Formal Logic
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Propositional and predicate logic done symbolically: connectives, truth tables, natural deduction, quantifier rules and their restrictions, proof strategy, modal basics, and the soundness and completeness results. For students taking a first symbolic logic course.
- What makes an argument valid?
- It is impossible for all the premises to be true while the conclusion is false. Validity depends on the form of the argument, not on whether the premises happen to be true.
- Valid but unsound: what does one look like?
- An argument whose form guarantees the conclusion but whose premises are false. Validity survives any amount of falsehood in the premises.
- Can a valid argument have a false conclusion?
- Yes, if at least one premise is false. What validity rules out is only the combination of all true premises with a false conclusion.
- Semantic entailment
- A set of sentences entails a conclusion when every interpretation making all of them true also makes the conclusion true. It is defined by meaning, not by any set of rules.
- Syntactic derivability
- A conclusion is derivable from a set of premises when a finite proof of it can be built from those premises using only the rules of the system. By itself it says nothing about truth.
- Logical truth
- A sentence true under every interpretation. In propositional logic these are the tautologies; in predicate logic they are the sentences true in every model.
- Contingent sentence
- One that is true under at least one interpretation and false under at least one other. Most ordinary sentences are contingent.
- Consistency of a set of sentences
- A set is consistent when one single interpretation makes every member true at once. Consistency is a property of the whole set, not of any one member.
- Why does an inconsistent set entail everything?
- Because no interpretation makes all its members true, the condition for entailment is met vacuously: there is no case at all where the premises are true and the conclusion false.
- Logical form
- The pattern that remains when the specific content of a sentence is replaced by schematic letters. Formal logic judges arguments by form, so two arguments sharing a form stand or fall together.
- Object language vs metalanguage
- The object language is the formal language under study; the metalanguage is the language you use to talk about it. The turnstiles and the Greek letters belong to the metalanguage.
- Counterexample method for invalidity
- To show an argument invalid, describe one interpretation on which every premise is true and the conclusion false. A single such case settles the question.
- Vacuous truth
- A universal claim about an empty class is true because nothing falsifies it. Standard first-order logic makes all F are G true whenever nothing is F.
- Corresponding conditional of an argument
- The conditional whose antecedent is the conjunction of the premises and whose consequent is the conclusion. An argument is valid exactly when its corresponding conditional is a tautology.
- Sentence letter
- An atomic symbol standing for a complete declarative sentence with a truth value. Propositional logic never looks inside it.
- Truth-functional connective
- A connective whose output truth value is fixed by the truth values of its inputs alone. All five standard connectives are truth-functional.
- Negation
- Reverses truth value: the negation is true exactly when what it negates is false.
- Conjunction
- True only when both conjuncts are true. English and, but, however, although and yet all formalize as conjunction, because the extra suggestion they carry is not truth-functional.
- Inclusive disjunction
- True when at least one disjunct is true, including the case where both are. The wedge is always inclusive in standard logic.
- Express exclusive or with the standard connectives
- Say that one holds or the other but not both, or equivalently that the two differ in truth value.
- Material conditional
- False in exactly one case, true antecedent with false consequent, and true in the other three. It captures only the truth-functional core of if.
- Paradoxes of material implication
- Because the conditional is true whenever the antecedent is false or the consequent true, a false antecedent or a true consequent makes any conditional true, however unrelated the parts are.
- Biconditional
- True exactly when the two sides have the same truth value. It formalizes if and only if and is equivalent to a conjunction of two conditionals.
- Translate: P only if Q
- Only if introduces the consequent, so it becomes a conditional from P to Q. Q is the necessary condition.
- Translate: P unless Q
- Unless means if not, so it becomes a conditional from the negation of Q to P, which is equivalent to the disjunction of P and Q.
- Translate: neither P nor Q
- Both are denied, so it is the negation of the disjunction, equivalently the conjunction of the two negations.
- Not both P and Q vs both not P and not Q
- Not both denies only the combination, so at least one of them fails. Both not denies each one separately, which is a much stronger claim.
- Well-formed formula
- A string built by the formation rules: every sentence letter is a formula, and if two strings are formulas so are their negation, conjunction, disjunction, conditional and biconditional. Nothing else counts.
- Main connective
- The connective applied last as the formula is built, so it is the one governing the whole formula. It decides which rule applies to the formula in a proof.
- Scope of a connective
- The subformula a connective applies to. Parentheses fix it, and changing the scope usually changes the truth conditions completely.