Bar Course Aptitude Test (BCAT) Deduction Syllabus

Define the contrapositive of 'If P then Q', and state its logical relationship to the original.

The contrapositive is 'If not Q then not P'. It is logically EQUIVALENT to the original conditional: whenever one is true, so is the other.

Define the converse of 'If P then Q', and state whether it is equivalent to the original.

The converse is 'If Q then P'. It is NOT logically equivalent to the original; assuming the converse from the original is the fallacy of affirming the consequent.

Define the inverse of 'If P then Q' and give its equivalence.

The inverse is 'If not P then not Q'. It is not equivalent to the original conditional, but it IS equivalent to the converse.

Summarise the four equivalence relationships among a conditional, its converse, inverse, and contrapositive.

A conditional is equivalent to its contrapositive; the converse is equivalent to the inverse. A conditional is NOT equivalent to its converse or its inverse.

How do you negate 'If P then Q' (when treated as a material conditional)?

$$\neg(P \to Q) \equiv P \land \neg Q.$$ The conditional fails only in the single case where the antecedent is true and the consequent is false.

Apply contraposition: from 'All lawyers are graduates', what equivalent statement follows?

'All non-graduates are non-lawyers'. Contraposition of a universal: if something is not a graduate, it cannot be a lawyer.

What is the 'plausibility trap' in BCAT Deduction?

It is the error of marking a conclusion Valid because it seems believable or sensible, rather than because the statements logically force it. Plausibility must never override strict logical necessity.

How should you guard against letting plausibility override logic?

Ask only: 'Could the premises be true while this conclusion is false?' If yes, mark Invalid—no matter how reasonable or expected the conclusion appears.

Give an example of a plausible-but-invalid conclusion in categorical reasoning.

From 'All cats are animals' concluding 'All animals are cats' is plausibly tempting to some but is invalid—it illegitimately converts a universal. The premise does not constrain non-cat animals.

What is a 'quantifier scope error' in deductive reasoning?

It is misreading how a quantifier ranges over a statement, e.g. confusing 'every student read some book' (each may read a different book) with 'there is some book every student read' (one shared book).

Distinguish 'Everyone loves someone' from 'There is someone everyone loves'.

'Everyone loves someone': $\forall x\,\exists y\,L(x,y)$—each person has a (possibly different) beloved. 'There is someone everyone loves': $\exists y\,\forall x\,L(x,y)$—one single person loved by all. The second is stronger and does not follow from the first.

Does '$\forall x\, \exists y\, R(x,y)$' entail '$\exists y\, \forall x\, R(x,y)$'?

No. Swapping a universal-then-existential into existential-then-universal is invalid; it asserts a single shared witness that the original does not guarantee.

Does '$\exists y\, \forall x\, R(x,y)$' entail '$\forall x\, \exists y\, R(x,y)$'?

Yes. If one y works for every x, then for each x there certainly exists such a y. The existential-then-universal form is the stronger one and implies the weaker.

What does it mean to 'overlook what the premises do not establish'?

It is wrongly treating information that is merely absent or unstated as if it were settled. If the premises leave something open, no conclusion about it can be Valid.

If statements describe only A and B, and a conclusion mentions C, how should you treat it?

Unless C is logically linked to A or B by the statements, the conclusion about C is not established and is Invalid—silence in the premises proves nothing about C.

From 'Some students passed', can you validly conclude 'some students failed'?

Invalid. 'Some passed' is compatible with all passing; it does not establish that any failed. 'Some' means 'at least one', not 'only some'.

In logic as tested by BCAT, does 'some' ever imply 'not all'?

No. 'Some A are B' means at least one A is B and is fully consistent with 'all A are B'. Reading 'some' as 'some but not all' is a common scope/implicature error.

Walk through the step-by-step method for evaluating a BCAT Deduction item.

1) Read the statements and assume them true. 2) Note the quantifiers/conditionals precisely. 3) Identify what the conclusion claims. 4) Test: can premises be true and conclusion false? 5) If no such case exists, mark Valid; otherwise Invalid.

What is the single decisive test for validity you should apply to every item?

The counterexample test: try to construct any scenario where all the statements are true but the conclusion is false. If you can, it is Invalid; if it is impossible, it is Valid.

Worked deduction: Premises 'All barristers are advocates' and 'No advocates are judges'. Is 'No barristers are judges' valid?

Valid. Every barrister is an advocate, and no advocate is a judge, so no barrister can be a judge. (Form: All A are B; No B are C; therefore No A are C.)

Worked deduction: 'If the witness is reliable, the case proceeds' and 'the case did not proceed'. Conclusion 'the witness is not reliable'—valid?

Valid by modus tollens. The consequent (case proceeds) is false, so the antecedent (witness reliable) must be false.

Worked deduction: 'Some lawyers are cyclists' and 'All cyclists are fit'. Is 'Some lawyers are fit' valid?

Valid. At least one lawyer is a cyclist, and every cyclist is fit, so that lawyer is fit—hence some lawyers are fit.

Worked deduction: 'No reptiles are mammals' and 'Some pets are reptiles'. Is 'Some pets are not mammals' valid?

Valid. The pets that are reptiles cannot be mammals (no reptile is a mammal), so at least some pets are not mammals.

Worked deduction: 'All A are B' and 'Some C are A'. Is 'Some C are B' valid?

Valid. The C's that are A must also be B (since all A are B), so at least some C are B.

Worked deduction: 'If P then Q' and 'If Q then R'. Is 'If P then R' valid?

Valid—this is the hypothetical syllogism (transitivity of conditionals): $$(P \to Q) \land (Q \to R) \Rightarrow (P \to R).$$

Worked deduction: 'Either the claim is time-barred or it proceeds' and 'the claim is not time-barred'. Conclusion 'it proceeds'—valid?

Valid—disjunctive syllogism. Given an inclusive/exclusive 'or', if one disjunct is false the other must be true: $$ (P \lor Q),\ \neg P \Rightarrow Q.$$

Worked deduction: 'All A are B' and 'No C are B'. Is 'No C are A' valid?

Valid. Every A is a B, but no C is a B, so no C can be an A (by contraposition of the inclusion). Equivalent to: No A are C.

Worked deduction: 'Most jurors agreed' — can you validly infer 'All jurors agreed'?

Invalid. 'Most' guarantees only a majority, not the whole; some jurors may have disagreed. Strengthening a quantifier from 'most' to 'all' is never valid.

Compare validity versus soundness: which one does BCAT Deduction actually test?

BCAT tests VALIDITY only—whether the conclusion follows from the given statements assumed true. Soundness (validity PLUS premises being actually true) is not at issue, because you take the premises as given.

What overall mindset best maximises accuracy on the BCAT Deduction section?

Be a strict, literal logician: assume the statements, ignore outside knowledge and plausibility, read quantifiers and conditionals precisely, and mark Valid only when a counterexample is genuinely impossible.