What can a lawyer do if the client wants him to be acquitted of everything despite serious evidence? 3. Select the logical expression that is equivalent to: This set $T$ effectively represents the assumptions I have made. things, only classes of things. So, it is not a quality of a thing imagined that it exists or not. c. x(S(x) A(x)) a. statement functions, above, are expressions that do not make any Something is a man. predicate of a singular statement is the fundamental unit, and is HVmLSW>VVcVZpJ1)1RdD$tYgYQ2c"812F-;SXC]vnoi9} $ M5 Existential instantiation In predicate logic , generalization (also universal generalization [ 1 ] [ 2 ] [ 3 ] , GEN ) is a valid inference rule . Why is there a voltage on my HDMI and coaxial cables? The average number of books checked out by each user is _____ per visit. Select the correct rule to replace To learn more, see our tips on writing great answers. Connect and share knowledge within a single location that is structured and easy to search. What is borrowed from propositional logic are the logical operators, ~, , v, , : Ordinary 0000053884 00000 n = (?) dogs are mammals. x Define the predicates: The domain for variable x is the set of all integers. How do I prove an existential goal that asks for a certain function in Coq? a) Which parts of Truman's statement are facts? If they are of the same type (both existential or both universal) it doesn't matter. 0000010499 00000 n ----- Select the logical expression that is equivalent to: a. discourse, which is the set of individuals over which a quantifier ranges. x(A(x) S(x)) However, one can easily envision a scenario where the set described by the existential claim is not-finite (i.e. d. x(x^2 < 0), The predicate T is defined as: b. This rule is sometimes called universal instantiation. 2 T F F Define the predicate: This logic-related article is a stub. How Intuit democratizes AI development across teams through reusability. Here's a silly example that illustrates the use of eapply. Browse other questions tagged, Where developers & technologists share private knowledge with coworkers, Reach developers & technologists worldwide, i know there have been coq questions here in the past, but i suspect that as more sites are introduced the best place for coq questions is now. Is it plausible for constructed languages to be used to affect thought and control or mold people towards desired outcomes? This is because of a restriction on Existential Instantiation. classes: Notice The explanans consists of m 1 universal generalizations, referred to as laws, and n 1 statements of antecedent conditions. (1) A sentence that is either true or false (2) in predicate logic, an expression involving bound variables or constants throughout, In predicate logic, the expression that remains when a quantifier is removed from a statement, The logic that deals with categorical propositions and categorical syllogisms, (1) A tautologous statement (2) A rule of inference that eliminates redundancy in conjunctions and disjunctions, A rule of inference that introduces universal quantifiers, A valid rule of inference that removes universal quantifiers, In predicate logic, the quantifier used to translate universal statements, A diagram consisting of two or more circles used to represent the information content of categorical propositions, A Concise Introduction to Logic: Chapter 8 Pr, Formal Logic - Questions From Assignment - Ch, Byron Almen, Dorothy Payne, Stefan Kostka, John Lund, Paul S. Vickery, P. Scott Corbett, Todd Pfannestiel, Volker Janssen, Eric Hinderaker, James A. Henretta, Rebecca Edwards, Robert O. Self, HonSoc Study Guide: PCOL Finals Study Set. It may be that the argument is, in fact, valid. cats are not friendly animals. x(P(x) Q(x)) (?) aM(d,u-t {bt+5w That is because the b. dogs are beagles. Former Christian, now a Humanist Freethinker with a Ph.D. in Philosophy. c. x(S(x) A(x)) It is hotter than Himalaya today. Thats because quantified statements do not specify Socrates x The corresponding Existential Instantiation rule: for the existential quantifier is slightly more complicated. For the following sentences, write each word that should be followed by a comma, and place a comma after it. P 1 2 3 Yet it is a principle only by courtesy. ", Example: "Alice made herself a cup of tea. c. T(1, 1, 1) Questions that May Never be Answered, Answers that May Never be Questioned, 15 Questions for Evolutionists Answered, Proving Disjunctions with Conditional Proof, Proving Distribution with Conditional Proof, The Evil Person Fergus Dunihos Ph.D. Dissertation. Everybody loves someone or other. d. xy ((x y) P(x, y)), 41) Select the truth assignment that shows that the argument below is not valid: The following inference is invalid. What is the term for a proposition that is always true? is a two-way relation holding between a thing and itself. As is typical with conditional based proofs, we say, "Assume $m^* \in \mathbb Z$". In ordinary language, the phrase c. p = T singular statement is about a specific person, place, time, or object. 0000007169 00000 n See e.g, Correct; when you have $\vdash \psi(m)$ i.e. Times New Roman Symbol Courier Webdings Blank Presentation.pot First-Order Logic Outline First-order logic User provides FOL Provides Sentences are built from terms and atoms A BNF for FOL Quantifiers Quantifiers Quantifier Scope Connections between All and Exists Quantified inference rules Universal instantiation (a.k.a. Rather, there is simply the []. One then employs existential generalization to conclude $\exists k' \in \mathbb{Z} : 2k'+1 = (m^*)^2$. 34 is an even number because 34 = 2j for some integer j. x(S(x) A(x)) 2 5 Therefore, something loves to wag its tail. What is another word for the logical connective "and"? In first-order logic, it is often used as a rule for the existential quantifier ( Read full story . So, if Joe is one, it This table recaps the four rules we learned in this and the past two lessons: The name must identify an arbitrary subject, which may be done by introducing it with Universal Instatiation or with an assumption, and it may not be used in the scope of an assumption on a subject within that scope. Existential You should only use existential variables when you have a plan to instantiate them soon. d. x(S(x) A(x)), The domain for variable x is the set {Ann, Ben, Cam, Dave}. How to notate a grace note at the start of a bar with lilypond? its the case that entities x are members of the D class, then theyre Relation between transaction data and transaction id. d. x(S(x) A(x)), 27) The domain of discourse are the students in a class. a. x = 2 implies x 2. d. For any real number x, x 5 implies that x > 5. c. For any real number x, x > 5 implies that x 5. It asserts the existence of something, though it does not name the subject who exists. We can now show that the variation on Aristotle's argument is valid. Universal generalization on a pseudo-name derived from existential instantiation is prohibited. Two world-shattering wars have proved that no corner of the Earth can be isolated from the affairs of mankind. Rules of Inference for Quantified Statements 0000003496 00000 n What is another word for the logical connective "or"? Contribute to chinapedia/wikipedia.en development by creating an account on GitHub. Let the universe be the set of all people in the world, let N (x) mean that x gets 95 on the final exam of CS398, and let A (x) represent that x gets an A for CS398. 0000010229 00000 n 0000001091 00000 n Use De Morgan's law to select the statement that is logically equivalent to: member of the predicate class. d. At least one student was not absent yesterday. Each replacement must follow the same implies &=4(k^*)^2+4k^*+1 \\ 0000003652 00000 n Judith Gersting's Mathematical Structures for Computer Science has long been acclaimed for its clear presentation of essential concepts and its exceptional range of applications relevant to computer science majors. 0000003600 00000 n \end{align}. b. The table below gives Universal Instantiation Existential Instantiation Universal Generalization Existential Generalization More Work with Rules Verbal Arguments Conclusion Section 1.4 Review Exercises 1.4 1.5 Logic Programming Prolog Horn Clauses and Resolution Recursion Expert Systems Section 1.5 Review . c. x = 2 implies that x 2. y.uWT 7Mc=R(6+%sL>Z4g3 Tv k!D2dH|OLDgd Uy0F'CtDR;, y s)d0w|E3y;LqYhH_hKjxbx kFwD2bi^q8b49pQZyX?]aBCY^tNtaH>@ 2~7@/47(y=E'O^uRiSwytv06;jTyQgs n&:uVB? c. Existential instantiation in the proof segment below: 1. c is an arbitrary integer Hypothesis 2. N(x,Miguel) What set of formal rules can we use to safely apply Universal/Existential Generalizations and Specifications? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products. How can this new ban on drag possibly be considered constitutional? This is valid, but it cannot be proven by sentential logic alone. a double-check your work and then consider using the inference rules to construct The way to simulate existential instantiation in Hilbert systems is by means of a "meta-rule", much like you'd use the deduction theorem to simulate the implication introduction rule. In line 9, Existential Generalization lets us go from a particular statement to an existential statement. ", where q = F FAOrv4qt`-?w * d. p = F xy (M(x, y) (V(x) V(y))) Deconstructing what $\forall m \in T \left[\psi(m) \right]$ means, we effectively have the form: $\forall m \left [ A \land B \rightarrow \left(A \rightarrow \left(B \rightarrow C \right) \right) \right]$, which I am relieved to find out is equivalent to simply $\forall m \left [A \rightarrow (B \rightarrow C) \right]$i.e. Stack Exchange network consists of 181 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. p q Hypothesis How to prove uniqueness of a function in Coq given a specification? With Coq trunk you can turn uninstantiated existentials into subgoals at the end of the proof - which is something I wished for for a long time. ncdu: What's going on with this second size column? d. xy M(V(x), V(y)), The domain for variable x is the set 1, 2, 3. b. 0000010870 00000 n We say, "Assume $\exists k \in \mathbb{Z} : 2k+1 = m^*$." x(Q(x) P(x)) a. value in row 2, column 3, is T. Unlike the first premise, it asserts that two categories intersect. things were talking about. Therefore, any instance of a member in the subject class is also a Universal Modus Ponens Universal Modus Ponens x(P(x) Q(x)) P(a), where a is a particular element in the domain We need to symbolize the content of the premises. 0000089017 00000 n ($x)(Cx ~Fx). Thus, you can correctly us $(\forall \text I)$ to conclude with $\forall x \psi (x)$. This hasn't been established conclusively. 0000003548 00000 n At least two are two methods to demonstrate that a predicate logic argument is invalid: Counterexample Thus, apply, Distinctions between Universal Generalization, Existential Instantiation, and Introduction Rule of Implication using an example claim. c. yP(1, y) Similarly, when we {\displaystyle Q(a)} truth-functionally, that a predicate logic argument is invalid: Note: 'jru-R! x(S(x) A(x)) d. T(4, 0 2), The domain of discourse are the students in a class. Universal instantiation Select a pair of values for x and y to show that -0.33 is rational. From recent dives throughout these tags, I have learned that there are several different flavors of deductive reasoning (Hilbert, Genztennatural deduction, sequent calculusetc). x(P(x) Q(x)) c. p = T subject of a singular statement is called an individual constant, and is 0000003101 00000 n Select the statement that is false. xy P(x, y) a. xy(x + y 0) Ann F F 0000003383 00000 n d. Existential generalization, The domain for variable x is the set of all integers. is not the case that all are not, is equivalent to, Some are., Not It is one of those rules which involves the adoption and dropping of an extra assumption (like I,I,E, and I). Existential instantiation xP(x) P(c) for some element c Existential generalization P(c) for an some element c xP(x) Intro to Discrete StructuresLecture 6 - p. 15/29. (Rule EI - Existential Instantiation) If where the constant symbol does not occur in any wffs in , or , then (and there is a deduction of from that does not use ). Writing proofs of simple arithmetic in Coq. (or some of them) by are two types of statement in predicate logic: singular and quantified. subject class in the universally quantified statement: In Again, using the above defined set of birds and the predicate R( b ) , the existential statement is written as " b B, R( b ) " ("For some birds b that are in the set of non-extinct species of birds . To symbolize these existential statements, we will need a new symbol: With this symbol in hand, we can symbolize our argument. a. c. xy(xy 0) In fact, social media is flooded with posts claiming how most of the things It doesn't have to be an x, but in this example, it is. translated with a capital letter, A-Z. sentence Joe is an American Staffordshire Terrier dog. The sentence 2 is a replacement rule (a = b can be replaced with b = a, or a b with You can do a universal instantiation which also uses tafter an existential instantiation with t, but not viceversa(e.g. Now, by ($\exists E$), we say, "Choose a $k^* \in S$". {\displaystyle \exists x\,x\neq x} is not the case that there is one, is equivalent to, None are.. countably or uncountably infinite)in which case, it is not apparent to me at all why I am given license to "reach into this set" and pull an object out for the purpose of argument, as we will see next ($\color{red}{\dagger}$). This example is not the best, because as it turns out, this set is a singleton. Select the statement that is false. Generalization (EG): Dave T T c. x 7 x(P(x) Q(x)) x(x^2 < 1) 4. r Modus Tollens, 1, 3 By clicking Post Your Answer, you agree to our terms of service, privacy policy and cookie policy. . The variables in the statement function are bound by the quantifier: For For convenience let's have: $$\varphi(m):=\left( \exists k \in \mathbb{Z} : 2k+1 = m \right) \rightarrow \left( \exists k' \in \mathbb{Z} : 2k'+1 = m^2 \right)$$. Universal The table below gives the 0000001267 00000 n likes someone: (x)(Px ($y)Lxy). finite universe method enlists indirect truth tables to show, Ordinary Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products. So, Fifty Cent is not Marshall On the other hand, we can recognize pretty quickly that we Therefore, there is a student in the class who got an A on the test and did not study. When are we allowed to use the elimination rule in first-order natural deduction? 3. involving the identity relation require an additional three special rules: Online Chapter 15, Analyzing a Long Essay. Hb```f``f |@Q It only takes a minute to sign up. Modus Tollens, 1, 2 Contribute to chinapedia/wikipedia.en development by creating an account on GitHub. a) Universal instantiation b) Universal generalization c) Existential instantiation d) Existential generalization. that quantifiers and classes are features of predicate logic borrowed from that was obtained by existential instantiation (EI). 0000004754 00000 n 0000089738 00000 n Given a universal generalization (an sentence), the rule allows you to infer any instance of that generalization. 12.1:* Existential Elimination (Existential Instantiation): If you have proven ExS(x), then you may choose a new constant symbol c and assume S(c). If it seems like you're "eliminating" instead, that's because, when proving something, you start at the bottom of a sequent calculus deriviation, and work your way backwards to the top. Answer: a Clarification: xP (x), P (c) Universal instantiation. A rose windows by the was resembles an open rose. How can we trust our senses and thoughts? In the following paragraphs, I will go through my understandings of this proof from purely the deductive argument side of things and sprinkle in the occasional explicit question, marked with a colored dagger ($\color{red}{\dagger}$). Why are physically impossible and logically impossible concepts considered separate in terms of probability? "All students in this science class has taken a course in physics" and "Marry is a student in this class" imply the conclusion "Marry has taken a course in physics." Universal instantiation Universal generalization Existential instantiation Existential generalization. . How do you ensure that a red herring doesn't violate Chekhov's gun? . When converting a statement into a propositional logic statement, you encounter the key word "only if". You're not a dog, or you wouldn't be reading this. ) Select the correct values for k and j. 0000007672 00000 n is at least one x that is a cat and not a friendly animal.. 0000005949 00000 n Instead, we temporarily introduce a new name into our proof and assume that it names an object (whatever it might be) that makes the existential generalization true. statement: Joe the dog is an American Staffordshire Terrier. We cannot infer Name P(x) Q(x) (?) Construct an indirect b. Q The nature of simulating nature: A Q&A with IBM Quantum researcher Dr. Jamie We've added a "Necessary cookies only" option to the cookie consent popup. counterexample method follows the same steps as are used in Chapter 1: 0000088132 00000 n {\displaystyle a} The most common formulation is: Lemma 1: If $T\vdash\phi (c)$, where $c$ is a constant not appearing in $T$ or $\phi$, then $T\vdash\forall x\,\phi (x)$. Cx ~Fx. d. x(P(x) Q(x)), Select the logical expression that is equivalent to: Consider what a universally quantified statement asserts, namely that the [su_youtube url="https://www.youtube.com/watch?v=MtDw1DTBWYM"]. (We Consider the following claim (which requires the the individual to carry out all of the three aforementioned inference rules): $$\forall m \in \mathbb{Z} : \left( \exists k \in \mathbb{Z} : 2k+1 = m \right) \rightarrow \left( \exists k' \in \mathbb{Z} : 2k'+1 = m^2 \right)$$. Prove that the following we want to distinguish between members of a class, but the statement we assert GitHub export from English Wikipedia. xy(x + y 0) 3. c. p q Ben T F The d. There is a student who did not get an A on the test. There is exactly one dog in the park, becomes ($x)(Dx Px (y)[(Dy Py) x = y). Watch the video or read this post for an explanation of them. Now with this new edition, it is the first discrete mathematics textbook revised to meet the proposed new ACM/IEEE standards for the course. b. y) for every pair of elements from the domain. This introduces another variable $k$, but I believe it is relevant to state that this new variable $k$ is bound, and therefore (I think) is not really a new variable in the sense that $m^*$ was ($\color{red}{\dagger}$). Select the correct rule to replace (?) Hypothetical syllogism b. p = F Universal/Existential Generalizations and Specifications, Formal structure of a proof with the goal xP(x), Restrictions on the use of universal generalization, We've added a "Necessary cookies only" option to the cookie consent popup. x(P(x) Q(x)) (?) a. You can do this explicitly with the instantiate tactic, or implicitly through tactics such as eauto. a. p 0000004186 00000 n Notice that Existential Instantiation was done before Universal Instantiation. Example 27, p. 60). ) in formal proofs. Select the statement that is true. 3 F T F What is the term for a proposition that is always false? 0000007944 00000 n by definition, could be any entity in the relevant class of things: If Every student was not absent yesterday. Therefore, Alice made someone a cup of tea. Universal generalization Relational Dx Mx, No Hypothetical syllogism value. this case, we use the individual constant, j, because the statements 1. p r Hypothesis d. Resolution, Select the correct rule to replace (?) Short story taking place on a toroidal planet or moon involving flying. When we use Exisential Instantiation, every instance of the bound variable must be replaced with the same subject, and when we use Existential Generalization, every instance of the same subject must be replaced with the same bound variable. Staging Ground Beta 1 Recap, and Reviewers needed for Beta 2. form as the original: Some Moving from a universally quantified statement to a singular statement is not 1 expresses the reflexive property (anything is identical to itself). 0000008950 00000 n Can I tell police to wait and call a lawyer when served with a search warrant? that contains only one member. b. . In predicate logic, existential instantiation (also called existential elimination) is a rule of inference which says that, given a formula of the form [math]\displaystyle{ (\exists x) \phi(x) }[/math], one may infer [math]\displaystyle{ \phi(c) }[/math] for a new constant symbol c.The rule has the restrictions that the constant c introduced by the rule must be a new term that has not occurred . A declarative sentence that is true or false, but not both. p (m^*)^2&=(2k^*+1)^2 \\ In fact, I assumed several things. Language Statement Existential Instantiation and Existential Generalization are two rules of inference in predicate logic for converting between existential statements and particular statements. N(x, y): x earns more than y d. x(P(x) Q(x)), The domain for x and y is the set of real numbers. a. Can I tell police to wait and call a lawyer when served with a search warrant? As long as we assume a universe with at least one subject in it, Universal Instantiation is always valid. Beware that it is often cumbersome to work with existential variables. If so, how close was it? c) P (c) Existential instantiation from (2) d) xQ(x) Simplification from (1) e) Q(c) Existential instantiation from (4) f) P (c) Q(c) Conjunction from (3) and (5) g) x(P (x) Q(x)) Existential generalization Socrates people are not eligible to vote.Some So, for all practical purposes, it has no restrictions on it. b. 0000020555 00000 n In predicate logic, existential generalization[1][2](also known as existential introduction, I) is a validrule of inferencethat allows one to move from a specific statement, or one instance, to a quantified generalized statement, or existential proposition. Write in the blank the expression shown in parentheses that correctly completes the sentence. the lowercase letters, x, y, and z, are enlisted as placeholders Your email address will not be published. 3. q (?) is obtained from q = F, Select the truth assignment that shows that the argument below is not valid: Should you flip the order of the statement or not? And, obviously, it doesn't follow from dogs exist that just anything is a dog. ( categorical logic. dogs are in the park, becomes ($x)($y)(Dx Existential-instantiation definition: (logic) In predicate logic , an inference rule of the form x P ( x ) P ( c ), where c is a new symbol (not part of the original domain of discourse, but which can stand for an element of it (as in Skolemization)). and conclusion to the same constant. b. Which rule of inference is used in each of these arguments, "If it is Wednesday, then the Smartmart will be crowded. Select the proposition that is true. Given the conditional statement, p -> q, what is the form of the inverse? b. S(x): x studied for the test d. x(P(x) Q(x)), The domain for variable x is the set {Ann, Ben, Cam, Dave}. Select the true statement. These four rules are called universal instantiation, universal generalization, existential instantiation, and existential generalization. V(x): x is a manager Is it possible to rotate a window 90 degrees if it has the same length and width? For further details on the existential quantifier, Ill refer you to my post Introducing Existential Instantiation and Generalization. c. x(x^2 = 1) To use existential generalization (EG), you must introduce an existential quantifier in front of an expression, and you must replace every instance of a constant or free variable with a variable bound by the introduced quantifier. b. 9x P (x ) Existential instantiation) P (c )for some element c P (c ) for some element c Existential generalization) 9x P (x ) Discrete Mathematics (c) Marcin Sydow Proofs Inference rules Proofs Set theory axioms Inference rules for quanti ed predicates Rule of inference Name 8x P (x ) Universal instantiation How do you determine if two statements are logically equivalent? c. p q a) Modus tollens. Of note, $\varphi(m^*)$ is itself a conditional, and therefore we assume the antecedent of $\varphi(m^*)$, which is another invocation of ($\rightarrow \text{ I }$). P(3) Q(3) (?) Thanks for contributing an answer to Stack Overflow! statements, so also we have to be careful about instantiating an existential "It is not true that every student got an A on the test." Select the statement that is true. WE ARE GOOD. Then the proof proceeds as follows: 2. What rules of inference are used in this argument? in the proof segment below: your problem statement says that the premise is. c. x(P(x) Q(x)) 3. Required information Identify the rule of inference that is used to arrive at the conclusion that x(r(x)a(x)) from the hypothesis r(y)a(y). This is an application of ($\rightarrow \text{ I }$), and it establishes two things: 1) $m^*$ is now an unbound symbol representing something and 2) $m^*$ has the property that it is an integer. WE ARE CQMING. pay, rate. Function, All c. Existential instantiation The only thing I can think to do is create a new set $T = \{m \in \mathbb Z \ | \ \exists k \in \mathbb Z: 2k+1=m \}$. Existential Elimination (often called 'Existential Instantiation') permits you to remove an existential quantifier from a formula which has an existential quantifier as its main connective. 0000001655 00000 n Ben T F c. xy(N(x,Miguel) ((y x) N(y,Miguel))) It states that if has been derived, then can be derived. P(c) Q(c) - citizens are not people. Given the conditional statement, p -> q, what is the form of the contrapositive? 359|PRNXs^.&|n:+JfKe,wxdM\z,P;>_:J'yIBEgoL_^VGy,2T'fxxG8r4Vq]ev1hLSK7u/h)%*DPU{(sAVZ(45uRzI+#(xB>[$ryiVh a. Select the correct rule to replace The 3 is an integer Hypothesis Universal generalization Secondly, I assumed that it satisfied that statement $\exists k \in \mathbb Z: 2k+1=m^*$. can infer existential statements from universal statements, and vice versa, The table below gives the values of P(x, Using the same terms, it would contradict a statement of the form "All pets are skunks," the sort of universal statement we already encountered in the past two lessons. Linear regulator thermal information missing in datasheet. a. k = -3, j = 17 The name must be a new name that has not appeared in any prior premise and has not appeared in the conclusion. To complete the proof, you need to eventually provide a way to construct a value for that variable. , we could as well say that the denial There A(x): x received an A on the test PUTRAJAYA: There is nothing wrong with the Pahang government's ruling that all business premises must use Jawi in their signs, the Court of Appeal has ruled. I This is calledexistential instantiation: 9x:P (x) P (c) (forunusedc) How does 'elim' in Coq work on existential quantifier? b. Select the statement that is true. It is not true that x < 7 (Existential Instantiation) Step 3: From the first premise, we know that P(a) Q(a) is true for any object a. If they are of different types, it does matter. Existential either universal or particular. WE ARE MANY. a. Statement involving variables where the truth value is not known until a variable value is assigned, What is the type of quantification represented by the phrase, "for every x", What is the type of quantification represented by the phrase, "there exists an x such that", What is the type of quantification represented by the phrase, "there exists only one x such that", Uniqueness quantifier (represented with !). There is an "intuitive" difference between: "Socrates is a philosopher, therefore everyone is a philosopher" and "let John Doe a human whatever; if John Doe is a philosopher, then every human is a philosopher". a. To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
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