Skip to main content
note
This page was translated from the German original, partly by machine. Some passages may read awkwardly or contain inaccuracies. When in doubt, please read the original.

29. Propositional Logic - Symbolese

 

Summary​

After several episodes on categorical logic, this video circles back to a different family of propositions: truth-functional or propositional logic. Like categorical logic, it is concerned with the form or structure of arguments rather than their content. To capture that structure as clearly as possible, propositional logic uses its own artificial language, which Di Donato playfully calls Symbolese, reassuring viewers that it is far easier to learn than Latin.

Statements as Building Blocks​

Whereas the fundamental elements of categorical logic were terms (subject, predicate, middle), the fundamental elements of propositional logic are statements or propositions, hence the name. Statements can be simple (also called atomic), such as "fast foods tend to be unhealthy," or compound, such as "Chuck Berry plays the guitar and Fats Domino plays the piano," which joins two atomic statements with the operator and.

The label truth-functional captures the key idea: the truth value of a compound proposition depends entirely on the truth values of its component parts. The conjunction above is true only if both atomic statements are true. As always, the focus is on validity, a property of form rather than content. Form does not guarantee truth, but it preserves truth as long as we start with true premises. By representing only statement forms and argument forms, symbolic logic makes formal relationships visible and removes confusion caused by the wording of a proposition.

The Vocabulary of Symbolese​

The language has several kinds of symbols:

  • Statement variables: lowercase letters p through z stand for any proposition and are used to build general statement forms (e.g., p · q). Substituting specific content gives a substitution instance.
  • Statement abbreviations: uppercase letters A through Z abbreviate specific propositions. "I am Batman" can become B, no matter how long or complex the original sentence is.
  • The conclusion indicator (the tri-dot, which Di Donato calls the "Predator sight") translates as therefore or so.

The Operators​

Operators connect or modify statements and determine the overall truth value:

  • Negation (the tilde, ~) means not or "it is not the case that," reversing the truth value of a proposition.
  • The hypothetical or conditional (horseshoe ⊃ or arrow →) translates if p then q and expresses material implication.
  • The disjunction (the wedge ∨) joins disjuncts to express either p or q.
  • The conjunction (the dot · or ampersand &) joins conjuncts to express both p and q.
  • The biconditional (the tri-bar ≡ or double-sided arrow ↔) expresses material equivalence, p if and only if q.
  • Brackets (parentheses) are essential for removing ambiguity in compound statements with several operators.

How Each Operator Behaves​

Each operator has its own truth conditions:

  • A hypothetical is always true except when the antecedent is true and the consequent is false.
  • A disjunction is true as long as at least one disjunct is true; it is false only when both are false.
  • A conjunction is true only when both conjuncts are true.
  • A biconditional is true only when both components share the same truth value.

Di Donato also notes that some compound propositions are not truth-functional. With connectives like because, believes that, or hopes that, the truth of the whole is not fixed by the truth of the parts. "Joe failed because he didn't turn in his homework" requires a genuine causal link, and "Joe believed the sun revolves around the earth" is true so long as Joe holds that belief, regardless of whether the belief itself is true.

Translating English into Symbolese​

A large part of the lesson is learning which English words signal which operator. For hypotheticals, the word if introduces the antecedent and then the consequent. The tricky phrase only if points instead to the consequent, so "p only if q" is p ⊃ q. A necessary condition is the consequent; a sufficient condition is the antecedent. Phrases like implies, entails, given that, provided that, and on the condition that all point to the antecedent, while is implied by or is entailed by point to the consequent.

For disjunctions, watch for either…or, plain or, and the counter-intuitive unless ("p unless q" equals p ∨ q). Conjunctions are signaled not only by and and both…and but also by although, but, even though, however, nevertheless, and yet. The biconditional appears as if and only if, a necessary and sufficient condition, and just in case.

Worked Examples​

Di Donato closes with practice translations:

  • "The ringing of the bell is sufficient for the dog to start salivating" → R ⊃ D (the sufficient item is the antecedent).
  • "Mark watches TV unless Jim goes to the bank" → M ∨ J.
  • "It's not the case that both Sean and Deirdre are dancers" → ~(S · D).
  • "Neither Bill nor Jeff play football" → ~(B ∨ J), a negated disjunction.
  • "Jules likes pizza even though Luigi likes lasagna just in case either Moshe likes mozzarella or Henry likes Limburger" → (J · L) ≡ (M ∨ H), showing how brackets keep complex forms clear.

The episode ends by previewing the next topic: using these elements to examine the different types of syllogisms built from these propositions.