Thinking Skills Assessment (TSA) Foundational Logic and Numeracy Syllabus

Foundational Logic and Numeracy syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Foundational Logic and Numeracy in Thinking Skills Assessment (TSA), not a summary of it.

1. 1

   Logic Essentials

   4 topics

   • Conditional statements
     • If-then reasoning and its direction
     • Converse, inverse and contrapositive
   • Necessary and sufficient conditions
     • Telling the two apart in arguments
     • Common confusions in everyday claims
   • Quantifiers and scope
     • All, some, none and their negations
     • Reasoning validly with quantified claims
   • Deductive versus inductive reasoning
     • Certainty versus probability of conclusions
     • When each form is appropriate
2. 2

   Numerical Fluency

   4 topics

   • Fractions, decimals and percentages
     • Converting fluently between forms
     • Percentage change, increase and decrease
   • Ratio and proportion
     • Sharing in a given ratio
     • Direct and inverse proportion
   • Averages and basic statistics
     • Mean, median, mode and range
     • Weighted averages in problems
   • Estimation and approximation
     • Sense-checking magnitudes
     • Rounding to speed up calculation
3. 3

   Data Handling Without a Calculator

   3 topics

   • Mental maths shortcuts
     • Useful tricks for fast computation
     • Recognising common factors and multiples
   • Reading data displays quickly
     • Extracting key figures from a chart
     • Comparing categories at a glance
   • Probability basics
     • Simple and combined events
     • Expressing chance as fraction, decimal or percentage

Foundational Logic and Numeracy flashcards for Thinking Skills Assessment (TSA)

25 of 51 cards from the Foundational Logic and Numeracy deck — real questions with worked answers.

1. In the conditional statement "If $P$, then $Q$", what are the standard names for $P$ and $Q$?

   $P$ is the antecedent (hypothesis); $Q$ is the consequent (conclusion). The statement is written $P \to Q$ and is only false when $P$ is true and $Q$ is false.

2. Write the converse, inverse and contrapositive of $P \to Q$.

   Converse: $Q \to P$. Inverse: $\neg P \to \neg Q$. Contrapositive: $\neg Q \to \neg P$. Only the contrapositive is logically equivalent to the original.

3. Why is the contrapositive a valid argument form but the converse a fallacy?

   $P \to Q$ is logically equivalent to its contrapositive $\neg Q \to \neg P$, so it preserves truth. The converse $Q \to P$ is not equivalent; assuming it commits the fallacy of affirming the consequent.

4. Under what single combination of truth values is the conditional $P \to Q$ false?

   Only when the antecedent $P$ is true and the consequent $Q$ is false. In all other cases (including whenever $P$ is false) the conditional is true (vacuously true when $P$ is false).

5. Name and state the two valid inference rules associated with conditionals.

   Modus ponens: from $P \to Q$ and $P$, infer $Q$. Modus tollens: from $P \to Q$ and $\neg Q$, infer $\neg P$.

6. What are the two classic formal fallacies involving conditionals?

   Affirming the consequent: from $P \to Q$ and $Q$, wrongly inferring $P$. Denying the antecedent: from $P \to Q$ and $\neg P$, wrongly inferring $\neg Q$. Both are invalid.

7. Express "$P$ only if $Q$" as a conditional. Is it the same as "$P$ if $Q$"?

   "$P$ only if $Q$" means $P \to Q$ ($Q$ is necessary for $P$). "$P$ if $Q$" means $Q \to P$. They are converses, not the same statement.

8. In the biconditional "$P$ if and only if $Q$", what must hold for it to be true?

   $P \leftrightarrow Q$ is true exactly when $P$ and $Q$ have the same truth value (both true or both false). It is the conjunction of $P \to Q$ and $Q \to P$.

9. Define a necessary condition and give the conditional form.

   $Q$ is necessary for $P$ if $P$ cannot be true without $Q$ being true; equivalently $P \to Q$. Without the necessary condition, the outcome cannot occur.

10. Define a sufficient condition and give the conditional form.

    $P$ is sufficient for $Q$ if the truth of $P$ guarantees the truth of $Q$; equivalently $P \to Q$. Having $P$ is enough to ensure $Q$, though $Q$ might occur by other means too.

11. In $P \to Q$, which variable is the sufficient condition and which is the necessary condition?

    $P$ (the antecedent) is sufficient for $Q$; $Q$ (the consequent) is necessary for $P$. "Sufficient implies necessary": the arrow points from sufficient to necessary.

12. What does it mean for a condition to be both necessary and sufficient?

    $P$ is necessary and sufficient for $Q$ when $P \leftrightarrow Q$: $P$ guarantees $Q$ and $Q$ cannot occur without $P$. The two conditions are logically equivalent.

13. "Being 18 or over is necessary but not sufficient for buying alcohol." Explain the logic.

    Necessary: you cannot buy it unless you are 18+, so purchase $\to$ age $\geq 18$. Not sufficient: being 18+ does not by itself guarantee a purchase (you also need money, stock, etc.), so age $\geq 18 \not\to$ purchase.

14. Identify the necessary and sufficient conditions in "Scoring 70% is sufficient to pass."

    Scoring 70% is a sufficient condition for passing: $\text{score} \geq 70\% \to \text{pass}$. It is not stated to be necessary; one might also pass another way (e.g. a lower threshold).

15. What are the universal and existential quantifiers, and their symbols?

    Universal quantifier $\forall$ ("for all / every") asserts a property holds for all members of a domain. Existential quantifier $\exists$ ("there exists / some") asserts at least one member has the property.

16. State the negation rules for quantifiers.

    $\neg(\forall x\, P(x)) \equiv \exists x\, \neg P(x)$ and $\neg(\exists x\, P(x)) \equiv \forall x\, \neg P(x)$. Negation flips the quantifier and negates the predicate.

17. What is the logical negation of "All swans are white"?

    "There exists at least one swan that is not white" ($\exists x:\ \text{swan}(x) \wedge \neg\text{white}(x)$). A single counterexample falsifies a universal claim; it is NOT "No swans are white".

18. Why does the order of mixed quantifiers matter? Contrast $\forall x \exists y$ with $\exists y \forall x$.

    $\forall x\,\exists y\,P(x,y)$ allows $y$ to depend on $x$ (each $x$ has some $y$). $\exists y\,\forall x\,P(x,y)$ demands one fixed $y$ that works for every $x$. The second is stronger and implies the first, but not vice versa.

19. In logic, what is meant by the "scope" of a quantifier?

    The scope is the portion of the formula to which the quantifier applies (the part it binds). A variable within the scope is bound by that quantifier; outside it, the variable is free. Brackets determine scope.

20. How does scope change the meaning of "Everyone loves someone"?

    "For each person, there is someone they love" ($\forall x\,\exists y\,\text{loves}(x,y)$) differs from "there is one person everyone loves" ($\exists y\,\forall x\,\text{loves}(x,y)$). The ambiguity comes from quantifier scope/order.

21. Define deductive reasoning and the key property of a valid deductive argument.

    Deductive reasoning moves from premises to a conclusion that follows with logical necessity. In a valid deduction, if the premises are true the conclusion must be true; it is truth-preserving and the conclusion contains no new information beyond the premises.

22. Define inductive reasoning and how its strength is judged.

    Inductive reasoning generalises from specific observations to a probable conclusion. It is judged strong or weak (not valid/invalid): true premises make the conclusion likely but not guaranteed. It is ampliative—the conclusion goes beyond the premises.

23. Distinguish a valid argument from a sound argument.

    Valid: the conclusion follows logically from the premises (correct form), regardless of whether premises are true. Sound: the argument is valid AND all its premises are actually true. Soundness implies a true conclusion.

24. Give the three-part structure of a classic categorical syllogism.

    Major premise, minor premise, and conclusion, e.g. "All M are P; all S are M; therefore all S are P." It links a subject and predicate through a shared middle term M.

25. What is abductive reasoning and how does it differ from deduction and induction?

    Abduction is inference to the best explanation: from an observation, you infer the most plausible cause. Unlike deduction it is not guaranteed, and unlike broad induction it specifically selects the hypothesis that best accounts for the evidence.

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