Državna Matura Logic common mistakes
Državna Matura Logic · Understand the traps and how to avoid them.
When asked to define a pojam, the candidate lists examples or properties instead of stating the genus and differentia specifica, e.g. defining "trokut" by drawing three kinds of triangles or saying "ima tri stranice i tri kuta" without naming the broader class (lik u ravnini) and the specific difference (omeđen trima dužinama).
Everyday definitions work ostensively (by pointing at instances), and textbooks introduce terms through examples, so candidates never practise the two-part structure. Under time pressure the example list feels safer because it is easy to produce, and the candidate cannot tell that a list of properties is not a definition at all - it is only a description.
What to do instead
Before writing, force the sentence into the frame "X je [rodni pojam] koji [specifična razlika]". Check two things: the genus must be the nearest broader class (not "stvar" or "pojam"), and the difference must apply to all and only the definiendum. Then run the substitution test: swap definiens for definiendum in a sentence and see whether meaning is preserved. Practise with 20 terms (kvadrat, pravednost, broj, živo biće) and mark for each the genus and the difference separately.
The candidate treats a definition that is too wide or too narrow as correct - e.g. accepting "Ptica je životinja koja leti" (too narrow: no penguins; too wide: bats fly) or "Čovjek je društveno biće" (too wide: ants) - and cannot diagnose which direction the definition fails.
Candidates check definitions only against typical examples (vrabac, golub) and never against counterexamples at the edges. The exam often supplies exactly those edge cases, and without a habit of testing both directions the student judges by familiarity rather than by extension of the term.
What to do instead
For every proposed definition, deliberately construct two test objects: one that satisfies the definiens but is not the definiendum (catches too-wide), and one that is the definiendum but fails the definiens (catches too-narrow). Write the verdict as "preširoka / preuska / preširoka i preuska". Keep a personal list of classic edge cases: pingvin, kit, slijepi miš, šišmiš, noj, umjetna inteligencija.
The candidate confuses logical relations between suds - treating kontrarnost (A and E: cannot both be true, can both be false) as identical to kontradikcija (A and O: exactly one true, one false) - and then draws wrong inferences about truth values in the square of opposition.
The Croatian terms kontrarno, kontradiktorno, subkontrarno and subalterno sound alike, and the square of opposition is usually memorised as a picture without the truth-table meaning behind each side. Students who memorise the diagram but not the "can both be false / cannot both be true" criterion collapse the two relations under exam pressure.
What to do instead
Rebuild the square from the truth conditions: for "Svi S su P" (A), "Nijedan S nije P" (E), "Neki S su P" (I), "Neki S nije P" (O), write for each pair whether they can both be true and whether they can both be false. Memorise the one-line tests: kontrarno = ne mogu oba biti istinita; subkontrarno = ne mogu oba biti lažna; kontradiktorno = uvijek različite vrijednosti; subalterno = istinitost općeg povlači istinitost posebnog, ne obrnuto. Then drill with concrete sentences about students and athletes.
The candidate evaluates a compound formula by reading it left to right, ignoring operator precedence - reading ¬p ∧ q as ¬(p ∧ q), or p ∨ q ∧ r as (p ∨ q) ∧ r - and so assigns the wrong truth value or builds the wrong truth table.
In ordinary language "nije... i..." is ambiguous, and in mathematics classes the student mostly met simple two-part formulas. Precedence (¬ before ∧ before ∨ before → before ↔) is a formal convention that is rarely drilled, so the eye follows the order of words in the sentence instead of the order of binding.
What to do instead
Adopt a fixed order and say it aloud: negacija veže najjače, zatim konjunkcija, pa disjunkcija, pa implikacija, a bikondicional najslabije. Whenever a formula has three or more operators, first bracket it fully (¬p ∧ q becomes (¬p) ∧ q; p → q → r is not well-formed without brackets - check whether the task intends right association). Then evaluate bottom-up, one operator per line, and keep a column for each subformula in the truth table rather than one final column.
The candidate treats material implication p → q as if it asserted a causal or content connection, marking it false whenever p is false ("Ako je 2+2=5, onda je Zagreb glavni grad" judged false), and mis-evaluating the vacuous cases of the truth table.
In everyday Croatian, "ako... onda..." suggests a real link between antecedent and consequent, so the false-antecedent rows feel wrong. The truth-functional definition is a stipulation, and students who never saw the rationale memorise the table but revert to intuition when the content is absurd.
What to do instead
Learn the definition through its only false case: implikacija je lažna samo kad je antecedent istinit, a konsekvent lažan. Test with a promise analogy: "Ako položiš ispit, kupit ću ti bicikl" is broken only if you pass and get no bicycle; if you fail, the promise is not broken. Then do ten exercises with absurd contents (neparne rečenice, lažni antecedenti) until the evaluation is mechanical, and note that p → q is equivalent to ¬p ∨ q.
When translating a quantified sentence, the candidate uses the wrong quantifier or the wrong connective - rendering "Neki studenti polažu logiku" as ∃x(S(x) → L(x)) instead of ∃x(S(x) ∧ L(x)), or "Svi psi laju" as ∀x(P(x) ∧ L(x)) instead of ∀x(P(x) → L(x)).
The pattern is counter-intuitive at first: universal statements take →, particular statements take ∧. Students translate word by word ("svi" = ∀, "i" absent, so any connective will do) and do not see that the wrong choice changes the truth conditions - ∀x(S(x) ∧ L(x)) says everything in the domain is a student who passes.
What to do instead
Memorise the two templates and the reason for them: ∀x(S(x) → L(x)) because the universal claim says nothing about non-students, so non-students must not falsify it; ∃x(S(x) ∧ L(x)) because the particular claim asserts that some object is both a student and a passer. Test by translating back into Croatian and checking whether the formula claims too much. Also drill negated quantifiers: ¬∀x S(x) ≡ ∃x ¬S(x) and ¬∃x S(x) ≡ ∀x ¬S(x).
The candidate judges an argument valid because the conclusion is true or the premises are true, or invalid because the conclusion is false - conflating valjanost with istinitost and failing to separate the two when the exam uses a sound argument with a true conclusion as a distractor.
In everyday argumentation we usually endorse conclusions we believe, so truth and validity blur together. Textbooks emphasise that validity is about form, but under exam conditions the candidate reads the content, recognises a familiar true statement, and stops there.
What to do instead
Apply the counterexample test explicitly: assume all premises true and ask whether the conclusion could still be false. If yes, invalid; if no, valid. Write the form in letters (Modus ponens: p → q, p, therefore q) and ignore what the letters stand for. Practise with deliberately false-premise but valid arguments and true-conclusion but invalid ones (affirming the consequent) until the two dimensions are clearly separate in your notes.
The candidate classifies an argument by its surface wording rather than its form, so "Ako pada kiša, cesta je mokra. Cesta je mokra. Dakle, pada kiša" is accepted as valid because it sounds like modus ponens, and other cases of affirming the consequent or denying the antecedent pass unnoticed.
Modus ponens and modus tollens are the two forms students remember, and both contain an implication, so any argument with "ako... onda..." plus a second premise and a conclusion feels familiar. The difference lies in whether the second premise affirms the antecedent or the consequent - a detail that is skipped when reading for content.
What to do instead
Reduce every argument to symbols before judging: label the antecedent and consequent, then write premise 2 as either p or q, or as ¬p or ¬q. Build a four-row table of the valid and invalid patterns: p → q, p ⊢ q (valid); p → q, ¬q ⊢ ¬p (valid); p → q, q ⊢ p (invalid, affirming the consequent); p → q, ¬p ⊢ ¬q (invalid, denying the antecedent). Recite the invalid two as "potvrđivanje konsekventa" and "nijekanje antecedenta" and give each a real-world example.
The candidate mistakes a fallacy label for a refutation, answering "to je ad hominem" without explaining why the move fails, or mislabels arguments - calling any appeal to authority "ad hominem", any emotional appeal "slamka", or treating a valid appeal to expert testimony as a fallacy.
Fallacy names are taught as a vocabulary list with one-line definitions, and exams often ask only for the name, so students never practise the diagnostic step. Similar-sounding Latin labels (ad hominem, ad verecundiam, ad populum) get attached to the nearest emotional feature of the passage.
What to do instead
For each fallacy, learn the structure it violates rather than its name: ad hominem attacks the arguer instead of the argument; ad verecundiam cites authority outside its competence or against overwhelming evidence; slamka substitutes a distorted version of the opponent's claim. For every exercise, write one sentence of the form "argument je pogrešan jer [premisa ne podupire zaključak]", then add the label. Distinguish legitimate appeals to relevant expertise from fallacious ones.
In modal and non-classical items, the candidate reads □p as "p is true" and ◇p as "p might be true in this world", collapsing necessity into actuality and possibility into mere uncertainty, and so judges □p → p and p → □p as equally acceptable or equally suspect.
The boxes and diamonds are new notation, and students map them onto the only modal words they know from ordinary speech ("mora" and "može"), which are used loosely for probability and permission. The possible-worlds semantics with an accessibility relation is usually presented briefly, and the distinction between truth at a world and truth at all accessible worlds is never made operational.
What to do instead
Fix the semantics in one sentence each: □p is true at world w when p is true at every world accessible from w; ◇p is true at w when p is true at some accessible world. Then test the key formulas on a two-world model you draw yourself: □p → p holds (reflexivity), p → □p fails (draw a world where p is true but an accessible world where it is false). Also distinguish alethic modality from deontic and epistemic uses, and note that in intuitionistic logic ¬¬p → p fails while p → ¬¬p holds.