brainscape_PH112_full_detailed Flashcards
(29 cards)
What are the three components of a verbal definition?
- <b>Definiendum</b>: the term being defined<br></br>2. <b>Definition sign</b>: typically ‘is’, ‘means’, or ‘iff’<br></br>3. <b>Definiens</b>: the defining expression, typically giving necessary and sufficient conditions.
What is the difference between ostensive and verbal definitions?
<b>Ostensive definitions</b> define by example, typically through pointing or showing (e.g., ‘this is a dog’).<br></br><b>Verbal definitions</b> use language to specify meaning, e.g., ‘a triangle is a three-sided polygon.’<br></br>Ostensive definitions are useful in learning but often ambiguous or incomplete.
What are the six rules (R1–R6) for defining predicates in formal logic? (BAVPPF)
R1. Use a biconditional (↔) as the definition sign<br></br>R2. The definiendum must be atomic (no connectives or quantifiers)<br></br>R3. The definiendum must only contain variables, not constants<br></br>R4. Variables in the definiendum must be pairwise distinct<br></br>R5. The predicate being defined must not appear in the definiens<br></br>R6. All free variables in the definiens must occur in the definiendum
What is the schema for a formally correct predicate definition?
A definition is formally correct if it follows:<br></br>∀x₁…xₙ(Px₁…xₙ ↔ φ[x₁…xₙ])<br></br>Where φ contains no free variables other than x₁…xₙ and P does not appear in φ.
What are ‘eliminability’ and ‘conservativeness’ in definitions?
<b>Eliminability</b>: the definition allows substitution of the definiendum with its definiens without changing truth values.<br></br><b>Conservativeness</b>: adding the definition to a theory doesn’t lead to new theorems about old terms (i.e., no unintended consequences).
What is the difference between a set and its elements?
A set is a collection of distinct objects. Its elements are the individual objects within it. For example, in the set {1, 2, 3}, the numbers 1, 2, and 3 are elements.
What are subset and proper subset?
X ⊆ Y if every element of X is also in Y (subset).<br></br>X ⊂ Y if X is a subset of Y but X ≠ Y (proper subset).
What is a function in set theory?
A function f from X to Y is a set of ordered pairs such that for every x ∈ X, there is exactly one y ∈ Y with (x, y) ∈ f.
What are the components of a structure in model theory?
A structure M = (D, I) consists of:<br></br>1. A domain D (non-empty set)<br></br>2. Interpretations I:<br></br> - For each constant: a specific element of D<br></br> - For each predicate: a relation over D
What does it mean for φ to be true in a structure?
A sentence φ is true in structure M (written M ⊨ φ) if φ is satisfied under the interpretation given by M.
What do ◻ϕ and ◇ϕ mean? for the worlds too.
◻ϕ: ‘It is necessary that ϕ’ – true in all accessible worlds.<br></br>◇ϕ: ‘It is possible that ϕ’ – true in at least one accessible world.
What are the key modal principles?
(K) ◻(ϕ → ψ) → (◻ϕ → ◻ψ)<br></br>(D) ◻ϕ → ◇ϕ<br></br>(T) ◻ϕ → ϕ<br></br>(S4) ◻ϕ → ◻◻ϕ<br></br>(S5) ◇ϕ → ◻◇ϕ
What is Lewis’s truth condition for ϕ ⟶ ψ?
ϕ ⟶ ψ is true at world w iff:<br></br>1. ψ is true at the closest ϕ-world(s) to w, or<br></br>2. there are no ϕ-worlds (vacuously true).
Why can’t counterfactuals be modeled with → or ◻(ϕ → ψ)?
They violate formal properties like:<br></br>- Antecedent strengthening<br></br>- Contraposition<br></br>- Transitivity
State the Kolmogorov axioms.
- Pr(W) = 1 (Normalization)<br></br>2. Pr(A) ≥ 0 for all A<br></br>3. If A ∩ B = ∅, then Pr(A ∪ B) = Pr(A) + Pr(B)
What is Bayes’ Theorem?
Pr(H|E) = [Pr(E|H) · Pr(H)] / Pr(E)
What is inductive strength?
The conditional probability of the conclusion given the premises: Pr(C | P1 ∧ … ∧ Pn)
What is the gambler’s fallacy?
The mistaken belief that a streak of one outcome lowers its probability on the next trial, even if outcomes are independent.
What are the two core principles of Bayesianism?
- <b>Coherence</b>: credences must satisfy probability axioms<br></br>2. <b>Conditionalisation</b>: update beliefs using Prnew(A) = Prold(A|E)
What is the Dutch Book Argument?
If your credences violate probability axioms (ie. coherenece), you are susceptible to a DB: a set of bets where you’re guaranteed to lose money no matter what the truth turns out to be.
It’s not rational to accept a DB, so for agent = rational, their degrees of deblief must satisfy axioms of probability.
What is the lottery paradox?
If we believe each ticket will lose (high probability), and believe the lottery must be won, then our beliefs are inconsistent (closed under conjunction).
What is the Lockean Thesis?
We believe a proposition iff its probability exceeds a threshold (e.g., > 0.5). But this can lead to paradoxes like the lottery paradox.
What is the gambler’s fallacy? (Explain in detail with example)
<ul><li>The gambler’s fallacy is the mistaken belief that the likelihood of an outcome decreases after a streak of that same outcome.</li><li>It arises from misunderstanding statistical independence.</li><li><b>Definition of independence:</b> Events A and B are independent iff Pr(A ∩ B) = Pr(A) · Pr(B).</li><li><b>Example:</b> A fair die is rolled three times and lands on 6 each time. One might falsely believe the next roll is less likely to be 6.</li><li>In reality, the probability remains 1/6 each time — outcomes are independent.</li></ul>
What are the problems with the finite frequentist interpretation of probability?
<ul><li><b>Accidental frequencies:</b> A finite sample may not represent the true tendency due to chance.</li><li><b>Change over time:</b> Frequencies may not remain constant — e.g., changing exam procedures or disease rates.</li><li><b>Single-case probabilities:</b> It gives no guidance on the probability of a one-off event like ‘this patient has cancer’.</li></ul>
- Definition: Probability = limiting relative frequency in a hypothetical infinite sequence of trials.
- Improves: Avoids accidental frequency and time-dependence problems.
- Limitation: Still struggles with single-case probability (e.g. what is the probability of this particular coin toss?).
- Definition: Probability = degree of belief of a rational agent.
- Can handle single-case probabilities and belief updates via conditionalisation.
- Problems: Depends on priors, which can be arbitrary; lacks objectivity.
- Claim: Metaphysical realism assumes a unique mapping from language to world, but this mapping is underdetermined.
- For any model that satisfies a theory, we can permute the domain and generate an equally valid but different interpretation.
- Example: 'Cats are animals' could be reinterpreted as 'tables are plants' under a permuted model, yet the theory remains satisfied.
- Conclusion: The idea that terms must refer to unique real-world entities collapses under model-theoretic indeterminacy.
- Accepts the idea that reference is not fixed by formal structure alone.
- Emphasises epistemic humility: we should be anti-realist about reference to unobservables.
- Focus on empirical adequacy rather than metaphysical truth.
- Critique: Doesn’t fully rescue metaphysical realism; instead, shifts away from it.
- Old evidence problem: If E is already known, and T is developed to explain E, then Pr(E|T) = Pr(E), so E doesn’t confirm T.
- Solution attempt: Modify priors retrospectively, but this is ad hoc and undermines conditionalisation.
- Dependence on priors: Confirms relative to prior beliefs; lacks objectivity.
- Confirmation holism: Evidence affects networks of beliefs, not just isolated hypotheses — hard to model precisely.
