7.7 Disjunctive and Hypothetical Syllogisms
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Chrysippus's Dog As a Case Study in Non-Linguistic Cognition
Chrysippus’s Dog as a Case Study in Non-Linguistic Cognition Michael Rescorla Abstract: I critique an ancient argument for the possibility of non-linguistic deductive inference. The argument, attributed to Chrysippus, describes a dog whose behavior supposedly reflects disjunctive syllogistic reasoning. Drawing on contemporary robotics, I urge that we can equally well explain the dog’s behavior by citing probabilistic reasoning over cognitive maps. I then critique various experimentally-based arguments from scientific psychology that echo Chrysippus’s anecdotal presentation. §1. Language and thought Do non-linguistic creatures think? Debate over this question tends to calcify into two extreme doctrines. The first, espoused by Descartes, regards language as necessary for cognition. Modern proponents include Brandom (1994, pp. 145-157), Davidson (1984, pp. 155-170), McDowell (1996), and Sellars (1963, pp. 177-189). Cartesians may grant that ascribing cognitive activity to non-linguistic creatures is instrumentally useful, but they regard such ascriptions as strictly speaking false. The second extreme doctrine, espoused by Gassendi, Hume, and Locke, maintains that linguistic and non-linguistic cognition are fundamentally the same. Modern proponents include Fodor (2003), Peacocke (1997), Stalnaker (1984), and many others. Proponents may grant that non- linguistic creatures entertain a narrower range of thoughts than us, but they deny any principled difference in kind.1 2 An intermediate position holds that non-linguistic creatures display cognitive activity of a fundamentally different kind than human thought. Hobbes and Leibniz favored this intermediate position. Modern advocates include Bermudez (2003), Carruthers (2002, 2004), Dummett (1993, pp. 147-149), Malcolm (1972), and Putnam (1992, pp. 28-30). -
On Basic Probability Logic Inequalities †
mathematics Article On Basic Probability Logic Inequalities † Marija Boriˇci´cJoksimovi´c Faculty of Organizational Sciences, University of Belgrade, Jove Ili´ca154, 11000 Belgrade, Serbia; [email protected] † The conclusions given in this paper were partially presented at the European Summer Meetings of the Association for Symbolic Logic, Logic Colloquium 2012, held in Manchester on 12–18 July 2012. Abstract: We give some simple examples of applying some of the well-known elementary probability theory inequalities and properties in the field of logical argumentation. A probabilistic version of the hypothetical syllogism inference rule is as follows: if propositions A, B, C, A ! B, and B ! C have probabilities a, b, c, r, and s, respectively, then for probability p of A ! C, we have f (a, b, c, r, s) ≤ p ≤ g(a, b, c, r, s), for some functions f and g of given parameters. In this paper, after a short overview of known rules related to conjunction and disjunction, we proposed some probabilized forms of the hypothetical syllogism inference rule, with the best possible bounds for the probability of conclusion, covering simultaneously the probabilistic versions of both modus ponens and modus tollens rules, as already considered by Suppes, Hailperin, and Wagner. Keywords: inequality; probability logic; inference rule MSC: 03B48; 03B05; 60E15; 26D20; 60A05 1. Introduction The main part of probabilization of logical inference rules is defining the correspond- Citation: Boriˇci´cJoksimovi´c,M. On ing best possible bounds for probabilities of propositions. Some of them, connected with Basic Probability Logic Inequalities. conjunction and disjunction, can be obtained immediately from the well-known Boole’s Mathematics 2021, 9, 1409. -
7.1 Rules of Implication I
Natural Deduction is a method for deriving the conclusion of valid arguments expressed in the symbolism of propositional logic. The method consists of using sets of Rules of Inference (valid argument forms) to derive either a conclusion or a series of intermediate conclusions that link the premises of an argument with the stated conclusion. The First Four Rules of Inference: ◦ Modus Ponens (MP): p q p q ◦ Modus Tollens (MT): p q ~q ~p ◦ Pure Hypothetical Syllogism (HS): p q q r p r ◦ Disjunctive Syllogism (DS): p v q ~p q Common strategies for constructing a proof involving the first four rules: ◦ Always begin by attempting to find the conclusion in the premises. If the conclusion is not present in its entirely in the premises, look at the main operator of the conclusion. This will provide a clue as to how the conclusion should be derived. ◦ If the conclusion contains a letter that appears in the consequent of a conditional statement in the premises, consider obtaining that letter via modus ponens. ◦ If the conclusion contains a negated letter and that letter appears in the antecedent of a conditional statement in the premises, consider obtaining the negated letter via modus tollens. ◦ If the conclusion is a conditional statement, consider obtaining it via pure hypothetical syllogism. ◦ If the conclusion contains a letter that appears in a disjunctive statement in the premises, consider obtaining that letter via disjunctive syllogism. Four Additional Rules of Inference: ◦ Constructive Dilemma (CD): (p q) • (r s) p v r q v s ◦ Simplification (Simp): p • q p ◦ Conjunction (Conj): p q p • q ◦ Addition (Add): p p v q Common Misapplications Common strategies involving the additional rules of inference: ◦ If the conclusion contains a letter that appears in a conjunctive statement in the premises, consider obtaining that letter via simplification. -
Methods of Proof Direct Direct Contrapositive Contradiction Pc P→ C ¬ P ∨ C ¬ C → ¬ Pp ∧ ¬ C
Proof Methods Methods of proof Direct Direct Contrapositive Contradiction pc p→ c ¬ p ∨ c ¬ c → ¬ pp ∧ ¬ c TT T T T F Section 1.6 & 1.7 T F F F F T FT T T T F FF T T T F MSU/CSE 260 Fall 2009 1 MSU/CSE 260 Fall 2009 2 How are these questions related? Proof Methods h1 ∧ h2 ∧ … ∧ hn ⇒ c ? 1. Does p logically imply c ? Let p = h ∧ h ∧ … ∧ h . The following 2. Is the proposition (p → c) a tautology? 1 2 n propositions are equivalent: 3. Is the proposition (¬ p ∨ c) is a tautology? 1. p ⇒ c 4. Is the proposition (¬ c → ¬ p) is a tautology? 2. (p → c) is a tautology. Direct 5. Is the proposition (p ∧ ¬ c) is a contradiction? 3. (¬ p ∨ c) is a tautology. Direct 4. (¬ c → ¬ p)is a tautology. Contrapositive 5. (p ∧ ¬ c) is a contradiction. Contradiction MSU/CSE 260 Fall 2009 3 MSU/CSE 260 Fall 2009 4 © 2006 by A-H. Esfahanian. All Rights Reserved. 1- Formal Proofs Formal Proof A proof is equivalent to establishing a logical To prove: implication chain h1 ∧ h2 ∧ … ∧ hn ⇒ c p1 Premise p Tautology Given premises (hypotheses) h1 , h2 , … , hn and Produce a series of wffs, 2 conclusion c, to give a formal proof that the … . p1 , p2 , pn, c . p k, k’, Inf. Rule hypotheses imply the conclusion, entails such that each wff pr is: r . establishing one of the premises or . a tautology, or pn _____ h1 ∧ h2 ∧ … ∧ hn ⇒ c an axiom/law of the domain (e.g., 1+3=4 or x > x+1 ) ∴ c justified by definition, or logically equivalent to or implied by one or more propositions pk where 1 ≤ k < r. -
Chapter 9: Answers and Comments Step 1 Exercises 1. Simplification. 2. Absorption. 3. See Textbook. 4. Modus Tollens. 5. Modus P
Chapter 9: Answers and Comments Step 1 Exercises 1. Simplification. 2. Absorption. 3. See textbook. 4. Modus Tollens. 5. Modus Ponens. 6. Simplification. 7. X -- A very common student mistake; can't use Simplification unless the major con- nective of the premise is a conjunction. 8. Disjunctive Syllogism. 9. X -- Fallacy of Denying the Antecedent. 10. X 11. Constructive Dilemma. 12. See textbook. 13. Hypothetical Syllogism. 14. Hypothetical Syllogism. 15. Conjunction. 16. See textbook. 17. Addition. 18. Modus Ponens. 19. X -- Fallacy of Affirming the Consequent. 20. Disjunctive Syllogism. 21. X -- not HS, the (D v G) does not match (D v C). This is deliberate to make sure you don't just focus on generalities, and make sure the entire form fits. 22. Constructive Dilemma. 23. See textbook. 24. Simplification. 25. Modus Ponens. 26. Modus Tollens. 27. See textbook. 28. Disjunctive Syllogism. 29. Modus Ponens. 30. Disjunctive Syllogism. Step 2 Exercises #1 1 Z A 2. (Z v B) C / Z C 3. Z (1)Simp. 4. Z v B (3) Add. 5. C (2)(4)MP 6. Z C (3)(5) Conj. For line 4 it is easy to get locked into line 2 and strategy 1. But they do not work. #2 1. K (B v I) 2. K 3. ~B 4. I (~T N) 5. N T / ~N 6. B v I (1)(2) MP 7. I (6)(3) DS 8. ~T N (4)(7) MP 9. ~T (8) Simp. 10. ~N (5)(9) MT #3 See textbook. #4 1. H I 2. I J 3. -
Argument Forms and Fallacies
6.6 Common Argument Forms and Fallacies 1. Common Valid Argument Forms: In the previous section (6.4), we learned how to determine whether or not an argument is valid using truth tables. There are certain forms of valid and invalid argument that are extremely common. If we memorize some of these common argument forms, it will save us time because we will be able to immediately recognize whether or not certain arguments are valid or invalid without having to draw out a truth table. Let’s begin: 1. Disjunctive Syllogism: The following argument is valid: “The coin is either in my right hand or my left hand. It’s not in my right hand. So, it must be in my left hand.” Let “R”=”The coin is in my right hand” and let “L”=”The coin is in my left hand”. The argument in symbolic form is this: R ˅ L ~R __________________________________________________ L Any argument with the form just stated is valid. This form of argument is called a disjunctive syllogism. Basically, the argument gives you two options and says that, since one option is FALSE, the other option must be TRUE. 2. Pure Hypothetical Syllogism: The following argument is valid: “If you hit the ball in on this turn, you’ll get a hole in one; and if you get a hole in one you’ll win the game. So, if you hit the ball in on this turn, you’ll win the game.” Let “B”=”You hit the ball in on this turn”, “H”=”You get a hole in one”, and “W”=”you win the game”. -
Philosophy 109, Modern Logic Russell Marcus
Philosophy 240: Symbolic Logic Hamilton College Fall 2014 Russell Marcus Reference Sheeet for What Follows Names of Languages PL: Propositional Logic M: Monadic (First-Order) Predicate Logic F: Full (First-Order) Predicate Logic FF: Full (First-Order) Predicate Logic with functors S: Second-Order Predicate Logic Basic Truth Tables - á á @ â á w â á e â á / â 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 0 0 1 1 0 1 0 0 1 0 0 0 0 1 0 1 1 0 1 1 0 0 1 0 0 0 0 0 0 0 1 0 0 1 0 Rules of Inference Modus Ponens (MP) Conjunction (Conj) á e â á á / â â / á A â Modus Tollens (MT) Addition (Add) á e â á / á w â -â / -á Simplification (Simp) Disjunctive Syllogism (DS) á A â / á á w â -á / â Constructive Dilemma (CD) (á e â) Hypothetical Syllogism (HS) (ã e ä) á e â á w ã / â w ä â e ã / á e ã Philosophy 240: Symbolic Logic, Prof. Marcus; Reference Sheet for What Follows, page 2 Rules of Equivalence DeMorgan’s Laws (DM) Contraposition (Cont) -(á A â) W -á w -â á e â W -â e -á -(á w â) W -á A -â Material Implication (Impl) Association (Assoc) á e â W -á w â á w (â w ã) W (á w â) w ã á A (â A ã) W (á A â) A ã Material Equivalence (Equiv) á / â W (á e â) A (â e á) Distribution (Dist) á / â W (á A â) w (-á A -â) á A (â w ã) W (á A â) w (á A ã) á w (â A ã) W (á w â) A (á w ã) Exportation (Exp) á e (â e ã) W (á A â) e ã Commutativity (Com) á w â W â w á Tautology (Taut) á A â W â A á á W á A á á W á w á Double Negation (DN) á W --á Six Derived Rules for the Biconditional Rules of Inference Rules of Equivalence Biconditional Modus Ponens (BMP) Biconditional DeMorgan’s Law (BDM) á / â -(á / â) W -á / â á / â Biconditional Modus Tollens (BMT) Biconditional Commutativity (BCom) á / â á / â W â / á -á / -â Biconditional Hypothetical Syllogism (BHS) Biconditional Contraposition (BCont) á / â á / â W -á / -â â / ã / á / ã Philosophy 240: Symbolic Logic, Prof. -
Proposi'onal Logic Proofs Proof Method #1: Truth Table Example
9/9/14 Proposi'onal Logic Proofs • An argument is a sequence of proposi'ons: Inference Rules ² Premises (Axioms) are the first n proposi'ons (Rosen, Section 1.5) ² Conclusion is the final proposi'on. p p ... p q TOPICS • An argument is valid if is a ( 1 ∧ 2 ∧ ∧ n ) → tautology, given that pi are the premises • Logic Proofs (axioms) and q is the conclusion. ² via Truth Tables € ² via Inference Rules 2 Proof Method #1: Truth Table Example Proof By Truth Table s = ((p v q) ∧ (¬p v r)) → (q v r) n If the conclusion is true in the truth table whenever the premises are true, it is p q r ¬p p v q ¬p v r q v r (p v q)∧ (¬p v r) s proved 0 0 0 1 0 1 0 0 1 n Warning: when the premises are false, the 0 0 1 1 0 1 1 0 1 conclusion my be true or false 0 1 0 1 1 1 1 1 1 n Problem: given n propositions, the truth 0 1 1 1 1 1 1 1 1 n table has 2 rows 1 0 0 0 1 0 0 0 1 n Proof by truth table quickly becomes 1 0 1 0 1 1 1 1 1 infeasible 1 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 1 3 4 1 9/9/14 Proof Method #2: Rules of Inference Inference proper'es n A rule of inference is a pre-proved relaon: n Inference rules are truth preserving any 'me the leJ hand side (LHS) is true, the n If the LHS is true, so is the RHS right hand side (RHS) is also true. -
Stoic Propositional Logic: a New Reconstruction
Stoic propositional logic: a new reconstruction David Hitchcock McMaster University [email protected] Stoic propositional logic: a new reconstruction Abstract: I reconstruct Stoic propositional logic, from the ancient testimonies, in a way somewhat different than the 10 reconstructions published before 2002, building especially on the work of Michael Frede (1974) and Suzanne Bobzien (1996, 1999). In the course of reconstructing the system, I draw attention to several of its features that are rarely remarked about, such as its punctuation-free notation, the status of the premisses of an argument as something intermediate between a set and a sequence of propositions, the incorrectness of the almost universal translation of the Greek label for the primitives of the system as indemonstrable arguments, the probable existence of an extended set of primitives which accommodates conjunctions with more than two conjuncts and disjunctions with more than two disjuncts, the basis for the system’s exclusion of redundant premisses, and the reason why the hypothetical syllogisms of Theophrastus are not derivable in the system. I argue that, though sound according to its originator’s (Chrysippus’s) conception of validity, the system as reconstructed is not complete according to that conception. It is an open problem what one needs to add to the system in order to make it Chrysippean-complete, or even whether it is possible to do so without making it Chrysippean-unsound. Key words: Stoicism, logic, history of logic, Stoic logic, Chrysippus, reconstruction, propositional logic, soundness, completeness 0. Introduction In Aristotle’s Earlier Logic (Woods 2001), John Woods finds in Aristotle’s earliest logical writings considerable grist for his ongoing sophisticated defence of classical validity against contemporary relevantist objections. -
Introduction to Logic and Critical Thinking
Introduction to Logic and Critical Thinking Version 1.4 Matthew J. Van Cleave Lansing Community College Introduction to Logic and Critical Thinking by Matthew J. Van Cleave is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. Table of contents Preface Chapter 1: Reconstructing and analyzing arguments 1.1 What is an argument? 1.2 Identifying arguments 1.3 Arguments vs. explanations 1.4 More complex argument structures 1.5 Using your own paraphrases of premises and conclusions to reconstruct arguments in standard form 1.6 Validity 1.7 Soundness 1.8 Deductive vs. inductive arguments 1.9 Arguments with missing premises 1.10 Assuring, guarding, and discounting 1.11 Evaluative language 1.12 Evaluating a real-life argument Chapter 2: Formal methods of evaluating arguments 2.1 What is a formal method of evaluation and why do we need them? 2.2 Propositional logic and the four basic truth functional connectives 2.3 Negation and disjunction 2.4 Using parentheses to translate complex sentences 2.5 “Not both” and “neither nor” 2.6 The truth table test of validity 2.7 Conditionals 2.8 “Unless” 2.9 Material equivalence 2.10 Tautologies, contradictions, and contingent statements 2.11 Proofs and the 8 valid forms of inference 2.12 How to construct proofs 2.13 Short review of propositional logic 2.14 Categorical logic 2.15 The Venn test of validity for immediate categorical inferences 2.16 Universal statements and existential commitment 2.17 Venn validity for categorical syllogisms Chapter 3: Evaluating inductive arguments and probabilistic and statistical fallacies 3.1 Inductive arguments and statistical generalizations 3.2 Inference to the best explanation and the seven explanatory virtues 3.3 Analogical arguments 3.4 Causal arguments 3.5 Probability 3.6 The conjunction fallacy 3.7 The base rate fallacy 3.8 The small numbers fallacy 3.9 Regression to the mean fallacy 3.10 Gambler’s fallacy Chapter 4: Informal fallacies 4.1 Formal vs. -
MGF1121 Course Outline
MGF1121 Course Outline Introduction to Logic............................................................................................................. (3) (P) Description: This course is a study of both the formal and informal nature of human thought. It includes an examination of informal fallacies, sentential symbolic logic and deductive proofs, categorical propositions, syllogistic arguments and sorites. General Education Learning Outcome: The primary General Education Learning Outcome (GELO) for this course is Quantitative Reasoning, which is to understand and apply mathematical concepts and reasoning, and analyze and interpret various types of data. The GELO will be assessed through targeted questions on either the comprehensive final or an outside assignment. Prerequisite: MFG1100, MAT1033, or MAT1034 with a grade of “C” or better, OR the equivalent. Rationale: In order to function effectively and productively in an increasingly complex democratic society, students must be able to think for themselves to make the best possible decisions in the many and varied situations that will confront them. Knowledge of the basic concepts of logical reasoning, as offered in this course, will give students a firm foundation on which to base their decision-making processes. Students wishing to major in computer science, philosophy, mathematics, engineering and most natural sciences are required to have a working knowledge of symbolic logic and its applications. Impact Assessment: Introduction to Logic provides students with critical insight -
Module 3: Proof Techniques
Module 3: Proof Techniques Theme 1: Rule of Inference Let us consider the following example. Example 1: Read the following “obvious” statements: All Greeks are philosophers. Socrates is a Greek. Therefore, Socrates is a philosopher. This conclusion seems to be perfectly correct, and quite obvious to us. However, we cannot justify it rigorously since we do not have any rule of inference. When the chain of implications is more complicated, as in the example below, a formal method of inference is very useful. Example 2: Consider the following hypothesis: 1. It is not sunny this afternoon and it is colder than yesterday. 2. We will go swimming only if it is sunny. 3. If we do not go swimming, then we will take a canoe trip. 4. If we take a canoe trip, then we will be home by sunset. From this hypothesis, we should conclude: We will be home by sunset. We shall come back to it in Example 5. The above conclusions are examples of a syllogisms defined as a “deductive scheme of a formal argument consisting of a major and a minor premise and a conclusion”. Here what encyclope- dia.com is to say about syllogisms: Syllogism, a mode of argument that forms the core of the body of Western logical thought. Aristotle defined syllogistic logic, and his formulations were thought to be the final word in logic; they underwent only minor revisions in the subsequent 2,200 years. We shall now discuss rules of inference for propositional logic. They are listed in Table 1. These rules provide justifications for steps that lead logically to a conclusion from a set of hypotheses.