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Philosophy of Science: Introduction to Logic

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- 1. AN INTRODUCTIONTO PHILOSOPHY OF SCIENCE PART 1 John Ostrowick john@ostrowick.com
- 2. INTRODUCTION • What is PhilSci? • Is science ﬁnding truth, or is it just a pragmatic approach to coping with the world? • Is there a distinction between science and non-science? • How or why does science achieve its success? • Is science a force for “good”? Does it have political underpinnings?
- 3. INTRODUCTION • Examples of application • Deciding whether a hypothesis is scientiﬁc • Theism question • Skepticism and “woo” • Evidence-based interventions • Political decisions (more later)
- 4. INTRODUCTION • What is a theory? • A theory, for our purposes, will not mean a “best guess”. It will mean a proposition (series of statements, or statement), which has a rigorous mathematical basis for its statement(s). It is derived from testing a hypothesis; that is, a theory is a hypothesis which has been supported with evidence, or tested experimentally. A ‘hypothesis’ means the same as ‘theory’, except it is called a ‘hypothesis’ prior to testing.
- 5. FORMAL LOGIC • Why? • In order to show the context of how the debates of PhilSci arise, we need to understand some basics of logic, and why in particular one form of logic is considered problematic.
- 6. FORMAL LOGIC • Some technical terms - 1 of 4 • Proposition • Premise • Enthymeme • “The case” means “it is true that, or real that”. So,“It is the case that it is raining” means “it is raining, really.” Contrarily,“just in case” means something like “if”.
- 7. FORMAL LOGIC • Some technical terms - 2 of 4 • A proposition is necessarily true if the premises guarantee the conclusion.The symbol for “necessarily that” is a square: ☐ • An argument or proposition is sufﬁcient if it alone guarantees the truth of some other proposition. • Jointly necessary and sufﬁcient • The opposite of necessary is “contingent”.This means “could fail to be the case” or “could fail to exist” or “could fail to happen” or “could have been otherwise”.
- 8. FORMAL LOGIC • Some technical terms - 3 of 4 • “Possible universe” or “possible world”. Do not confuse with MUT • There are different modes of necessity. Something can be logically necessary, but not causally necessary, or ontologically necessary. • Probably (P) and Possibly (♢). • Likely. Likelihood L is not the same as probability. L is a measure of how surprising a piece of evidence is, assuming a hypothesis is true. Likelihood is an indication of our trust in the model (hypothesis) that we have, rather than an indication of how probable the outcomes are per se.
- 9. FORMAL LOGIC • Some technical terms - 4 of 4 • There are different modes of possibility: logical and ontological. • Analytic versus Synthetic.A statement is “analytically” true if it is true by deﬁnition. Contrarily, a statement is “synthetically” true if we have to check its truth against the known world. So,“John is not a bachelor” is a synthetic truth.A popular example that is given in the literature is “the current president of the USA is a man” or “is an actor” and so on. • A priori versus A posteriori. • We can get a priori analytic truths, a priori synthetic truths, and a posteriori synthetic truths.We cannot get a posteriori analytic truths.
- 10. FORMAL LOGIC • Premises and Conclusions • When we present an argument, it is said to have premises and conclusions. Consider the following argument All men are mortal Socrates is a man Therefore Socrates is mortal
- 11. FORMAL LOGIC • Premises and Conclusions • When we present an argument, it is said to have premises and conclusions. Consider the following argument All men are mortal Socrates is a man Therefore Socrates is mortal • Symbols and operators: ∧ & v ~ ¬ – ➝ ⊃ ∀ ∃
- 12. FORMAL LOGIC • Examples of symbolism Socrates can die and he’s a man: 1. (∀x)(Mx ➝ Dx) Premise 2. Ms Premise 3. Ms ➝ Ds 1 4. Ds 2, 3 Driving a car and using petrol: D ⊃ P ; D ; ∴ P
- 13. FORMAL LOGIC • Transitivity and Closure • A > B > C, entails that A > C • if p entails q, and p, then q. • One direction. Bidirectional is iff.
- 14. FORMAL LOGIC Example of implication type Symbolism Interpretation If a fruit is an apple, then John will eat it. A → J If it’s an apple, then John eats Only if a fruit is an apple, will John eat. J → A If John eats, then it is an apple John will eat, if and only if it is an apple. J ⟷ A if John’s eating, it’s an apple, if; it’s an apple, John’s eating it
- 15. FORMAL LOGIC • LEM and PNC • A v ¬A . Either it’s raining or it’s not raining. • ¬(A & ¬A) . It’s false that (it’s raining and not raining). • But = and
- 16. FORMAL LOGIC • Abduction, Induction and Deduction • Abduction: argument to best explanation. Science. A form of induction. • Induction: future resembles past; things correlate. • Deduction: logically valid.
- 17. FORMAL LOGIC • Deduction • Valid: logically correct. E.g. John is a froop, all froops are bleen, therefore John is bleen. • Sound: logically correct AND premises are true. John is a man.All men are male.Therefore John is male.
- 18. FORMAL LOGIC • Modus Ponens P → Q P ∴ Q • If I drive my car, then I use petrol. I drove my car, therefore I used petrol.
- 19. FORMAL LOGIC • Modus Tollens P → Q ¬Q ∴ ¬P • If I drive my car, then I use petrol. I did not use petrol, therefore I did not drive my car.
- 20. FORMAL LOGIC • Logical Fallacies: Afﬁrming the Consequent: P → Q Q ∴ P • If I drive my car, I use petrol. I used petrol, therefore I drove my car.This is false. I may have used petrol to start a barbecue.
- 21. FORMAL LOGIC • Logical Fallacies: Denying the Antecedent: P → Q ¬P ∴ ¬Q • If I drive my car, I use petrol. I did not use drive, therefore I did not use petrol.Well, that’s false. I may have used petrol to start a barbecue, so I can both use petrol and not drive my car.
- 22. FORMAL LOGIC • Truth Tables P Example Q Example P ⊃ Q Example T I drive my car T I use petrol T I drive my car then I use petrol T I drive my car F I don’t use petrol F I drive my car and don’t use petrol (clearly false) F I don’t drive my car T I do use petrol (for something) T If I drove my car I would use petrol (true, even if I am not driving now) F I don’t drive my car F I don’t use petrol T If I drove my car I would use petrol (true, even if I am not driving now)
- 23. FORMAL LOGIC • Truth Tables • Different truth tables for different operations, e.g. AND, OR, NOT, XOR, etc. • Important point is it is possible to compute or automatically check if an argument is valid.
- 24. FORMAL LOGIC • Venn Diagrams • Cows,Tables, intersect on “things with four legs” (AND) • Socrates, Men, union on ‘bearded things’ (OR)
- 25. FORMAL LOGIC • Strong vs Cogent: Inductive arguments • Often can’t get a valid (formal) argument • Have to rely on evidence • Evidence supports: strong argument • Premises true and evidence supports: cogent

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