Making Sense of Logic        Tibor G Molnar     info@brainwaves.com.au     The Philosophy Corner        3rd August, 2011
Contents1.   Where is Logic?                                                                   p.032.   What is Logic?    ...
Where is Logic? (1)In our search for Logic, we must first distinguish between two different notions of “existence”:       ...
Where is Logic? (2)Logic is clearly not any kind of “physical” stuff, so it must reside somewhere within metaphysics:Metap...
Where is Logic? (3)The Conceptual Domain – the internal, “mental” world of thoughts and ideas        Semiotics – the study...
What is Logic? (1)Logic is a consistent/coherent set of axiom-based rules/constraints that define/describe the nature of “...
What is Logic? (2)Indeed, Logic must be universal. Whatever are the rules of Logic, there can only be one coherent,univers...
Elements of Logic (1)Quantifiers   Existential quantifier (semiotic declarative): “some …” (∃)   Universal quantifier: “al...
SRVVLEOQRWQHFHVVDULOQRW
¸
HWFConnectives   Unary Operators (modifying the sense of one declaration/assertion/proposition)       Negation            ...
Joint denial             (NOR)       Conjunction                 (AND), (Λ)                                           Λ   ...
Elements of Logic (2)Statements/Propositions                          [Quantifier] [Subject] [Operator] [Object|Predicate]...
Elements of Logic (3)    Connectives – Material (Philoan1) Implication 2 (p ⊃ q), (p :T
aS V q), (~(p Λ ~q))            Known also as the “material conditional”, material implication is a binary “truth-function...
is false – it cannot be the case that both            (p is the case) and (q is not the case). But that’s the only one tha...
T                      T                   T                                           T                      F           ...
Elements of Logic (4)Connectives – Material Implication (cont.)        For example, take as a “rule” the following implica...
Elements of Logic (5)Connectives – Material Implication (cont.)        For the fourth row, substituting 3 for n gives a ca...
Elements of Logic (6)Connectives – Material Implication (cont.)        However, truth-functionality comes at a price, and ...
Elements of Logic (7)Connectives – Material Implication (cont.)        Moreover, having fish for dinner on a day other tha...
Elements of Logic (8)    Connectives – Material Implication (cont.)              And things just keep getting worse! Given...
WKHQZHKDYHILVKIRUGLQQHU
evaluates as true! And yet, it’s meaningless gibberish… in no sense at all is there anything “true”              about sta...
Elements of Logic (9)    Paraconsistent Logic 4             Nevertheless, logicians are unhappy about these problems, and ...
Elements of Logic (10)    Paraconsistent Logic (cont.)             Relevance Logic 7 8             This is a kind of non-c...
Elements of Logic (11)Connectives – Other Kinds of Implication       Material implication is not the only way to formulate...
Elements of Logic (12)     Predicate/Propositional Calculus               These are the rules of syntax and grammar that d...
Elements of Logic (13)Axioms (cont.)       Axioms of Pure Predicate Calculus – of Description and Classification          ...
Elements of Logic (14)     Axioms (cont.)              The axioms listed so far are all that are required for the developm...
Elements of Logic (15)Categorical Syllogisms – The formal structure of implication and inference    Deduction (or logical ...
Elements of Logic (16)Categorical Syllogisms (cont.):    Deduction (cont.)                        Modus Tollendo Ponens – ...
Elements of Logic (17)Categorical Syllogisms (cont.):    Induction (or ampliative inference):            Minor Premise/Ass...
Elements of Logic (18)Categorical Syllogisms (cont.):    Subsumption (or Logical Entailment) – incorporating something und...
Elements of Logic (19)Categorical Syllogisms (cont.):    Altogether, there are a total of 256 different syllogistic formul...
Elements of Logic (20)Syllogistic Fallacies        Syllogisms may be logically invalid for all sorts of reasons; here are ...
Elements of Logic (21)Syllogistic Fallacies (cont.)                  Fallacy of four terms (fallacy by equivocation):     ...
Elements of Logic (22)Syllogistic Fallacies (cont.)                  Fallacy of Illicit Treatment of Major or Minor Term: ...
Elements of Logic (23)Syllogistic Fallacies (cont.)                  Fallacy of Inferring a Negative Conclusion from Affir...
Elements of Logic (24)Syllogistic Fallacies (cont.)                  Fallacy of Necessity – (i.e., ignoring unspecified al...
“Kinds” of Logic (1)Modal Logic       An extension of classical logic that deals with the various “modes” of truth; viz.: ...
³ER[´                         possibly:                (◊) “diamond”                                                   ◊  ...
impossibly:              (~◊)                                                    ◊       Modal logic allows us to work wit...
“Kinds” of Logic (2)Fuzzy Logic       From Aristotle’s Principles of Non-Contradiction and the Excluded Middle we are led ...
“Kinds” of Logic (3)     Fuzzy Logic (cont.)              In line with these principles, “Boolean” logic 12 admits the pos...
“Kinds” of Logic (4)     Fuzzy Logic (cont.)              Second, consider a computer program designed to recognise 2D geo...
“Kinds” of Logic (5)Fuzzy Logic (cont.)        Now imagine that we enhance the software by adding circles to its repertoir...
“Kinds” of Logic (6)     Fuzzy Logic (cont.)               In any case, this “fuzzy” approach is useful in another respect...
“Kinds” of Logic (7)Paraconsistent Logic        We’ve already spoken about paraconsistent logics; they are logical systems...
“Kinds” of Logic (8)Moral Logic – a.k.a. Ethics        Though some or other “reasoning” is involved, ethics is not really ...
“Kinds” of Logic (9)     Moral Logic – a.k.a. Ethics (cont.)              Deontological Ethics – [Gk: deon = obligation, d...
“Kinds” of Logic (10)Moral Logic – a.k.a. Ethics (cont.)        Consequentialist or Teleological Ethics – [Gk: telos, tele...
“Kinds” of Logic (11)     Moral Logic – a.k.a. Ethics (cont.)              Consequentialist or Teleological Ethics (cont.)...
“Kinds” of Logic (12)     Moral Logic – a.k.a. Ethics (cont.)              Consequentialist or Teleological Ethics (cont.)...
“Kinds” of Logic (13)Moral Logic – a.k.a. Ethics (cont.)        The difference between all these various ethical framework...
“Kinds” of Logic (14)Mathematical Logic       Mathematical logic (also known as “symbolic logic”) is mostly set theory on ...
“Kinds” of Logic (15)Mathematical Logic (cont.)       Mathematical Logic became an independent branch of logic/mathematics...
“Kinds” of Logic (16)Mathematical Logic (cont.)               Friedrich Ludwig Gottlob Frege [1848-1925] – German mathemat...
“Kinds” of Logic (17)Mathematical Logic (cont.)               Alfred Tarski [1901/2-1983]:                   Worked on the...
“Kinds” of Logic (18)The Logic of Scientific Explanation        A Deductive-Nomological (D-N) model of Scientific Explanat...
“Kinds” of Logic (19)The Logic of Scientific Explanation (cont.)        A Deductive-Statistical (D-S) model of Scientific ...
“Kinds” of Logic (20)The Logic of Scientific Explanation (cont.)        Unificationist models of Scientific Explanation   ...
“Kinds” of Logic (21)     Semantic Logic              Semantic logic is the study of the semantics and grammar of formal (...
Ps and Qs – Symbolic Representation (1)If logic is the grammar of “symbolic representation”, then its inputs are not world...
Ps and Qs – Symbolic Representation (2)In any spatio-temporally extended domain (real or imaginary), the things that can b...
Ps and Qs – Symbolic Representation (3)Entities    Entities have up to four kinds of properties:                      exte...
Ps and Qs – Symbolic Representation (4)     The Dilemma of Identity: The Ship of Theseus             Theseus, son of Aegeu...
Ps and Qs – Symbolic Representation (5)     The Dilemma of Identity: The Ship of Theseus (cont.)              But then Hob...
Ps and Qs – Symbolic Representation (6)     Fundamental (elementary) entities:              “Atoma” – physical/phenomenal ...
Ps and Qs – Symbolic Representation (7)     Ontological Commitment (E!)              Some entities exist “physically” – i....
Ps and Qs – Symbolic Representation (8)     Observables vs Beables              In the physical domain, we need to disting...
Ps and Qs – Symbolic Representation (9)     Events              If “entities” are what extant objects are, then “events” a...
Thinking and Reasoning (1)     Schopenhauer’s “Rules of Thought”              Schopenhauer notes that, for logical reasoni...
Thinking and Reasoning (2)Schopenhauer’s Fourfold Root of the Principle of Sufficient Reason        Schopenhauer identifie...
Thinking and Reasoning (3)     Two Kinds of “Reason”               Immanuel Kant [1724-1804] identified two distinct kinds...
Thinking and Reasoning (4)Belief         To assert proposition “p” requires one merely to hold the belief that p is true. ...
Thinking and Reasoning (4)Fact and Truth       “Facts” are actual states-of-affairs in the world – it is how things actual...
Thinking and Reasoning (5)     Fact and Truth (cont.)              The truth of a proposition may be asserted, inter alia,...
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  1. 1. Making Sense of Logic Tibor G Molnar info@brainwaves.com.au The Philosophy Corner 3rd August, 2011
  2. 2. Contents1. Where is Logic? p.032. What is Logic? p.063. Elements of Logic p.08 Quantifiers; Connectives (Operators); Statements/Propositions Material Implication; Predicate/Propositional Calculus Axioms; Categorical Syllogisms; Syllogistic Fallacies; Paraconsistent Logics4. “Kinds” of Logic p.32 Modal; Fuzzy; Paraconsistent; Intuitionistic; Moral (Ethical); Mathematical Logic; The Logic of Scientific Explanation5. Ps and Qs – Symbolic Representation p.53 Entities, identity, events, ontological commitment, observables vs beables6. Thinking and Reasoning p.626. Complexity p.69 Feedback, unpredictability, Mandelbrot Sets © Copyright 2011 Tibor G Molnar All rights reserved 2
  3. 3. Where is Logic? (1)In our search for Logic, we must first distinguish between two different notions of “existence”: Physical existence Non-physical existenceSpatio-temporally extended Not spatio-temporally extended Exists “in Space and Time”Physically active/interactive Not physically active/interactive Causally efficacious Causally inert, “epiphenomenal”Matters of fact – of how the world actually is Matters of thought and imagination – e.g., ideas, imaginings, declarations, assertions, propositions, etc. May be “factual” – i.e., about facts – but not facts in themselvesPhysics – [Gk: phusis = nature] – the study of the structure and Metaphysics – literally “beyond nature” [Gk: meta- = beyond]dynamics of our actual, “physical” environment (universe) – the study of what lies “beyond” our physical environment © Copyright 2011 Tibor G Molnar All rights reserved 3
  4. 4. Where is Logic? (2)Logic is clearly not any kind of “physical” stuff, so it must reside somewhere within metaphysics:Metaphysics – literally “beyond nature” [Gk: meta- = beyond] – the study of what lies “beyond” the physical domain There are various different ways of being “beyond” the physical; for example: Spiritual/Supernatural – other-worldly, but somehow “real” or actual Theological – e.g., gods, miracles Mystical – e.g., spirits, afterlife, ghosts Pseudo-scientific – e.g., time travel, thought transfer, teleportation Non-spiritual/Natural but imaginary – not necessarily other-worldly, but declaredly not “real” Fantasy, Fiction Conceptual – the internal, “mental” world of thoughts and ideasLogic isn’t exactly “supernatural” or “imaginary”, so perhaps it resides somewhere within this conceptualdomain… © Copyright 2011 Tibor G Molnar All rights reserved 4
  5. 5. Where is Logic? (3)The Conceptual Domain – the internal, “mental” world of thoughts and ideas Semiotics – the study of symbolic representation/reference and language Linguistics – the study of language Semantics – the study of “meaning” Reference – what symbols/signs “refer” or point to: – to entities – objects/things – the units of existence – to events – happenings – the units of change – to properties, descriptors, or “kinds” Optional ontological commitment (E!) to the physical existence of entities/events J. Karel Lambert [1928-]: Free Logic and the Concept of Existence (1967) Significance – what “meaning” symbols/signs communicate Truth – correspondence to, or consistency with, some or other (external) “fact” Syntactics – the study of signs/symbols, and the grammar of their manipulation Logic – the rules of syntax and grammar that define/describe the valid manipulation of symbolic representation and reference © Copyright 2011 Tibor G Molnar All rights reserved 5
  6. 6. What is Logic? (1)Logic is a consistent/coherent set of axiom-based rules/constraints that define/describe the nature of “right”thinking – i.e., the rules for correctly manipulating thoughts (about material objects and/orimmaterial/abstract/imaginary ideas), their relationships and their interactions.Logic has nothing to do with the actual “having” of thoughts and ideas – there is nothing about the sheerconceptualisation of an idea that is a candidate for logical scrutiny. Nor is Logic contingent upon existence– e.g., though there are no actual unicorns, there is nothing inherently “illogical” about imagining them.But thinking about what unicorns might be like, and about what Unicorns might do, is somehow more thansimply imagining or inventing them. Thinking/reasoning about unicorns – about what magical powers theymight have and what magical things they might do – is answerable to logic. For example, could a unicorn bein two places at once, or influence the movement/behaviour of remote objects? If it could, then how? And ifnot, why not? Deliberations on questions of this kind are candidates for logical evaluation.So, could a unicorn add two plus two and get five (without “five” being the unicorn’s word for “four”)?No, it could not – at least not if the unicorn was thinking “logically”! The rules of logic define what is“valid” or “correct” – even for unicorns! Breaking the rules of logic renders all thinking invalid –“illogical”, or just plain wrong – regardless of whether it is the thinking of humans, unicorns, orsuperhuman aliens. Even Gods can’t add two plus two and get five! © Copyright 2011 Tibor G Molnar All rights reserved 6
  7. 7. What is Logic? (2)Indeed, Logic must be universal. Whatever are the rules of Logic, there can only be one coherent,universal set of them. For individual things to each have their own logic – each inconsistent with everyother logic – is tantamount to there to be no logic at all!Logic is also essential. Without symbolic representation and the grammar of Logic, there can not only beno Set Theory and no Mathematics, but also no Language and no Thinking!Logic is to the world of ideas what Physics is to the world of nature – the formal description of themechanics of thought; the specification of the rules for the valid manipulation of symbolic representationand reference.Logic works like a “meat-grinder” – it does to statements and propositions what a meat-grinder does totopside steak. First, it is strictly GIGO – the quality of what comes out depends on the quality of what is put in: It is strictly truth-preserving (but not truth-generating): Conformity with Logic assures validity/legitimacy, but not factuality/actuality or truth. Second, it is unidirectional – it doesn’t work when you crank the handle backwards! From “if P then Q” (P ⇒ Q), it does not necessarily follow that “if Q then P” (Q ⇒ P). © Copyright 2011 Tibor G Molnar All rights reserved 7
  8. 8. Elements of Logic (1)Quantifiers Existential quantifier (semiotic declarative): “some …” (∃) Universal quantifier: “all …” (∀) Numerical quantifiers: one, all, none, every, many, few, some, most, etc. Modal quantifiers: necessarily (‘
  9. 9. SRVVLEOQRWQHFHVVDULOQRW
  10. 10. ¸
  11. 11. HWFConnectives Unary Operators (modifying the sense of one declaration/assertion/proposition) Negation (not) (¬) (~) Tautology (T) Contradiction (⊥) ⊥ Binary Operators (joining two declarations/assertions/propositions) Equality (=) Alternative denial (NAND) Inequality (
  12. 12. Joint denial (NOR) Conjunction (AND), (Λ) Λ Biconditional (XNOR) (IFF) (≡) (↔) Disjunction (OR), (V) Membership/Containment (∈) Exclusive disjunction (XOR), (⊕) Implication (⊃), (:) © Copyright 2011 Tibor G Molnar All rights reserved 8
  13. 13. Elements of Logic (2)Statements/Propositions [Quantifier] [Subject] [Operator] [Object|Predicate] If the sense of its operator is static, then the statement/proposition is a declaration/description of the state of some entity/ies or states-of-affairs: i.e., an account of how things are, what things are “like”, and how they are spatially related to each other. e.g.: ∃(2) ∃(4): ((2 + 2) = 4) If the sense of the operators is dynamic: then the statement/proposition is a description of some event: i.e., an account of what things do, what leads to what, what makes what, how entities/states- of-affairs change, and how things are causally connected/related to each other. e.g.: ∃(hydrogen) ∃(oxygen): ((hydrogen + oxygen) : water) © Copyright 2011 Tibor G Molnar All rights reserved 9
  14. 14. Elements of Logic (3) Connectives – Material (Philoan1) Implication 2 (p ⊃ q), (p :T
  15. 15. aS V q), (~(p Λ ~q)) Known also as the “material conditional”, material implication is a binary “truth-function” according to which the argument (p :T) is true whenever (a), the consequent (q) is true; or (b), the antecedent/s (p) is false (or irrelevant). Thus, the material conditional asserts that: If (p is the case) then (q is the case) In other words, the expression “if p then not q” (p :aT
  16. 16. is false – it cannot be the case that both (p is the case) and (q is not the case). But that’s the only one that’s false – the other three combinations of p and q are valid implications, and evaluate as true. Here’s the whole “truth table”: MATERIAL IMPLICATION p q (p :T
  17. 17. T T T T F F F T T F F T1 First suggested by Philo, an ancient philosopher of the Megarian school, in the fourth century BCE.2 Peter Suber (Dept of Philosophy, Earlham College): http://www.earlham.edu/~peters/courses/log/mat-imp.htm (accessed 24/07/2011). © Copyright 2011 Tibor G Molnar All rights reserved 10
  18. 18. Elements of Logic (4)Connectives – Material Implication (cont.) For example, take as a “rule” the following implicative statement (p :T), which is obviously true for any n: (n is a perfect square) : (n is not prime) Now, the first row of our truth table works perfectly: substituting any perfect square, say 4, for n gives a case where p and q are both true, and the rule – the whole implication (p :T) – is also true; viz.: (4 is a perfect square) : (4 is not prime) In fact, it can never be otherwise, and I can offer no example of a value of n for which p is true (n is a perfect square) and q is false (n is prime). Clearly, the second row of the truth table is also correct – the implication (p :aT) breaks the rule: it is always, necessarily false. Substituting 6 for n gives a case where p is false (6 is not a perfect square) and q is true (6 is not prime); viz.: (6 is a perfect square) : (6 is not prime) Nevertheless, the whole implicative statement – the compound conditional (~p : T) – still evaluates as true. Never mind that 6 is not a perfect square, the implication (~p :T) is still valid, because 6 isn’t prime. The rule still stands – if 6 were a perfect square, then it really would not be a prime number – and so the third row of the truth table is also correct. © Copyright 2011 Tibor G Molnar All rights reserved 11
  19. 19. Elements of Logic (5)Connectives – Material Implication (cont.) For the fourth row, substituting 3 for n gives a case where p and q are both false: (3 is a perfect square) : (3 is not prime) Here both the antecedent and consequent are false; and yet, as before, the whole implication – the “rule”, the compound conditional (~p : T) – still evaluates as true: for were 3 a perfect prime, then it certainly would not be a prime number. Truth-functionality As the above example shows, we may safely conclude that the rule of material implication (at least in this Philoan formulation) is strictly “truth-functional”. In other words, in accordance with the truth table associated with the rule of material implication, we may confidently compute the truth value of a material implicative statement from the truth-values of its antecedent/s (protasis) and its consequent (apodosis). Truth-functionality is a handy thing to have, for it means that material implication my be employed as part of a consistent, “truth-preserving” system of logical inference; just the kind of thing we need for automated/computerised deliberative/evaluative processes. © Copyright 2011 Tibor G Molnar All rights reserved 12
  20. 20. Elements of Logic (6)Connectives – Material Implication (cont.) However, truth-functionality comes at a price, and a high price at that! For it turns out that strictly applied, truth-functionality does not ensure “meaning-functionality”. Maintaining truth- functionality requires us to sacrifice what we might call semantic “relevance”. The problem arises in row 3 of the truth table, which states that the rule of material implication evaluates (p :T) as true when p is false. This is fair enough… the rule (p :T) is not broken by instances where p is not the case. So, although the rule of material implication asserts that q follows from p, it does not assert that q can only follow from p – q may also occur when not-p. This makes material implication paradoxical – at least in the ancient sense of the word “paradox”. It some cases it violates our everyday intuitions – indeed, it is often not how we ordinarily understand the conditional “if … then …” in everyday English usage. For example, consider the rule expressed by the following compound conditional “If today is Friday then we have fish for dinner”. In material implicative form, this becomes: MC1: (Today is Friday) :(we have fish for dinner) This is fine as far as it goes… if p is true (i.e., today is Friday), then according to the rule (MC1) of material implication, it follows that q must also be true (i.e., we have fish for dinner). © Copyright 2011 Tibor G Molnar All rights reserved 13
  21. 21. Elements of Logic (7)Connectives – Material Implication (cont.) Moreover, having fish for dinner on a day other than Friday does not render the implication false – having fish on some other night does not violate the rule (MC1) which states that we have fish for dinner on Fridays. When we have fish for dinner on a Tuesday, say, we’re certainly not doing it because it is Friday; but that doesn’t matter… the rule that we have fish on Fridays still stands. In fact, according to the material implication MC1, all conditionals of the form If (any antecedent at all, true or false) then (we have fish for dinner) evaluate as true. The rule that we have fish on Fridays stands regardless of when else, why else, or even why not else, we might have fish for dinner. There is no way for MC1 to be false – for the rule to be broken – other than by failing to have fish for dinner on some Friday. And here’s the rub: since strictly speaking any falsehood is as false as any other, this means that we may substitute any falsehood at all into the above conditional without affecting its overall validity/truth. Indeed, according to rule MC1, even this conditional: If (2+2=5) then (we eat fish for dinner). evaluates as true! Now, maybe this is strictly logically correct, but it is absurd nevertheless! We may feel that MC1 is a fine rule, but evaluating this conditional as true is certainly not fine! © Copyright 2011 Tibor G Molnar All rights reserved 14
  22. 22. Elements of Logic (8) Connectives – Material Implication (cont.) And things just keep getting worse! Given that all contradictions (t Λ ~t) are necessarily false, it follows that according to any rule of material implication, any contradiction necessarily (materially) implies any consequent at all! In other words, for any rule (p :T), any conditional of the form ((t Λ ~t) :q), however absurd, will always evaluate as true! For example, according to our earlier material implication rule MC1, even If ((2+2=4) Λ (2+2
  23. 23. WKHQZHKDYHILVKIRUGLQQHU
  24. 24. evaluates as true! And yet, it’s meaningless gibberish… in no sense at all is there anything “true” about statements like this! The bottom line is that “classical” logic is about validity and not truth – whilst perfectly truth- functional and truth-preserving, it all-too-readily unravels when faced with falsehoods and counterfactuals.3 And this is probably as it should be: classical binary true/false logic isn’t equipped to handle real-world situations in which, though there is only one way for things to be true, there are many ways for them to be false.3 A “counterfactual” is a conditional statement of which the antecedent is contrary to fact; for example, “If I were smarter than I am, I might be able to understand counterfactuals.” © Copyright 2011 Tibor G Molnar All rights reserved 15
  25. 25. Elements of Logic (9) Paraconsistent Logic 4 Nevertheless, logicians are unhappy about these problems, and are searching for ways to render logic “paraconsistent”. Paraconsistent logic is one in which contradictions do not imply any and every consequent – i.e., in which the logical principle that “anything follows from contradictory premises” (ex contradictione quodlibet (ECQ)) does not hold. There are today a number of competing formulations, but a problem-free, truth-functional “paraconsistent” logic has so far proved elusive. Many-Valued Logic Perhaps the simplest way of generating a paraconsistent logic is to devise a “many-valued” logic.5 Classically, there are exactly two truth values – “T” and “not-T” (“F”). The many-valued approach relaxes this constraint, allowing more than two “truth” values. Any number of truth-values is permitted, but the simplest multi-valued logics evaluate formulas in terms of just three: “True”, “False”, and “Both true and false”.64 The term “paraconsistent” was coined by Miró Quesada at the Third Latin America Conference on Mathematical Logic in 1976. The prefix “para” in English has two meanings: ‘quasi’ (as in ‘similar to’) and ‘beyond’. Quesada probably had the first meaning in mind; but many exponents of paraconsistent logic now take it to mean the second. [http://plato.stanford.edu/entries/logic- paraconsistent/ (accessed 30/07/2011)].5 First proposed by the Argentinian logician F. G. Asenjo in his PhD dissertation, published as “A Calculus of Antinomies”, Notre Dame Journal of Formal Logic, Vol. XVI (1966), pp.103-5.6 [http://plato.stanford.edu/entries/logic-paraconsistent/ (accessed 30/07/2011)]. © Copyright 2011 Tibor G Molnar All rights reserved 16
  26. 26. Elements of Logic (10) Paraconsistent Logic (cont.) Relevance Logic 7 8 This is a kind of non-classical logic requiring that the antecedent and consequent of implications be relevantly related. This ensures paraconsistency by declaring ((p Λ ~p) :q) as false/underivable. In other words, to be a candidate for truth evaluation in relevance logic, contradictory antecedents in conditionals must share propositional and/or predicate “letters” with their consequents. Non-Adjunctive Logic 9 A relaxation of the requirement that the adjunct of p and q (p Λ p) must necessarily follow from the separate truths of p and q. Non Truth-Functional Logics A logic is truth-functional (i.e., truth-preserving) if true antecedents necessarily lead only to true consequents. Non truth-functional logic relaxes this requirement, allowing valid arguments from true premises to false conclusions.7 Proposed in 1928 by the Russian philosopher Ivan E. Orlov [1886-1936]; see http://plato.stanford.edu/entries/logic-relevance/.8 Anderson, Alan R. and Belnap Jr., Nuel D.: Entailment: The Logic of Relevance and Necessity, (Princeton University Press, Vol. 1, 1975, Vol. 2 (with J. Michael Dunn), 1993).9 Horacio Arló Costa: Non-Adjunctive Inference and Classical Modalities (Carnegie Mellon University, 2003) © Copyright 2011 Tibor G Molnar All rights reserved 17
  27. 27. Elements of Logic (11)Connectives – Other Kinds of Implication Material implication is not the only way to formulate “implication”. Here are some other ways: Strict Implication This restricts material implication such that (p : T) is true only if p necessarily implies q. In other words, ‘p :T) Strict implication is central to modal logic. Relevant Implication This restricts material implication (p :T) to cases where p is relevant to q. Non-Material (Semantic) Implication, or “entailment” (⇒), (~(p Λ ~q)) ⇒ A subtle variant of material implication, which states that p cannot now be the case without q also now being the case. Converse Implication (q ⊂ p) The implication that, if (q is now the case) then (p must have been the prior case). © Copyright 2011 Tibor G Molnar All rights reserved 18
  28. 28. Elements of Logic (12) Predicate/Propositional Calculus These are the rules of syntax and grammar that describe the proper, truth-functional manipulation of statements/propositions; expressed in terms of (a), a set of definitions/declarations/axioms; plus (b), a set of rules/algorithms/syllogisms that specify the relationships between, and proper truth- functional handling of, these axioms. Axioms – Self-Defining, Conceptually Irreducible, Logical “Simples” 10 Axioms of Semiotics – of Symbolic Representation and Reference Axioms of existence; distinct variables; variable substitution; quantifier introduction and substitution; identity; equality. Many of these axioms have been with us since ancient times; for example: Principle of Affirmation/Negation: Something is either the case, or it is not the case. Aristotle [382-322 BCE]: Categories, sect. 3, part 10 Principle of Non-Contradiction: Something is either the case or not the case, but not both. Aristotle: The Metaphysics, Bk Gamma III, 1005b Principle of the Excluded Middle: Something is either the case or not the case, but not neither. Aristotle: The Metaphysics, Bk Gamma VII, 1011b10 A complete definition/elaboration of these axioms is available at http://us.metamath.org/index.html. © Copyright 2011 Tibor G Molnar All rights reserved 19
  29. 29. Elements of Logic (13)Axioms (cont.) Axioms of Pure Predicate Calculus – of Description and Classification Axioms of quantified implication, quantified negation and quantifier commutation; specialisation and generalisation. Axioms of Zermelo-Fraenkel Set Theory with Choice (ZFC) Axioms of extensionality; replacement; union; regularity; power sets; infinity and choice. Axioms of Propositional Calculus – of Explanation Axioms of simplification, distribution, contraposition and material implication (modus ponens). From these axioms we can derive principles of haecceity, quiddity, and mereological assembly. © Copyright 2011 Tibor G Molnar All rights reserved 20
  30. 30. Elements of Logic (14) Axioms (cont.) The axioms listed so far are all that are required for the development of all of mathematics, and of all semiotic, semantic and logical systems. However, they are all analytic (a priori) axioms, and as such have no purchase on the physical domain. Thus, for reference to and description/explanation of physical phenomena, the following additional (empirical) axioms/principles may be added: Axiom of ontological commitment; of dynamism/change; heterogeneity/difference; and “ex nihilo nihil fit” (nothing comes from nothing). Principles of spatial contiguity, isotropy and uniformity; of temporal continuity and constancy; and of locality – of contiguity/differentiability of action, and interaction by contact. Principles of equipoise, impetus/momentum, and least action.1111 The origins of these principles may be traced back through William Rowan Hamilton [1805-1865], Pierre Louis Moreau de Maupertuis [1698-1759], Isaac Newton [1642-1727] and Jean Buridan (Iohannes Buridanus) [c.1295-c.1360] to Philoponus (John the Grammarian) of Alexandria [c.490-c.570], Aristotle [384-322 BCE], and even Anaximander of Miletus [c.610-c.546 BCE]. Today, all of contemporary physics – including all of quantum mechanics – is still answerable to these elementary principles. © Copyright 2011 Tibor G Molnar All rights reserved 21
  31. 31. Elements of Logic (15)Categorical Syllogisms – The formal structure of implication and inference Deduction (or logical implication/inference): Major Premise/Assumption Quantity (Middle term) are (Predicate) plus Minor Premise/Assumption Quantity (Subject) are (Middle Term) Deduction/Implication/Inference Quantity (Subject) are (Predicate) Modus Ponendo Ponens – [L: Mode which, by affirming, affirms.] (Affirming the Antecedent) General (Major) Premise: Whenever p is true, q is also true. Particular (Minor) Premise: It happens that, in this case, p is true. Deductive Inference/Conclusion: Therefore, in this case, q is also true. Modus Tollendo Tollens – [L: Mode which, by denying, denies.] (Contrapositive Inference, or Denying the Consequent) General Premise: Whenever p is true, q is also true. Particular Premise: It happens that, in this case, q is false. Deductive Inference/Conclusion: Therefore, in this case, p must be false. © Copyright 2011 Tibor G Molnar All rights reserved 22
  32. 32. Elements of Logic (16)Categorical Syllogisms (cont.): Deduction (cont.) Modus Tollendo Ponens – [L: mode which, by denying, affirms.] (Affirmation by Denial – A Disjunctive Syllogism) General Premise: Either p or q is generally true. Particular Premise: It happens that, in this case, p is false. Deductive Inference/Conclusion: Therefore, in this case, q must be true. © Copyright 2011 Tibor G Molnar All rights reserved 23
  33. 33. Elements of Logic (17)Categorical Syllogisms (cont.): Induction (or ampliative inference): Minor Premise/Assumption Quantity (Subject) are (Middle Term) plus Empirical Observation/s Quantity (Subject) are (Predicate) Ampliative Inference All (Middle term) are (Predicate) Ampliative Inference Particular Premise: This is a swan. Empirical Observation: This swan is white. (Empirical Observation: This other swan is also white.) (Empirical Observation: This third swan is also white.) … … Ampliative Inference/Generalisation: All swans are white. © Copyright 2011 Tibor G Molnar All rights reserved 24
  34. 34. Elements of Logic (18)Categorical Syllogisms (cont.): Subsumption (or Logical Entailment) – incorporating something under a more general category: Major Premise/Assumption Quantity (Middle term) are (Predicate) Particular Conclusion Quantity (Subject) are (Predicate) Subsumption/Entailment Minor Premise – Quantity (Subject) are (Middle term) Subsumption Major Premise: Non-existent entities are not observable. Particular Conclusion: This apple is observable. Subsumed/Entailed Minor Premise: Therefore, this apple exists. © Copyright 2011 Tibor G Molnar All rights reserved 25
  35. 35. Elements of Logic (19)Categorical Syllogisms (cont.): Altogether, there are a total of 256 different syllogistic formulations; of which just 19 are valid! The other 237 are not valid; here is one example: Affirming the Consequent General Premise: All parrots have two legs. Particular Premise: Socrates had two legs. Inferred Conclusion: Therefore, Socrates was a parrot. This is rather obviously wrong – it fails to account for the possibility that other things, apart from parrots, might also have two legs. Clearly, having two legs is a necessary condition of being a parrot, but it is not a sufficient condition. (Caution: some other examples are much less obvious.) And, in accordance with the GIGO principle, even perfectly valid syllogisms can sometimes be nonsense: General Premise: All kookaburras have iPods. Particular Premise: Socrates was a kookaburra. Inference/Conclusion: Socrates had an iPod. Alan Saunders (ABC, March 2006) © Copyright 2011 Tibor G Molnar All rights reserved 26
  36. 36. Elements of Logic (20)Syllogistic Fallacies Syllogisms may be logically invalid for all sorts of reasons; here are some more examples: Fallacy of the Undistributed Middle (e.g., affirming the consequent): General Premise: All A are C. Particular Premise: B is C. Inference/Conclusion: Therefore, B is A. (Invalid if not all C are A) Fallacy of Affirming a Disjunct: General Premise: A is B or C. Particular Premise: A is B. Inference/Conclusion: Therefore, A is not C. (Invalid if A is both B and C) © Copyright 2011 Tibor G Molnar All rights reserved 27
  37. 37. Elements of Logic (21)Syllogistic Fallacies (cont.) Fallacy of four terms (fallacy by equivocation): General Premise: Nothing is better than eternal happiness. Particular Premise: One beer is better than nothing. Inference/Conclusion: Therefore, one beer is better than eternal happiness. (Equivocation on meaning of “nothing”.) Fallacy of Exclusive premises (where both premises are negative): Negative Premise 1: No A are B. Negative Premise 2: Some B are not C. Inference/Conclusion: Therefore, some C are not A. Fallacy of Accident – (i.e., applying a general premise to an irrelevant situation): General Premise: Cutting people with knives is a crime. Particular Premise: Surgeons cut people with knives. Inference/Conclusion: Therefore, surgeons are criminals. © Copyright 2011 Tibor G Molnar All rights reserved 28
  38. 38. Elements of Logic (22)Syllogistic Fallacies (cont.) Fallacy of Illicit Treatment of Major or Minor Term: Major Premise: All A are C. Minor Premise: No C are A. Inference/Conclusion: Therefore, no C are B. Major Premise: All A are B. Minor Premise: All A are C. Inference/Conclusion: Therefore, all C are B. Fallacy of Inferring an Affirmative Conclusion from Negative Premises: Negative Premise 1: No A are B. Negative Premise 2: No B are C. Positive Inference/Conclusion: Therefore, all A are C. © Copyright 2011 Tibor G Molnar All rights reserved 29
  39. 39. Elements of Logic (23)Syllogistic Fallacies (cont.) Fallacy of Inferring a Negative Conclusion from Affirmative Premises: General Affirmative Premise 1: All A is B. General Affirmative Premise 2: All B is C. Negative Inference/Conclusion: Therefore, some C is not A. (The above is valid if A ∈ B ∈ C, but invalid if A ≡ B ≡ C). Existential Fallacy – (i.e., where subject and/or middle term is not instantiated): General Universal Premise 1: All aliens are hostile. General Particular Premise 2: All Martians are aliens. Particular Inference: Therefore, some Martians are hostile. (But, are there any Martians? In Boolean logic, universal premises are not required to have members; but particular premises are.) © Copyright 2011 Tibor G Molnar All rights reserved 30
  40. 40. Elements of Logic (24)Syllogistic Fallacies (cont.) Fallacy of Necessity – (i.e., ignoring unspecified alternatives): General Premise: We must reduce car accidents. Particular Premise: Closing roads eliminates car accidents. Inference/Conclusion: Therefore, we must close all roads. (What about other ways to reduce car accidents?). And the list goes on… and on… © Copyright 2011 Tibor G Molnar All rights reserved 31
  41. 41. “Kinds” of Logic (1)Modal Logic An extension of classical logic that deals with the various “modes” of truth; viz.: Necessity and non-necessity (contingency) Possibility and non-possibility (impossibility) To achieve this, modal logic introduces two additional unary operators: necessarily: (‘
  42. 42. ³ER[´ possibly: (◊) “diamond” ◊ and hence: contingently: (~‘
  43. 43. impossibly: (~◊) ◊ Modal logic allows us to work with counterfactuals – of what would be the case if circumstances were somehow as we imagine them. For example, ◊(‘water Λ life) Water is essential for life to be possible ~(Mercury Λ water) There is no water on Mercury ergo ~◊(Mercury Λ life) ◊ Life is impossible on Mercury © Copyright 2011 Tibor G Molnar All rights reserved 32
  44. 44. “Kinds” of Logic (2)Fuzzy Logic From Aristotle’s Principles of Non-Contradiction and the Excluded Middle we are led to conclude that, ultimately, something is either the case (and hence “true”), or it is not (and hence “false”). Between the assertions that “p is the case”, and its logical opposite that “p is not the case”, there is nowhere to insert a third “neither/nor” proposition. As Aristotle wrote in his ‘Metaphysics’, there is no middle ground between two contradictory propositions: “… there can be nothing intermediate to an assertion and a denial. We must assert or deny any single predicate of any single subject. The quickest way to show this is by defining truth and falsity. … [F]alsity is the assertion that that which is, is not; or that which is not, is. And truth is the assertion that that which is, is; and that which is not, is not.” [The Metaphysics, Book Gamma VII, 1011b] [Tr. Hugh Lawson-Tancred (Penguin Classics, 1998)] Thus, according to Aristotle, “truth” asserts a correspondence between a statement and some fact, and “falsity” asserts the absence of such correspondence. And correspondence is either present or it is absent – it cannot be half-present or half-absent – ultimately, things either correspond or they don’t. © Copyright 2011 Tibor G Molnar All rights reserved 33
  45. 45. “Kinds” of Logic (3) Fuzzy Logic (cont.) In line with these principles, “Boolean” logic 12 admits the possibility of two and only two truth values: the semantic content of an assertion (whether predicative or propositional) must be either “true” or “false”. And yet, so-called “fuzzy” logic seems to fly square in the face of this ancient wisdom; asserting instead that, contra Aristotle, the space between truth and falsity is a kind of continuum – that there are valid logical positions between true and false. To fuzzy logicians, truth is a matter of degree, and statements can be, say, 10% true and 90% false at the same time. I want to argue that this fuzziness of “truth” and “falsity” is illusory; and that the notions of multivalent/fractional truths within “fuzzy” logic do not contradict the original Aristotelian bivalency. The name, “fuzzy logic”, is to be interpreted literally: for it is the logic rather than the truth/falsity distinction that is actually fuzzy. I offer two examples: First, if the tossing of a coin yields a head 50% of the time, fuzzy logic describes this as the predicate “yields a head” having a truth-value of one half. However, this does not make the singular predicate ‘half-true’; and we cannot infer from this formulation that it is ever possible to toss a coin and get half a head. Rather, it is just for computational convenience that implicitly plural predicates conflate “being true half the time” to “being half true”.12 Boole, George [1815-1864]: An Investigation on the Laws of Thought (1854). © Copyright 2011 Tibor G Molnar All rights reserved 34
  46. 46. “Kinds” of Logic (4) Fuzzy Logic (cont.) Second, consider a computer program designed to recognise 2D geometric shapes. A digital camera scans an image, and the software determines its shape, perhaps by employing the technique of best-fitting curves.13 Imagine that the software emulates this technique by inscribing and circumscribing certain standard shapes, and then computing a “degree of fit” by measuring the deviation from each superimposed shape – i.e., by comparing the areas of the regions that stick out (or in). Imagine further that this is an early prototype of the software, which uses only two standard shapes – squares and rectangles – and hence can only recognise squares and rectangles. We show it a circle, and the software correctly calculates that it is more like a square than a rectangle. The same software also correctly calculates that an ellipse is more like a rectangle than a square.13 Developed in 1795 by Karl Friedrich Gauss [1777-1855]. © Copyright 2011 Tibor G Molnar All rights reserved 35
  47. 47. “Kinds” of Logic (5)Fuzzy Logic (cont.) Now imagine that we enhance the software by adding circles to its repertoire of standard shapes, thereby enabling it to recognise circles as well. Now when we show it a circle, it correctly identifies the shape as a circle. But, when we show it an ellipse, we get an interesting situation. An ellipse is still more like a rectangle than a square, and it is also more like a circle than a square; but what would it mean if the program described an ellipse as more like a rectangle than a circle? Surprisingly, this answer might be very useful, not least as a graphic description of the ellipticity or degree of elongation of the ellipse under examination: a slightly oblate ellipse might be described as rectangle=5% and circle=95%, whereas a highly elongated ellipse might be described as rectangle=75% and circle=25%. Of course, there is a more precise, mathematical measure of ellipticity (ε): l 2 − h2 ε= l2 (where l = length and h = height), but as an aid to visualising a shape’s ellipticity, the “fuzzy” description may well be preferable over this mathematical one. Of course, an ellipse is ultimately not any kind of blend of circle and rectangle, but under certain circumstances this is a handy “fuzzy logical” metaphor. © Copyright 2011 Tibor G Molnar All rights reserved 36
  48. 48. “Kinds” of Logic (6) Fuzzy Logic (cont.) In any case, this “fuzzy” approach is useful in another respect. Most shapes are at least slightly irregular, so a computer program that could only identify perfectly formed squares, rectangles and circles would most often return the result, “No match found”. Isn’t it more informative to learn that a slightly irregular oblate ellipse is 95% circular and 5% rectangular rather than being told that it is of no identifiable shape? And that is why we have fuzzy logic – not because there are worldly affairs that are partly true and/or partly false, but because it enables us to construct/manipulate logical arguments composed of statements that embody a degree of approximation and/or uncertainty. So don’t be fooled – the existence of fuzzy logic does not mean that there are degrees/shades of truth and falsity. Fuzzy logic is simply probabilistic logic, and supervenes more-or-less simply upon a fundamentally bivalent Boolean logic. Fuzzy logic might tell you that I am 40% tall and 60% short, but that is just another way of saying that my height is precisely 173.4 centimetres! Based on a clever mathematical device that allows us to replace the predicate “is true half the time” with “is half true”,14 fuzzy logic is a logic with fractional truth values that enables the application of statistical/probabilistic techniques to the simultaneous evaluation of multiple approximate and/or imprecise propositions, each of which, if suitably precisely defined, is still simply either the case, or is not the case.14 The formal validity and truth-functionality of this substitution was developed by Dr Bart Kosko et al. in the 1970s-80s. © Copyright 2011 Tibor G Molnar All rights reserved 37
  49. 49. “Kinds” of Logic (7)Paraconsistent Logic We’ve already spoken about paraconsistent logics; they are logical systems devised to resolve the problem with material implication: i.e., that a contradiction implies any and everything consequent.Intuitionistic Logic Contrary to mathematical Platonism (the view that mathematical entities exist per se and per re, in their own right) and the then-prevailing formalism of David Hilbert [1862-1943], mathematical intuitionism holds that mathematical facts/truths are not discovered, but invented – i.e., that the subject-matter of mathematics is the constructions of mathematicians. Perhaps simplistically, its principal departure from classical logic is the denial of the axiomatic status of Aristotle’s Principle of the Excluded Middle. In other words, for the intuitionist, this principle is not a self-evident truth that simply must be just so, but rather a contingent (and negotiable) mathematical concept dreamt up by mathematicians. “Intuitionistic logic” was developed initially by the Dutch mathematician and logician, Arend Heyting [1898-1980], a student of the Dutch mathematician, Luitzen Egbertus Jan (Bertus) Brouwer [1881-1966]. I don’t believe it for a moment – for while I accept that mathematics is a man-made language like any other, its subject matter is not as arbitrary as intuitionists would have us believe. © Copyright 2011 Tibor G Molnar All rights reserved 38
  50. 50. “Kinds” of Logic (8)Moral Logic – a.k.a. Ethics Though some or other “reasoning” is involved, ethics is not really any kind of “logic” as such; rather, it is a calculus of moral values – a set of quasi-logically related guidelines for the assessment of good and evil, the evaluation of the rightness and moral worthiness of actions, and the resolution of tensions between conflicting/competing moral principles. There are several approaches to ethics, of which the most common are: Ethical Intuitionism Now largely out of favour, this is the view that ethical propositions are objectively true/false, and are known intuitively, independent of evaluative judgement. For example, George Edward (G.E.) Moore [1873-1958] held that the “good” things in life are certain (ineffable) wholes – consisting of aesthetics and/or love – and that “goodness” is a simple, unanalysable quality which, fortunately, just happens to be known to us by intuition. Aretaic Ethics – [Gk: arete = excellence, virtue] A virtue-based ethical framework dating back to the ancient Greeks, which assesses moral right and worth primarily in the context of what a particular action/behaviour might say about the virtue or excellence of character of the agent. For example, Aristotle sought to describe what characteristics a virtuous person would have, and then argued that people should act in accordance with those characteristics. © Copyright 2011 Tibor G Molnar All rights reserved 39
  51. 51. “Kinds” of Logic (9) Moral Logic – a.k.a. Ethics (cont.) Deontological Ethics – [Gk: deon = obligation, duty + -logia = reason] A duty- or rule-based ethical framework which judges the moral right or worth of an action based on the character of the behaviour itself, and its adherence to some or other rule/s. Immanuel Kant [1724-1804], for example, argued that the only absolutely good thing is a good Will, and so the single determining factor of whether an action is morally right is the will, or motive of the person doing it. The Scottish philosopher, William David Ross [1877-1971], listed seven right making features of moral action:15 * Duty of beneficence: to help other people (increase pleasure, improve character) * Duty of non-maleficence: to avoid harming other people. * Duty of justice: to ensure people get what they deserve. * Duty of self-improvement: to improve ourselves. * Duty of reparation: to recompense someone if we have acted wrongly towards them. * Duty of gratitude: to benefit people who have benefited us. * Duty of promise-keeping: to act according to explicit and implicit promises, including the implicit promise to tell the truth.15 Ross, William D.: The Right and the Good (Oxford: Clarendon Press, 1930), pp.26-7 © Copyright 2011 Tibor G Molnar All rights reserved 40
  52. 52. “Kinds” of Logic (10)Moral Logic – a.k.a. Ethics (cont.) Consequentialist or Teleological Ethics – [Gk: telos, teleos = end] A consequence- or outcome-based ethical framework according to which the rightness of an action is determined by the goodness of its consequences. There are several variants of consequentialism, each emphasising different kinds of consequences. Act vs Rule Utilitarianism Act (or case) utilitarianism states that, when faced with a choice, we must first consider the likely consequences of our potential actions; and, from that, choose to do whichever we believe will generate the most “pleasure”. Rule utilitarianism, by contrast, begins by looking at potential rules of action. To determine whether a rule should be followed, we must look at what would happen if it were constantly followed. If adherence to the rule produces more happiness than otherwise, it is a rule that morally must be followed at all times. The distinction between act and rule utilitarianism is therefore based on a difference over the proper object of consequential calculation – specific to single cases or generalised to rules. © Copyright 2011 Tibor G Molnar All rights reserved 41
  53. 53. “Kinds” of Logic (11) Moral Logic – a.k.a. Ethics (cont.) Consequentialist or Teleological Ethics (cont.) Hedonistic utilitarianism Hedonistic Utilitarianism holds that a good action is one that results in an increase in pleasure, and the best action is one that results in the most pleasure for the greatest number. What matters most for this form of utilitarianism is the aggregate happiness – the happiness of everyone – and not the happiness of any particular person. Jeremy Bentham [1748–1832] argued that the right act or policy was that which would cause the greatest good for the greatest number of people”. He also suggested a procedure for estimating the moral status of any action – a classification of 12 pains, 14 pleasures and a “felicific calculus” by which we might test the “happiness factor” of any action.16 Bentham’s student, John Stuart Mill [1748–1832], also proposed a hierarchy of pleasures, implying that the pursuit of certain kinds of pleasure is to be more highly valued than the pursuit of other pleasures.1716 Bentham, Jeremy: The Principles of Morals and Legislation (1789), Ch IV.17 Mill, John Stuart: Utilitarianism (Oxford: Oxford University Press 1998) © Copyright 2011 Tibor G Molnar All rights reserved 42
  54. 54. “Kinds” of Logic (12) Moral Logic – a.k.a. Ethics (cont.) Consequentialist or Teleological Ethics (cont.) Eudaimonic consequentialism has as its ultimate aim a full, flourishing life (which may or may not be the same as enjoying a great deal of pleasure). Aesthetic consequentialism holds that the ultimate aim is to produce beauty. Negative utilitarianism Most utilitarian theories advocate producing the greatest amount of good for the greatest number of people. By contrast, negative utilitarianism requires us to promote the least amount of evil or harm, or to prevent the greatest amount of suffering for the greatest number. Karl Popper [1902-1994] argued that this is a more effective ethical formula, since, they contend, the greatest harms are more consequential than the greatest goods.18 Pragmatism Pragmatic ethics holds that moral behaviour evolves socially over the course of many lifetimes (similar to scientific knowledge) – such that any currently explicated moral criterion is liable to be supplanted by some improved version discovered by future generations.18 Popper, Karl R.: The Open Society and its Enemies (London, 1945) © Copyright 2011 Tibor G Molnar All rights reserved 43
  55. 55. “Kinds” of Logic (13)Moral Logic – a.k.a. Ethics (cont.) The difference between all these various ethical frameworks lies more in their approach to moral dilemmas than in the actual moral conclusions reached. For example, a consequentialist may argue that lying is wrong because of its negative consequences – though a consequentialist may allow that certain foreseeable consequences might make lying acceptable. A deontologist might argue that lying is always wrong, regardless of any potential “good” that might occasionally come from it. A virtue ethicist would focus less on lying in any particular instance and instead consider what a decision to tell a lie or not tell a lie said about one’s character and moral standing. And an ethical pragmatist would say that, because moral progress has seemed to converge for a variety of reasons (some of which may be beyond our current awareness) on a view against lying, it is currently reasonable to treat lying as wrong. © Copyright 2011 Tibor G Molnar All rights reserved 44
  56. 56. “Kinds” of Logic (14)Mathematical Logic Mathematical logic (also known as “symbolic logic”) is mostly set theory on steroids. More formally, it’s a branch of logic/mathematics concerned primarily with the foundations of mathematics (the development of axiomatic frameworks for geometry, arithmetic, and computability) and philosophical logic. The project started out as an attempt to place logic and mathematics on a common set-theoretic foundation; and now includes both the study of the application of formal logic to mathematics and the mathematical study of logic. The unifying themes include the study of the “expressive power” of formal systems and the “deductive power” of formal proof systems. Mathematical logic is sometimes divided into sub-disciplines – set theory, model theory, recursion theory and proof theory – but these are not at all distinct. Rather, they are each different elaborations of aspects of the same basic results in logic – particularly first-order logic and definability. © Copyright 2011 Tibor G Molnar All rights reserved 45
  57. 57. “Kinds” of Logic (15)Mathematical Logic (cont.) Mathematical Logic became an independent branch of logic/mathematics in the 19th century. The main players were: George Boole [1815-1864]: An Investigation of the Laws of Thought (1854) Developed one of the first mathematical descriptions of logic. Developed so-called Boolean algebra – the algebra of binary truth values – which today forms the basis of all digital computing. Georg Cantor [1845-1918] – German mathematician and founder of Set Theory. Among many other achievements, discovered/defined infinite sets of different cardinality Cantor’s Theorems (1891): “The set of real numbers is non-denumerable (i.e., of cardinality greater than the natural numbers)”. “The power set P(S) (set of all subsets) of any set (S) is always greater than the set itself.” Cantor’s paradox: “The power set (S*) of the set of all sets (S) is simultaneously both greater than, and a subset of, the set (S).” Thus the set of all sets (S) cannot itself be a set-theoretic object. © Copyright 2011 Tibor G Molnar All rights reserved 46
  58. 58. “Kinds” of Logic (16)Mathematical Logic (cont.) Friedrich Ludwig Gottlob Frege [1848-1925] – German mathematician, logician and philosopher. In Begriffsschrift (1879), he formalised the foundation of modern logic and showed that mathematics grows out of logic. David Hilbert [1862-1943] – German mathematician and philosopher. Bertrand Arthur William Russell [1872-1970] Russell’s paradox – “the set of things x that are such that x is not a member of x” Rudolf Carnap [1891-1970] – German logical positivist Clarified the structure of mathematical and scientific language Developed “confirmation theory” (1950) – a formal way of evaluating the extent to which evidence supports a theory. This was later shown to be unviable; by Goodman and Hempel Nelson Goodman [1906-] – asked, among other things, “How do we select our preferred uniformities of Nature?” Carl Gustav Hempel [1905-1997] – pondered the question of symbolic reference – i.e., “how does something that is neither A nor B confirm that all As are B?” © Copyright 2011 Tibor G Molnar All rights reserved 47
  59. 59. “Kinds” of Logic (17)Mathematical Logic (cont.) Alfred Tarski [1901/2-1983]: Worked on the definition of “Truth” in formalised languages; on the mathematical theory of modelling; and on decidable/undecidable axiomatic systems Kurt Gödel [1906-1978] – Theorems of Undecidability and Incompleteness (1931) 1) Undecidability: In any sufficiently rich formal system, there are always true theorems that cannot be proven from within that system. 2) Incompleteness: In any sufficiently rich formal system in which decidability of all questions is required, there will always be contradictory statements. A formal system cannot be both consistent and complete. Alan Turing [1912-1954] – the father of modern computing Developed theorems of effective computation; defined a computable function as one that, for any given x, Turing machine halts on f(x) Alonso Church [1903-] – developed the thesis/intuition that every computable function is “general recursive”. This is still to be mathematically defined, but no one has yet found an exception. © Copyright 2011 Tibor G Molnar All rights reserved 48
  60. 60. “Kinds” of Logic (18)The Logic of Scientific Explanation A Deductive-Nomological (D-N) model of Scientific Explanation Scientific Explanation via deduction from deterministic laws: the explanandum must be a logical consequence of the explanans, and the sentences constituting the explanans must be true. The explanation must be Deductive: i.e, the explanation should take the form of a sound deductive argument in which the explanandum follows as a conclusion from the premises in the explanans. And the explanation must be Nomological (≈ lawful, in accordance some law): i.e., the explanans must contain as an essential premise at least one “law of nature”. But what is a “Law of Nature”? At a minimum, a Law of Nature is an “exceptionless generalisation”, not merely accidentally true But is this adequate as a definition of Natural laws? See Aspects of Scientific Explanation and Other Essays in the Philosophy of Science By Carl Hempel (New York: Free Press, 1965) © Copyright 2011 Tibor G Molnar All rights reserved 49
  61. 61. “Kinds” of Logic (19)The Logic of Scientific Explanation (cont.) A Deductive-Statistical (D-S) model of Scientific Explanation Scientific Explanation via deduction from statistical laws – the deduction of “a narrower statistical uniformity” from a more general set of premises, at least one of which involves a more general statistical law. A Inductive-Statistical (I-S) model of Scientific Explanation Scientific Explanation involving the subsumption of individual events under more general statistical/probabilistic laws. Motivated by the idea that explanation by appeal to statistical/probabilistic laws is inductive, not deductive. Karl Raimund Popper [1902-1994]: Discomfirmation theory: Verification is never assured, but falsification takes just one counter-example. A Statistical Relevance (S-R) model of Scientific Explanation Championed by Wesley C. Salmon [1925-2001] © Copyright 2011 Tibor G Molnar All rights reserved 50
  62. 62. “Kinds” of Logic (20)The Logic of Scientific Explanation (cont.) Unificationist models of Scientific Explanation Championed by Michael Friedman and Philip Kitcher A Causal Mechanical (C-M) model of Scientific Explanation The idea that Scientific Explanation consists in the elaboration of the causal mechanisms and processes underlying worldly affairs. In other words, to qualify as a scientific explanation is to make a material contribution towards answering this question: “What must the world be like, that it produce the phenomena we observe?” Perhaps surprisingly, this view of scientific explanation is by far not the most common; possibly because it is too demanding. Indeed, critics argue that we construct scientific explanations despite the fact that we cannot, in principle, glean what the world is really like – that the actual causal mechanisms underlying natural processes cannot be determined with any real degree of confidence. Though I understand their concerns, I still uphold this as my preferred model. By definition, anything less is mere story-telling. © Copyright 2011 Tibor G Molnar All rights reserved 51
  63. 63. “Kinds” of Logic (21) Semantic Logic Semantic logic is the study of the semantics and grammar of formal (and natural) languages. A formal “language” can be defined apart from any interpretation of it. This is done by designating a set of symbols (an “alphabet”) and a set of formation rules (a “formal grammar”) that determine which strings of symbols are well-formed formulas (WFFs). When transformation rules (“rules of inference”) are added and certain sentences are accepted as axioms, a logical system is formed. An interpretation of a formal language is (roughly) an assignment of meanings to its symbols and truth- conditions to its sentences.19 One example is “X-bar theory” – a component of linguistic theory which attempts to identify syntactic features presumably common to certain human languages. First proposed by Noam Chomsky20 and further developed by Ray Jackendoff,21 it claims that among their phrasal categories, those languages share certain structural similarities.19 The Cambridge Dictionary of Philosophy: Formal semantics20 Chomsky, N.: “Remarks on Mominalization”, in Reading in English Transformational Grammar, R. Jacobs and P. Rosenbaum (eds.), (Waltham: Ginn, 1970) pp.184-221.21 Jackendoff, R.: “X-bar-Syntax: A Study of Phrase Structure” Linguistic Inquiry Monograph 2 (Cambridge, MA: MIT Press, 1977) © Copyright 2011 Tibor G Molnar All rights reserved 52
  64. 64. Ps and Qs – Symbolic Representation (1)If logic is the grammar of “symbolic representation”, then its inputs are not worldly objects, eventsor states-of-affairs, but the symbols/statements/declarations/propositions that represent them.(There are logical symbols, true statements and valid propositions; but not logical planets, truepotatoes, or valid Wednesdays.)Logic applies only to those symbols/statements/propositions that represent something – that areabout something. For example, the following symbols each represent a heart, though they each sayvery different things about it: HEART (Literal) (Figurative) (Representative) (Schematic)Symbols/statements/propositions that do not represent anything – i.e., that have no semanticcontent or “meaning” – are not candidates for logical analysis. © Copyright 2011 Tibor G Molnar All rights reserved 53
  65. 65. Ps and Qs – Symbolic Representation (2)In any spatio-temporally extended domain (real or imaginary), the things that can be represented –about which meaningful things can be said – fall into two basic categories: Entities – the things that things “are” – objects, things, states-of-affairs. Events – the things that things “do” – changes, happenings, occurrences.In language, we use nouns to refer to the things that things “are”, and adjectives to describe howthey are. Similarly, we use verbs and gerunds to refer to the things that things “do”, and adverbs todescribe how they do them.For those who don’t remember, a gerund is “a noun made out of a verb” – a word that names anevent in just the same way as a noun names an object/thing. But events are not objects; and whilstgerunds are permissible/useful in language, treating events as objects (other than purelylinguistically) is an error of reification – literally, of “making real”. Strictly speaking, “rainbow”and “wave” are gerunds – names for events/happenings – in the sense that they are not things inthemselves, but things that other things do: e.g., diffract light and oscillate, respectively.Errors of reification are as abundant in Physics (e.g., electromagnetic and gravitational fields) asthey are in Philosophy (e.g., justice, beauty, love, and consciousness).Entities and events are manipulated within Logic via the rules of Predicate and Propositional“Calculus”; which together define/constitute the structure of description, inference andexplanation. © Copyright 2011 Tibor G Molnar All rights reserved 54
  66. 66. Ps and Qs – Symbolic Representation (3)Entities Entities have up to four kinds of properties: external: what they are “like”, e.g., shape, size and colour; internal: what they are made of, e.g., composition and structure; dynamic: how they change; and relational/interactive: how they relate to/interact with other things. Though, strictly speaking, relational/interactive properties are not properties of entities, but properties of the relationship/interaction between entities. Entities also have “identity”; as defined/underwritten by their: Endurance – spatio-temporal continuity or “one-and-the-sameness”. Quiddity – intrinsic nature, “whatness” – identity through enduring sameness of appearance or function – i.e., that which makes a thing the thing that it is. [L.: quidditas = “whatness”; from quid = what] Haecceity – particularity, “thisness” – identity through enduring sameness of constituent parts; i.e., why other things can never be this thing. [L.: haecceitas = “thisness”; from hic, haec = this] © Copyright 2011 Tibor G Molnar All rights reserved 55
  67. 67. Ps and Qs – Symbolic Representation (4) The Dilemma of Identity: The Ship of Theseus Theseus, son of Aegeus and the second big hero of the Greeks after Hercules, is the mythical youth who, in about the 13th century BCE, is said to have sailed to Crete and slain the Minotaur. As Plutarch 22 tells the story, the ship on which Theseus returned from Crete was placed on exhibit in the Athenian harbour, and preserved by the Athenians for many centuries. To keep the ship in good repair, the Athenians replaced each plank in the original ship as it decayed; until, eventually, there was not a single original plank left in the ship. Thus arose a dilemma concerning the ship’s identity: was the ship now on display in Athens the same ship as that on which Theseus sailed to Crete? To elucidate this dilemma of identity, the 17th-century Scottish philosopher, Thomas Hobbes, presented a variant of Plutarch’s story.23 In his version, Theseus sails homeward across the Mediterranean in his wooden galleon, carrying a cargo of timber. While still at sea, his ship develops a leak. He finds a faulty plank, so he quickly removes it and replaces it with a new plank from the hold. A short time later his ship springs another leak, so he replaces that plank too. And so on, until after a time, he has replaced every single plank in his ship. So far, Plutarch’s original dilemma is unchanged: after every plank had been replaced, is the ship on which Theseus returns to Athens the same ship as the one on which he set out from Athens?22 Plutarch [c. 45-120 CE]: Vita Thesei (75 CE) tr. John Dryden, pp.22-23 (available at the Internet Classics Archive: http://classics.mit.edu/Plutarch/theseus.html).23 Thomas Hobbes [1588-1679]: “Of Identity and Difference”, in De Corpore (1655) Pt 2, Ch. 11. © Copyright 2011 Tibor G Molnar All rights reserved 56
  68. 68. Ps and Qs – Symbolic Representation (5) The Dilemma of Identity: The Ship of Theseus (cont.) But then Hobbes goes on to describe how the problem with Theseus’s original ship was not with the planks themselves, but with the manner of their assembly; and how the ship’s engineer later took all the planks that had been removed from Theseus’s ship and carefully reassembled them, thereby reconstructing the original ship. And now there arises a more intriguing dilemma: which ship, if either, is the Ship of Theseus? Is it the ship with Theseus still standing at the helm, or is it the ship that had been reassembled from the planks removed from the original ‘Ship of Theseus’? As with most philosophical dilemmas, the answer is, “It depends…” – it depends on how one perceives the notion of “identity”. The Ship of Theseus is the repaired ship if we grant primacy of identity to its quiddity (structure and/or function), and it is the reconstructed ship if we grant primacy of identity to its haecceity (constitution). This dilemma is not resolved by Leibniz’s Law – the “Identity of Indiscernibles”24 – or even by its converse, the “Indiscernibility of Identicals”. Leibniz’s Law does not clearly state what it means for two things to be “identical”. Quiddically identical objects (e.g., perfect clones with every property identical) may be discrete/non-identical in (a), their spatio-temporal location; and (b), their relational properties; neither of which, strictly speaking, are properties of the objects themselves. Leibniz conflates quiddical and haecceitical identity by overlooking and/or misattributing these “indirect” properties.24 Leibniz, Gottfried Wilhelm [1646-1716]: Discourse on Metaphysics (1685), Section 9: “If two objects have all of their properties in common, then they are identical.”, or [∀F (F(a)↔F(b)) → (a=b)] © Copyright 2011 Tibor G Molnar All rights reserved 57
  69. 69. Ps and Qs – Symbolic Representation (6) Fundamental (elementary) entities: “Atoma” – physical/phenomenal “simples” 25 “Monads” – metaphysical “simples” 26 Today, our physical “atoma” are the subatomic particles of the Standard Model, or perhaps quantum bits, but the principle is the same. Macroscopic Agglomerations (Collectives, Aggregates) of Entities Mereology – the study of the relationship of part to whole Mereological reductionism – big things are made of little things Emergence/supervenience – little things combine to make big things Inheritability of properties – e.g., if bricks are made of clay, then brick houses are made of clay25 Democritus of Abdera [c.460 – c.370 BCE]26 Leibniz, Gottfried Wilhelm [1646-1716]: The Monadology (1714) tr. George MacDonald Ross (1999) © Copyright 2011 Tibor G Molnar All rights reserved 58
  70. 70. Ps and Qs – Symbolic Representation (7) Ontological Commitment (E!) Some entities exist “physically” – i.e., “when you stop thinking about them, they don’t go away”:27 e.g., reindeer Other entities do not exist “physically” – i.e., there would be no evidence for them had someone not dreamt them up: e.g., unicorn But the physical (ontological) status of some entities is less obvious: Do rainbows, triangles, numbers, Santa Claus – or even thoughts – exist “physically”?27 Philip K. Dick [1928-1982], American novelist, writing in 1972. © Copyright 2011 Tibor G Molnar All rights reserved 59
  71. 71. Ps and Qs – Symbolic Representation (8) Observables vs Beables In the physical domain, we need to distinguish between “observables” and “beables”:28 Observables – things (phenomena) that can be observed. Beables – things (objects) that can actually, physically, be. Perhaps remarkably, however, it is in the very nature of what it means to “make an observation”, that observables cannot be beables, and that beables cannot be observable! If beables are those things that actually (physically) exist; then only beables can be causally efficacious – i.e., only beables can actually do things, or cause things to happen. Even more remarkably, it follows from the above that all observables are necessarily epiphenomenal!28 Bell, John Stewart [1928-1990]: The Theory of Local Beables (Geneva: CERN, 1975), pp.1-2. © Copyright 2011 Tibor G Molnar All rights reserved 60
  72. 72. Ps and Qs – Symbolic Representation (9) Events If “entities” are what extant objects are, then “events” are “happenings” – the things that those extant objects do. The Principle of “Sufficient Reason” 29 – everything that happens does so for some reason/cause – declares that no state of affairs can obtain, and no statement can be true, unless there is sufficient reason why it should not be otherwise. In other words, “[…] nothing ever comes to pass without there being a cause or at least a reason determining it, that is, something to give an a priori reason why it is existent rather than non-existent, and in this wise rather than in any other.” 30 It is these reasons and causes that are candidates for expression in logical terms.29 Generally attributed to Gottfried Leibniz, but he was almost certainly not its originator. References to such a principle can be found in St Thomas Aquinas [c.1225-1274], and can be traced all the way back to Anaximander of Miletus [c.610 – c.547/6 BCE]30 Leibniz, Gottfried: Essays on the Justice of God and the Freedom of Man in the Origin of Evil, Part 1, Para. 44, p.147. © Copyright 2011 Tibor G Molnar All rights reserved 61
  73. 73. Thinking and Reasoning (1) Schopenhauer’s “Rules of Thought” Schopenhauer notes that, for logical reasoning to make sense – indeed to be possible at all – the following four principles must hold: Aristotle’s Principle of “Identity”: Every event, entity, and thing has particular attributes that make it the thing that itis Aristotle’s Principle of “Non-Contradiction”: Something may be either the case or not the case, but not both Aristotle’s Principle of the “Excluded Middle”: Something may be either the case or not the case, but not neither Leibniz’s Principle of “Sufficient Reason”:31 Every event has a necessary and sufficient cause31 Arthur Schopenhauer [1788-1860]: On the Fourfold Root of the Principle of Sufficient Reason (1813, tr. 1888) © Copyright 2011 Tibor G Molnar All rights reserved 62
  74. 74. Thinking and Reasoning (2)Schopenhauer’s Fourfold Root of the Principle of Sufficient Reason Schopenhauer identified four aspects/elements of sufficient reason/cause: Principle of Sufficient Reason of “Becoming”: Every state-of-affairs follows regularly from its preceding state. Principle of Sufficient Reason of “Knowing”: Knowledge must derive from sufficient “external” grounds – i.e., truth is the reference of a judgment to something outside itself. Principle of Sufficient Reason of “Being”: Every spatio-temporal location/motion is conditioned by prior spatio-temporal location/motion/interaction. Principle of Sufficient Reason of “Acting”: Every human decision is determined by some motive. If all of these principles hold, then on Schopenhauer’s definition of reason and case, the principle of Universal Causal Determinism necessarily holds. © Copyright 2011 Tibor G Molnar All rights reserved 63
  75. 75. Thinking and Reasoning (3) Two Kinds of “Reason” Immanuel Kant [1724-1804] identified two distinct kinds of “reason”: “Pure Reason” – Logical reasoning – thinking/“reasoning” things through.32 “Practical Reason” – Emotional reasoning – being “reasonable”.33 Simon Blackburn 34 describes “meaning” in similar terms: “the meaning of a signifier, word or statement is the set of thoughts and/or emotions conjured up by its use.” And so it is… things are meaningful to us (in the sense of “intelligible”) to the extent that we can apprehend their semantic content, and meaningful (in the sense of “imbued with import or value”) to the extent that we are emotionally engaged with them.32 Kant, I.: The Critique of Pure Reason (1787), tr. J. M. D. Meiklejohn (Project Gutenberg, 2007)33 Kant, I: The Critique of Practical Reason (1788), tr.T. K. Abbott (Project Gutenberg, 2004)34 Blackburn, S.: Oxford Dictionary of Philosophy, (OUP, 1996) © Copyright 2011 Tibor G Molnar All rights reserved 64
  76. 76. Thinking and Reasoning (4)Belief To assert proposition “p” requires one merely to hold the belief that p is true. To assert, somewhat more strongly, that “proposition p is true” requires one also to have evidence or grounds. On this interpretation, we may define “belief” as the acceptance of a proposition (of its “truth”), without regard for evidence or grounds. “Belief” qua ‘acceptance’, however, is not to be confused with “belief” qua ‘expectation’, which is a (probabilistic) projection/inference from (possibly incomplete) evidence. “Acceptance” is a true/false binary state – either we accept something or we don’t – whereas “expectation” is a probabilistic continuum – we may expect one thing more strongly than another. Strictly speaking, expectations are inferences – they are not, and do not involve, any act of ‘acceptance’ per se. For clarity, therefore, we might consider disambiguating these two senses of “belief” by referring only to acceptances as “beliefs”, and referring to expectations as “expectations”. On this interpretation, the forming and/or holding of beliefs does not fall within the domain of logic. © Copyright 2011 Tibor G Molnar All rights reserved 65
  77. 77. Thinking and Reasoning (4)Fact and Truth “Facts” are actual states-of-affairs in the world – it is how things actually are. Facts are the external referents that make propositions “true”. Facts are never true or false, they are just the facts. There can be no false facts – if something is not the case, then it is simply not a fact. For example, there may be fake apples, but there is no such thing as a false apple. “Truth”, then, is the quality predicated not of facts or states-of-affairs, but of statements/propositions whose semantic content (meaning) wholly correspond to the relevant fact(s). A “truth” is not, itself, a fact; rather it is a factual statement – a true statement about some fact. A “truth” is an epistemically determined (knowledge-bound) statement of what is claimed/accepted as proven from the available evidence. “Truth” is thus a semantic consideration, not a logical one. To assert that a proposition is true is to make some kind of existence claim for its semantic content. A “truth” is thus a proposition that asserts a fact. Our assertions can be in error; they are, at best, factual claims. With very few exceptions, our knowledge of the world is empirical, and our truth claims are contingent upon the accuracy of our experience. © Copyright 2011 Tibor G Molnar All rights reserved 66
  78. 78. Thinking and Reasoning (5) Fact and Truth (cont.) The truth of a proposition may be asserted, inter alia, on one or more of the following grounds: o correspondence o coherence o declaration o revelation o pragmatism o axiom o logical entailment necessity (synthetic a priori) o tautology From the Principle of the Excluded Middle, it follows that there are no coherent trivalent, or multivalent, systems of truth: Truth and falsehood belong to a two-valued logic system. Whatever interpretations may be found for the so-called truth-values in multi-valued logics such as those developed by Tarski and Lucasiewicz, it can hardly be considered that they are measurements of truth – what is therein defined as negation, though the definition be entirely legitimate with respect to the relevant system, will not correspond to anything that would ordinarily be understood by the term. 3535 Ayer, Alfred J. [1910-1989]: “Negation”, in Journal of Philosophy, Vol. 49 (1952), No. 26. © Copyright 2011 Tibor G Molnar All rights reserved 67

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