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Covers all topics for PH104 Formal Methods of Philosophical Argumentation. Based to cover all potential questions in the exam....

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PH104 Notes Classical Sentential Logic Two ways of criticising arguments: 1. Criticism concerning content 2. Criticism concerning form Validity: An argument is (logically) valid just in case its conclusion follows logically from its premises. A statement follows logically from other statements just in case it is impossible that the conclusion is false while the premises are true. Inductively Strong: An argument is inductively strong (to degree d) just in case (a) it is improbable (to degree d) that its conclusion is false given its premises are true, (b) but the argument is not logically valid. Sound Argument: An argument is sound just in case (a) it is logically valid and (b) all of its premises are true. Bad Logically Valid Arguments:  

Valid arguments with false premises (unsound arguments) Valid arguments of inadequate form (e.g. circular arguments (where the truth of the conclusion is assumed in the premises)).

Good logically invalid arguments:  

Some inductively strong arguments Some enthymemes (invalid arguments with a missing premise that would make it valid)

Syntax: deals with relationships between linguistic expressions and does so by referring only to linguistic expressions. Syntactic concepts (e.g., formula, derivability). Semantics: deals with the relationship between linguistic expressions and their meanings. Semantic concepts (e.g., truth, logical consequence) Tautology: A formula ϕis a tautology (logically true in CSL) iff ϕ has the value 1 under every truth assignment. Contradiction: A formula ϕis a contradiction (logically false in CSL) iff ϕ has the value 0 under all truth assignments. Contingent: A formula ϕis contingent in CSL iff (a) there is truth assignment under which ϕ takes the value 1 and (b) there is a truth assignment under which ϕtakes the value 0. Logical Consequence: A formula ϕfollows logically from (is a logical consequence of) a set of formulas {ψ1, . . . , ψn} in CSL iff there is no truth assignment under which ψ1, . . . , ψn all take the value 1 and ϕtakes the value 0

Validity in CSL: An argument form ψ1, . . . , ψn ∴ ϕis valid in CSL iff ϕ follows logically from {ψ1, . . . , ψn} in CSL. Logical Equivalence: A formula ϕis logically equivalent in CSL to a formula ψ iff ϕ takes the same value as ψ under each truth assignment Consistency: A set of formulas {ϕ1, . . ϕ . , n} is consistent in CSL iff there is a truth assignment under which each of the formulas ϕ1, . . ϕ . , n takes the value 1. Soundness of a deductive system D w.r.t. a semantics S: If ψ is derivable from T in D, then ψ follows logically from T according to S. (You cannot prove anything that is wrong) Completeness of a deductive system D w.r.t. a semantics S: If ψ follows logically from T according to S, then ψ is derivable from T in D. You can prove anything that is right.

Predicate Logic

Useful formalisation: ∃xMxe ∧ ∀y∀z(Mye ∧ Mze → y = z) Read as “The Earth has exactly one moon”. Scope: The scope of an occurrence of a quantifier expression ∃v or ∀v in a formula ϕis the smallest sub formula of ϕthat immediately follows this occurrence. Bound: An occurrence of a variable v in a formula ϕis bound iff 1. it immediately follows an occurrence of ∃ or ∀, or 2. it is within the scope of a quantifier expression ∃v or ∀v. Open Formula: ϕis an open formula iff at least one occurrence of a variable is free in ϕ (don’t have truth values) Closed Formula: ϕis a closed formula (a sentence) iff no occurrence of a variable is free in ϕ (do have truth values). Conditions for Universal Introduction of x in for y: 1. There is no premise in which y is free. 2. There is no undischarged assumption in which y is free. Conditions for Existential Elimination:

Semantics An interpretation establishes a link between language and world. Under an interpretation, each descriptive expression has an extension. 1. The extension of a singular term is the particular object it denotes.

2. The extension of a predicate is the set of object to which the predicate applies (and a two placed predicate is the set of ordered pairs). 3. The extension of a sentence is its truth value. The truth value of a sentence is determined by the extensions of the predicates and singular terms in the sentence. D,I is a CPL-interpretation just in case 1. D is a non-empty set of objects (D =/= ∅); 2. I is a function that associates 2.1 with each individual constant c an object I(c) ∈ D; 2.2 with each n-place predicate P a set I(P) of n-tuples of objects in D (i.e., I(P) ⊆ D n ). Semantic definition of truth a la Tarski: For all CPL-sentences ϕ: is true under an interpretation I iff ϕis true under I relative to every variable assignment (over the domain of I).

ϕfollows logically from a set of formulas T in CPL (T ϕ ) iff there is no CPL-interpretation D,I such that all sentences in T are true under D,I but ϕis false under D,I ϕis logically true/false in CPL iff ϕ is true/false under every CPL-interpretation. ϕis contingent in CPL iff ϕ is true under some CPL-interpretation and false under some other CPL-interpretation. ϕis logically equivalent to ψ in CPL iff ϕ and ψ have the same truth value under all CPLinterpretations. T is consistent in CPL iff there is some CPL-interpretation such that all sentences in T are true under that interpretation. Variable Assignment: α is a variable assignment (VA) over D iff α is a function that assigns to each variable v an element α(v) of D. a’ is a v-variant of α iff α 0 is a VA over D that differs from α at most in what it assigns to v. X is a subset of Y (short: X ⊆ Y ) iff every element of X is also an element of Y. X is a proper subset of Y (short: X ⊂ Y ) iff X is a subset of Y but X ≠ Y. The intersection of sets X and Y (short: X ∩ Y ) is the set of those things that are elements of both X and Y . That is, X ∩ Y = {x | x ∈ X and x ∈ Y }. The union of sets X and Y (short: X ∪Y ) is the set of those things that are elements of X or Y . That is, X ∪ Y = {x | x ∈ X or x ∈ Y }. The set difference of X and Y (short: X\Y ) is the set of those things that are elements of X but not elements of Y . That is, X\Y = {x | x ∈ X and x ∈/ Y }. The Cartesian product of sets X and Y (short: X × Y ) is the set of those ordered pairs hx, yi such that x is an element of X and y is an element of Y . That is, X × Y = {hx, yi | x ∈ X and y ∈ Y }.

Formal Methods Proposition are the meanings expressed by declarative sentences. Propositions are the contents of attitudes. Propositions are bearers of truth values. A sentence is true if and only if the proposition it expresses is true. Propositions play three crucial roles: 1. They are the meanings expressed by declarative sentences. 2. They are the contents of attitudes such as belief and desire. 3. They are bearers of truth values. Identity criterion for worlds For all possible worlds w, w’: w = w’ iff for all propositions X, X is true at w iff X is true at w’ Identity criterion for propositions For all propositions X, Y : X = Y iff for all possible worlds w, X is true at w iff Y is true at w Propositions are sets of possible worlds. Let W be a non-empty set of possible worlds. X is a proposition (over W ) iff X is a subset of W. Let X be a proposition (over W ) and w a world in W. Then X is true at w iff w ∈ X. For all propositions X and Y : • X logically entails Y iff X ⊆ Y . • Y logically follows from X iff X ⊆ Y . A set of sentences T is maximally consistent iff 1. T is consistent and 2. there is no set of sentences T + that properly extends T, i.e. T ⊂ T +, and is consistent

Modal notions can’t be proved with truth tables. E.g. It is possible that there is no Brexit deal. So, it is not necessary that there is a Brexit deal. Would just be formalised as p therefore not q.

These principles do not fit for all modalities. For instance: • Principle T does not make sense for deontic modalities. (“What is obligatory is actually the case.”) • Principle T does not make sense for doxastic modalities. (“What is believed is actually the case.”) • Principle S4 is controversial for knowledge. (“What is known is known to be known.”) • Principle S5 is controversial for knowledge. (“What is not known is known to be not known.”)

NEC: Rule of necessitation – If we can derive p then we can derive necessarily p System K = CSL + NEC + K + Defbox + Def♦ (basic system) System D = system K + D System T = system K + T System S4 = system K + T + S4 System S5 = system K + T + S5 A kripke model satisfies various principles if: 1. Principle D iff R is serial: For every w ∈ W , there is a w’ ∈ W such that wRw’. ) Every world has an accessible world).

2. Principle T iff R is reflexive: For every w ∈ W , wRw. 3. Principle S4 iff R is transitive: For every w1,w2,w3 ∈ W , if w1Rw2 and w2Rw3, then w1Rw3. 4. Principle S5 iff R is Euclidean: For every w,w1,w2 ∈ W , if wRw1 and wRw2, then w1Rw2

Counterfactuals

Counterfactuals are conditionals in subjunctive mood: • If ϕwere the case, then ψ would be the case. The Queen is not in Russia. But it doesn’t follow that if the Queen were in Russia, she would be in America. (ie the material conditional would have a truth value of one, but the counterfactual needs a truth value of 0 in this case). (Evidence of why a material conditional doesn’t work for a counterfactual condition).

Strict conditional – Box(p implies q)

It also allows contraposition – the negation of both parts of the conditional. Example: (a) If Alice had gone to the party, Bertrand would have gone. (b) If Bertrand had not gone, Alice would not have gone. Suppose that Bertrand would have been even more excited to attend if Alice had. Suppose Alice wanted to go but stayed away to avoid Bertrand. In this scenario, (a) is true and (b) is false. A counterfactual conditional is true iff it takes less of a departure from actuality to make the antecedent true along with the consequent than to make the antecedent true without the consequent Using the models, then it is true if: 1. There are no models where the antecedent is true - (vacuously true) 2. The closest model where the antecedent is true, the consequent is also true.

Naïve counterfactual definition of causation C causes E iff 1. C and E both occur and 2. if C had not occurred, E would not have occurred. Objection due to pre-emption: Two partisans conspire to assassinate a hated dictator, agreeing that one or other will shoot the dictator on a public occasion. Acting side-by-side, assassins A and B both take aim when the dictator appears. A pulls her trigger and fires a shot that hits its mark, but B desists from firing when he sees A pull her trigger. Clearly, A’s firing causes the dictator’s death. • But according to the naive definition of causation, A’s firing is not a cause of the dictator’s death. • After all, it is not the case that if A had not fired, the dictator would not have died. • For if A had not fired, B would have shot the dictator.

Lewis’ account of causation: C causes E iff there is a causal chain from C to E. A causal chain is a finite sequence of actual events E1, . . . , En such that if E2 causally depends on E1, E3 causally depends on E2 etc. An actual event E causally depends on an actual event C iff if C were not to occur E would not occur. Objection: Late pre-emption Billy and Suzy both throw their rocks at a window at the same time. Since Suzy stands a bit closer, her rock reaches the window first. The window shatters, and Billy’s rock goes flying through the now-empty window frame. On Lewis’ account, Suzy’s throw is not a cause of the shattering. • The shattering doesn’t causally depend on Suzy’s throw. Because even if Suzy hadn’t thrown the rock, the window would have shattered due to Billy’s throw. • There are no intermediate events between Suzy’s throw and the shattering forming a causal chain. Consider Suzy’s rock in mid-trajectory. The shattering does not depend on it, because even without it the window would still have shattered due to Billy’s throw. Objection: Causation isn’t transitive A climber is in the middle of a multi-pitch route. A rock becomes dislodged and falls towards her. She sees the rock and presses herself against the wall. The rock flies over her head. She survives.

On Lewis’ account, causation is transitive. However, this scenario suggests that causation is not transitive. • The rock’s becoming dislodged (C) caused the climber to press herself against the wall (D). And D caused the climber to survive (E). But the rock’s becoming dislodged didn’t cause the climber to survive. • So C causes D and D causes E, but C doesn’t cause E.

Belief Systems Belief is a propositional attitude. An agent a believes a proposition X iff a holds X to be true.

Corollary 1. If X ∈ Bel and X → Y ∈ Bel, then Y ∈ Bel. Corollary 2. If X ∈ Bel, then ¬X ∈/ Bel

The Preface Paradox An author has written a book in which she asserts propositions P1, . . . , Pn. • She rationally believes each one of these propositions. • Having learnt from experience, the author rationally believes that there are undetected errors in the book. • She rationally believes that some of the propositions are false. • So she rationally believes that P1 ∧ . . . ∧ Pn is false Let Bel be the author’s belief system. Then: PP1. P1 ∈ Bel, . . . , Pn ∈ Bel. PP2. ¬(P1 ∧ . . . ∧ Pn) ∈ Bel. Combine these with the postulates of rational belief, then you are screwed. Potential solutions: Remove closure under conjunction – however this could lead to inconsistent beliefs/contradictions. A paradox is a valid argument in which each premise is plausible, but the conclusion is absurd. Belief Revision

Expansion Bel + E Adding a new belief E to Bel without removing any beliefs. Contraction Bel ÷ E Removing a belief E to Bel and other beliefs that entail it. Revision Bel ∗ E Adding a belief E to Bel and possibly removing other beliefs to avoid inconsistencies in the resulting belief system. Let Cn be a consequence operation defined as follows: for any set of sentences (or propositions) T, Cn(T) is the set of all logical consequences of T. One says that T is logically closed iff T = Cn(T). Bel + E = Cn(Bel ∪ {E}) Rational Revision Operations: 1. Bel ∗ E = Cn(Bel ∗ E). (Closure) 2. E ∈ Bel ∗ E. (Success) 3. Bel ∗ E ⊆ Bel + E. (Inclusion) 4. If ¬E ∈/ Bel, then Bel ∗ E = Bel + E. (Preservation) 5. If E is not logically false, then Bel ∗ E is consistent. (Consistency) 6. If E1 is logically equivalent to E2, then Bel ∗ E1 = Bel ∗ E2. (Extensionality) When one has to drop old beliefs in the light of new evidence, there may be different beliefs that one could drop. Then it is underdetermined how one should rationally revise.

Probability Kolmogorov axioms: 1. The probability of the sample space is 1. E.g. if the two possibilities are p or q, then p or q = 1. 2. All probabilities are greater or equal to 0. 3. If two events (a and b) are mutually exclusive then pr(A and B) = pr(A) + pr(B)

General law of disjunctions = Pr(A ∨ B) = Pr(A) + Pr(B) − Pr(A ∧ B). Pr(A|B) := Pr(A ∩ B)/Pr(B) Pr(A ∩ B) = Pr(B|A) · Pr(A) Law of total probability: Pr(A) = Pr(A|B) · Pr(B) + Pr(A|¬B) · Pr(¬B) Bayes Theorem: Pr(H|E) = (Pr(E|H) · Pr(H))/Pr(E) Algebra of events is all combinations according to basic set theory (e.g. intersection, union etc) and includes the empty set. e.g. Specify the algebra of events for the first toss = {∅, {H}, {T}, {H,T}}

Classical Probability The classical probability of an event A = the number of possibilities where A occurs the total number of possibilities It assumes equal probability in events. Objections: It is circular It is too narrow Frequentism The probability of an event relies on the relative frequency of it. E.g. if an event has occurred 46 times out of 100, a probability of 0.46 is assigned to that event. Objections:

Accidental frequencies Change over time (indeterminate probabilities) Single case Hypothetical Frequencies Uses infinite sequences of frequencies 1. It is highly idealised and counterfactual. We can’t actually do infinitely many trials. 2. Every finite number of trials is consistent with any limit. Since we can only make finitely many trials, we can never know what the limit and, hence, the probability is. 3. It cannot deal with probabilities of unrepeatable events and single-case probabilities. Propensity A propensity is a physical disposition or tendency of a physical situation to yield certain (relative frequencies of) outcomes. Popper believes the propensity of an act lies in the entire experimental set up. Objections: 1. It is unclear what propensities exactly are. 2. It is more metaphysical than scientific in nature. 3. Problems with conditional propensities (Humphreys’ problem). Consider • the propensity that A gets a common cold tomorrow given A had contact with B yesterday. • the propensity that A had contact with B yesterday given that A gets a common cold tomorrow. Bayes’ theorem seems implausible for propensities. • Past propensities cannot depend on future propensities. (temporal asymmetry) • But Bayes’ theorem follows from the axioms of probability. • So propensities do not obey the axioms of probability Bayesianism Probability is degree of belief (credence) of a rational agent. Caveat: • The degrees of belief of actual humans sometimes (or often) violate the laws of probability. • For example, there are cases in which A entails B, but many people’s degrees of belief are Pr(A) > Pr(B) (“Linda case”). If A ⊆ B, then Pr(A) ≤ Pr(B). Core principles of Bayesianism:

Coherence - The degree of belief function of an inferentially perfectly rational agent satisfies the Kolmogorov axioms of probability. Conditionalisation - An inferentially perfectly rational agent with degree of belief function Prold , who becomes certain of a piece of evidence E, should shift to a new probability function Prnew related to Prold by: Prnew (A) = Prold (A|E), if Prold (E) > 0 Degrees of belief can be revealed in betting behaviour. A/B are the minimum odds under which you would accept a bet on X iff your credence in X is B/A+B . A Dutch Book is a set of bets in which you are guaranteed to lose money no matter what the truth turns out to be. A Dutch Book can be made against an agent iff the agent’s degrees of belief fail to satisfy the axioms of probability. Bayesian Confirmation Theory 1. E confirms T (relative to Pr) iff Pr(T|E) > Pr(T). 2. E disconfirms T (relative to Pr) iff Pr(T|E) < Pr(T). 3. E is neutral for T (relative to Pr) iff Pr(T|E) = Pr(T).

Corollary 1 If T entails E, then E confirms T. Corollary 2. If T entails ¬E, then E disconfirms T. Conf (T, E) := Pr(T|E) − Pr(T) −1 < Conf (T, E) < 1 Theories should be tested with a severe test, which is the prediction of a theory that is very unlikely, if we aren’t assuming the theory. This means PrE is very small, and so confirms the theory more. Paradox of the ravens Relies on: Nicod’s Criterion – A universal hypothesis of the form ∀x(Px → Qx) is confirmed by any positive instance of the form Pa ∧ Qa. The equivalence principle - Logically equivalent theories are confirmed by the same evidence Nicod’s criterion in conjunction with the equivalence principle leads to the counter-intuitive conclusion that, for any a, ∀x(Rx → Bx) is confirmed by ¬Ra ∧ ¬Ba. Solved as Bayesian confirmation theory shows it only weakly follows.

Inductive Logic If P1, . . . , Pn ∴ C is an argument that is not logically valid, then • the closer Pr(C|P1 ∧ . . . ∧ Pn) is to 1 and the closer Pr(¬C|P1 ∧ . . . ∧ Pn) is to 0, • the stronger the premises support the conclusion An argument P1, . . . , Pn ∴ C has inductive strength of degree d iff (a) Pr(C|P1 ∧ . . . ∧ Pn) = d, (b) but the argument is not logically valid.

Problems of Bayesianism The Principle of Conditionalisation: An inferentially perfectly rational agent with degree of belief function Prold , who becomes certain of a piece of evidence E, should shift to a new probability function Prnew related to Prold by: Prnew (A) = Prold (A|E), if Prold (E) > 0. This picture of updating degrees of belief is too restrictive. • It presupposes that we ar...