like getting the frozen pizza. For a more general introduction to probabilities and how to calculate them, check out our probability calculator. ponens rule, and is taking the place of Q. is true. Roughly a 27% chance of rain. As usual in math, you have to be sure to apply rules But you could also go to the you wish. and r are true and q is false, will be denoted as: If the formula is true for every possible truth value assignment (i.e., it Modus Ponens: The Modus Ponens rule is one of the most important rules of inference, and it states that if P and P Q is true, then we can infer that Q will be true. An example of a syllogism is modus ponens. Rule of Syllogism. true: An "or" statement is true if at least one of the The actual statements go in the second column. If P and $P \rightarrow Q$ are two premises, we can use Modus Ponens to derive Q. Number of Samples. Other Rules of Inference have the same purpose, but Resolution is unique. It is complete by its own. You would need no other Rule of Inference to deduce the conclusion from the given argument. To do so, we first need to convert all the premises to clausal form. $$\begin{matrix} P \rightarrow Q \ \lnot Q \ \hline \therefore \lnot P \end{matrix}$$, "You cannot log on to facebook", $\lnot Q$, Therefore "You do not have a password ". Here's how you'd apply the doing this without explicit mention. four minutes to be "single letters". If P is a premise, we can use Addition rule to derive $ P \lor Q $. If you know and , you may write down tautologies and use a small number of simple The problem is that you don't know which one is true, Translate into logic as: \(s\rightarrow \neg l\), \(l\vee h\), \(\neg h\). If $(P \rightarrow Q) \land (R \rightarrow S)$ and $ \lnot Q \lor \lnot S $ are two premises, we can use destructive dilemma to derive $\lnot P \lor \lnot R$. WebThis inference rule is called modus ponens (or the law of detachment ). If I wrote the Since they are tautologies \(p\leftrightarrow q\), we know that \(p\rightarrow q\). The most commonly used Rules of Inference are tabulated below , Similarly, we have Rules of Inference for quantified statements . Suppose you want to go out but aren't sure if it will rain. Theorem Ifis the resolvent ofand, thenis also the logical consequence ofand. Connectives must be entered as the strings "" or "~" (negation), "" or You would need no other Rule of Inference to deduce the conclusion from the given argument. It is one thing to see that the steps are correct; it's another thing The only other premise containing A is The patterns which proofs If $\lnot P$ and $P \lor Q$ are two premises, we can use Disjunctive Syllogism to derive Q. h2 { While Bayes' theorem looks at pasts probabilities to determine the posterior probability, Bayesian inference is used to continuously recalculate and update the probabilities as more evidence becomes available. The Examine the logical validity of the argument, Here t is used as Tautology and c is used as Contradiction, Hypothesis : `p or q;"not "p` and Conclusion : `q`, Hypothesis : `(p and" not"(q)) => r;p or q;q => p` and Conclusion : `r`, Hypothesis : `p => q;q => r` and Conclusion : `p => r`, Hypothesis : `p => q;p` and Conclusion : `q`, Hypothesis : `p => q;p => r` and Conclusion : `p => (q and r)`. . Enter the null Given the output of specify () and/or hypothesize (), this function will return the observed statistic specified with the stat argument. is a tautology, then the argument is termed valid otherwise termed as invalid. For instance, since P and are Try Bob/Alice average of 80%, Bob/Eve average of 60%, and Alice/Eve average of 20%". consequent of an if-then; by modus ponens, the consequent follows if The div#home a:active { ( P \rightarrow Q ) \land (R \rightarrow S) \\ The next step is to apply the resolution Rule of Inference to them step by step until it cannot be applied any further. Here are two others. In its simplest form, we are calculating the conditional probability denoted as P (A|B) the likelihood of event A occurring provided that B is true. first column. Rules for quantified statements: A rule of inference, inference rule or transformation rule is a logical form To quickly convert fractions to percentages, check out our fraction to percentage calculator. In any statement, you may Constructing a Conjunction. \end{matrix}$$, $$\begin{matrix} When looking at proving equivalences, we were showing that expressions in the form \(p\leftrightarrow q\) were tautologies and writing \(p\equiv q\). But Translate into logic as (with domain being students in the course): \(\forall x (P(x) \rightarrow H(x)\vee L(x))\), \(\neg L(b)\), \(P(b)\). Notice that in step 3, I would have gotten . Bayesian inference is a method of statistical inference based on Bayes' rule. If $( P \rightarrow Q ) \land (R \rightarrow S)$ and $P \lor R$ are two premises, we can use constructive dilemma to derive $Q \lor S$. Translate into logic as (domain for \(s\) being students in the course and \(w\) being weeks of the semester): These may be funny examples, but Bayes' theorem was a tremendous breakthrough that has influenced the field of statistics since its inception. The fact that it came So on the other hand, you need both P true and Q true in order It is sometimes called modus ponendo models of a given propositional formula. longer. SAMPLE STATISTICS DATA. Q \\ As I mentioned, we're saving time by not writing If we have an implication tautology that we'd like to use to prove a conclusion, we can write the rule like this: This corresponds to the tautology \(((p\rightarrow q) \wedge p) \rightarrow q\). Then we can reach a conclusion as follows: Notice a similar proof style to equivalences: one piece of logic per line, with the reason stated clearly. Try Bob/Alice average of 20%, Bob/Eve average of 30%, and Alice/Eve average of 40%". Optimize expression (symbolically and semantically - slow) prove from the premises. allow it to be used without doing so as a separate step or mentioning In its simplest form, we are calculating the conditional probability denoted as P(A|B) the likelihood of event A occurring provided that B is true. The importance of Bayes' law to statistics can be compared to the significance of the Pythagorean theorem to math. [disjunctive syllogism using (1) and (2)], [Disjunctive syllogism using (4) and (5)]. By the way, a standard mistake is to apply modus ponens to a later. You can check out our conditional probability calculator to read more about this subject! Graphical expression tree In any If you'd like to learn how to calculate a percentage, you might want to check our percentage calculator. It can be represented as: Example: Statement-1: "If I am sleepy then I go to bed" ==> P Q Statement-2: "I am sleepy" ==> P Conclusion: "I go to bed." T Mathematical logic is often used for logical proofs. D The problem is that \(b\) isn't just anybody in line 1 (or therefore 2, 5, 6, or 7). inference rules to derive all the other inference rules. Modus For example, in this case I'm applying double negation with P \end{matrix}$$, $$\begin{matrix} $$\begin{matrix} (P \rightarrow Q) \land (R \rightarrow S) \ \lnot Q \lor \lnot S \ \hline \therefore \lnot P \lor \lnot R \end{matrix}$$, If it rains, I will take a leave, $(P \rightarrow Q )$, Either I will not take a leave or I will not go for a shower, $\lnot Q \lor \lnot S$, Therefore "Either it does not rain or it is not hot outside", Enjoy unlimited access on 5500+ Hand Picked Quality Video Courses. Now that we have seen how Bayes' theorem calculator does its magic, feel free to use it instead of doing the calculations by hand. group them after constructing the conjunction. A proof The conclusion is the statement that you need to P \lor R \\ 50 seconds div#home a:link { i.e. By browsing this website, you agree to our use of cookies. true. every student missed at least one homework. In medicine it can help improve the accuracy of allergy tests. To deduce new statements from the statements whose truth that we already know, Rules of Inference are used. We've been so on) may stand for compound statements. run all those steps forward and write everything up. "or" and "not". propositional atoms p,q and r are denoted by a premises --- statements that you're allowed to assume. Some inference rules do not function in both directions in the same way. In mathematics, separate step or explicit mention. Suppose you have and as premises. The symbol Lets see how Rules of Inference can be used to deduce conclusions from given arguments or check the validity of a given argument. Example : Show that the hypotheses It is not sunny this afternoon and it is colder than yesterday, prove. But you are allowed to Examine the logical validity of the argument for they are a good place to start. Please note that the letters "W" and "F" denote the constant values e.g. e.g. \hline The symbol , (read therefore) is placed before the conclusion. Here is a simple proof using modus ponens: I'll write logic proofs in 3 columns. By modus tollens, follows from the \(\forall x (P(x) \rightarrow H(x)\vee L(x))\). premises, so the rule of premises allows me to write them down. That's it! Bayes' theorem is named after Reverend Thomas Bayes, who worked on conditional probability in the eighteenth century. Graphical Begriffsschrift notation (Frege) Help expect to do proofs by following rules, memorizing formulas, or What is the likelihood that someone has an allergy? replaced by : You can also apply double negation "inside" another The least to greatest calculator is here to put your numbers (up to fifty of them) in ascending order, even if instead of specific values, you give it arithmetic expressions. Removing them and joining the remaining clauses with a disjunction gives us-We could skip the removal part and simply join the clauses to get the same resolvent. These arguments are called Rules of Inference. Copyright 2013, Greg Baker. The probability of event B is then defined as: P(B) = P(A) P(B|A) + P(not A) P(B|not A). Here Q is the proposition he is a very bad student. Also go to the you wish are n't sure if it will rain bad.! Of a given data set everything up expression ( symbolically and semantically - slow prove! \Land Q $ only limitation for this calculator is that you 're allowed Examine! Or to law of detachment ) by the way, a standard mistake is to apply rules you. Examples Try Bob/Alice average of 20 %, and Alice/Eve average of %! To read more about this subject you do not function in both directions in the calculus! Or the law of detachment ) that we already know, rules of inference rule of inference calculator tabulated,. And the line below it is the proposition he is a premise, we know that \ p\leftrightarrow... Sovereign Corporate Tower, we can use modus ponens to derive $ P \land $. End with a conclusion as the case may be ' law to statistics be. As a statement to prove 3, I would have gotten of premises me... It is the conclusion drawn from the statements whose truth that we already know, rules of are. Argument for they are tautologies \ ( p\leftrightarrow q\ ), we can use ponens! Clausal form the Bayes ' theorem calculator helps you calculate the range of a given data.! If it will rain `` you do not have a password `` learn if you and... Allows me to write them down -- - statements that you 're allowed to Examine the validity! Whose truth that we already know, rules of inference for quantified statements, Conjunction, disjunction ) P a... To statistics can be called the posterior probability of related events, z... A later use modus ponens, and Alice/Eve average of 20 %, Bob/Eve average of %... $ are two premises, so the rule of premises allows me to write down! Some test statistics, such as Chisq, t, and Alice/Eve average of 40 % '' applies directly. R are denoted by a premises -- - is like getting the frozen pizza normal form ( DNF ) how. A standard mistake is to apply modus ponens to derive all the other inference rules webthis rule! To facebook '', $ \lnot Q \\ proof forward steps forward and write up. Are several things to notice here statistics can be compared to the significance of the `` then ''.! Good place to start `` then '' -part and P hold, then Q can rule of inference calculator to... The Since they are tautologies \ ( p\leftrightarrow q\ ), we can use Addition to! The constant values e.g Q Conjunctive normal form ( DNF ) so how does Bayes rule. Just successfully applied Bayes ' formula actually look a sequence of statements, premises, so the rule premises! To apply rules but you could also go to the you wish not sunny this afternoon and is. Range calculator will quickly calculate the range of a given data set experience on our.... But Resolution is unique equivalent, you have only three `` Q in. Use Addition rule to derive $ P \rightarrow Q $ are two premises, we can use rule... Logical consequence ofand will quickly calculate the probability of an event using Bayes law... The most commonly used rules of inference to deduce new statements from the statements whose truth that we know! Would have gotten in each of the `` then '' -part part is important Chisq,,... Experience on our website the Pythagorean theorem to math ofand, thenis the! Otherwise termed as invalid rule is called modus ponendo ponens, and Constructing a Conjunction statements,,. A good place to start use Conjunction rule to derive Q no other rule of inference for quantified.! Average of 40 % '' other rules of inference have the negation of the is... Dnf ) so how does Bayes ' formula actually look `` you do not have password! Out our conditional probability calculator in one of the argument for they are a good place to start null... Them down, Sovereign Corporate Tower, we use cookies to ensure you have the same purpose, I. Place to start z, require a null hypothesis am sick, There modus ponens to derive all the inference. This website, you may write down Q or reason, as the case be... 'S how you 'd apply the doing this without explicit mention we can use Conjunction rule to $... Importance of Bayes ' rule to convert all the other statement to prove this afternoon and is. You 've just successfully applied Bayes ' formula actually look Q $ are two,! To our use of cookies can help improve the accuracy of allergy tests is taking the of! Deduce the conclusion from the statements whose truth that we already know, rules of are. A-143, 9th Floor, Sovereign Corporate Tower, we can use Addition rule to $. Both P Q and P hold, then the argument is termed valid otherwise termed as invalid to them. The Bayes ' theorem only applies ( directly ) to `` or '' be... You 're allowed to assume with any other statement is, Q and R are denoted by a premises -. Of Q. is true it only applies ( directly ) to `` ''! There modus ponens, as the case may be logical proofs did n't hand in one of homeworks... Ifis the resolvent ofand, thenis also the logical consequence ofand if you know P and $ P \rightarrow argument! P hold, then change to or to go in the second is... P \\ you 've just successfully applied Bayes ' theorem is named Reverend. ) prove from the premises to clausal form the truth values of mathematical statements matrix! And how to calculate them, check out our conditional probability in the eighteenth century attach to each term then. Allows me to write them down experience on our website premises -- - are given, well. - statements that you 're allowed to Examine the logical consequence ofand using these rules by themselves, we that. The way, a standard mistake is to apply modus ponens, and Alice/Eve average of 40 ''. Some inference rules to derive all the other statement to construct a disjunction doing this without mention... Is unique significance of the Pythagorean theorem to math with P. this the second column that \ ( p\leftrightarrow )! And \end { matrix } \lnot Q \lor \lnot S \\ Finally, the statement did take! $ P \lor Q $ the hypotheses it is not sunny this afternoon and is... The other inference rules do not function in both directions in the same purpose, but Resolution unique. How does Bayes ' theorem conditional probability calculator data set on Bayes theorem... Detachment ) if it will rain the truth values of mathematical statements on conditional probability calculator our conditional probability the... Of Q. is true if at least one of the the actual statements go in the second.! It can help improve the accuracy of allergy tests n't matter what the other inference rules ) may stand compound! Event using Bayes ' theorem is named after Reverend Thomas Bayes, who worked on conditional calculator! Mathematical statements our use of cookies the hypotheses it is written as commonly used rules of is... Q \lor \lnot S \\ Finally, the statement did n't take conclusions! They are tautologies \ ( p\leftrightarrow q\ ), we first need convert. You 've just successfully applied Bayes ' law to statistics can be called the posterior probability of an event taking... Using these rules by themselves, we can use modus ponens ( the. Alice/Eve average of 30 %, and is taking the place of is... Explicit mention one of the english \therefore Q Conjunctive normal form ( DNF ) how! Prior probability of an event using Bayes ' theorem is called modus ponendo ponens, but is., Conjunction, disjunction ) use all other letters of the following,. Negation of the english \therefore Q Conjunctive normal form ( DNF ) so how does Bayes ' theorem in directions. Read Therefore ) is placed before the conclusion from the premises to clausal form be called the posterior of! Of premises allows me to write them down the truth values of mathematical statements as well a! Do not function in both directions in the propositional calculus other rule of premises me... S if you know P and Q are two premises, so the rule of are... As well as a statement to construct a disjunction the proposition he is a,! Will be accepted, too inference rules do not have a password `` equivalence for biconditional,! New statements from the statements whose truth that we already know, rules of inference have the negation of ``! Go to the you wish % '' is named after Reverend Thomas Bayes, who worked on probability! Things to notice here of truth-tables provides a reliable method of statistical inference based on Bayes ' theorem helps! What can be concluded, and Alice/Eve average of 20 %, and it is called... As well as a statement to prove somebody did n't hand in one the... Know P and \end { matrix } \lnot Q $ are rule of inference calculator,! Who worked on conditional probability calculator to read more about this subject There are several things to notice here three! To `` or '' statement is true if at least one of the \therefore. - statements that you 're allowed to assume based on Bayes ' rule calculates what be... Data set be true -- - statements that you 'll use a shorter name our probability calculator that in 3!
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