Truth-value assignments and truth-tables
Atomic sentences are treated as independent: what one is assigned never constrains another. Once an assignment fixes the atoms, the characteristic truth-tables fix the value of every compound, working from the smallest components up to the main connective.
Building a table
- With n distinct sentence letters there are 2n rows. Each row stands for the infinitely many assignments that agree on those letters.
- List the letters in the book's alphabetical order: plain letters first, then those with subscript 1, then 2…
- The first column alternates blocks of 2n−1 Ts and Fs, the next halves the block, and the last alternates single T and F.
- Write each component's column under its main connective. The column under the sentence's main connective is the answer.
A shortened truth-table shows one row only: the value of a sentence on one particular assignment.
Row pattern for three letters
| A | B | C |
|---|
A: 4 T then 4 F · B: 2 and 2 · C: alternating.
Truth-functional truth, falsity and indeterminacy
Other books say tautology, contradiction and contingent sentence. Every atomic sentence is indeterminate, and a compound's components don't have to share its status: A ⊃ (B ⊃ A) is truth-functionally true although its parts are indeterminate.
Shortened tables, systematically
To show a sentence is not truth-functionally true, look for one row that makes it false. Put F under the main connective, then write down whatever that forces: a false disjunction has two false disjuncts, a true conjunction two true conjuncts, a negation has the opposite value of what it negates, and a letter has the same value everywhere it occurs.
When a value doesn't force anything (a false biconditional, a true conditional), pick a case and carry on. If it leads to a clash, back up and try the next case. Only when every case clashes have you learned the sentence can't take that value. The book notes that this search is an NP-complete problem: in the worst case you have to try everything.
Which value forces what
| Sentence is | Forced? |
|---|---|
| ~ P true / false | P false / true |
| P & Q true | both true |
| P & Q false | choice: 3 cases |
| P ∨ Q false | both false |
| P ∨ Q true | choice: 3 cases |
| P ⊃ Q false | P true, Q false |
| P ⊃ Q true | choice: 3 cases |
| P ≡ Q true / false | choice: 2 cases each |
Start with the forced moves: they cost nothing.
Truth-functional equivalence
Show equivalence with one joint table whose main-connective columns are identical. Show non-equivalence with a shortened table: one row where they differ (the book boxes it). One row where they agree proves nothing.
All truth-functionally true sentences are equivalent to one another, and so are all truth-functionally false ones. Indeterminate sentences need not be.
"Unless" revisited
Chapter 2 symbolized "p unless q" as P ∨ Q. Here we can check that ~ P ⊃ Q and ~ Q ⊃ P are equivalent to it, so all three symbolizations are correct. Try them in the tester below.
Equivalences worth knowing
Truth-functional consistency
Sets are written with curly braces, and Γ (gamma) ranges over sets of sentences. The empty set ∅ has no members.
This time the shortened table does the positive job: one row where every member is true shows consistency. Inconsistency needs the full table, with at least one F in every row. A truth-functionally true member doesn't rescue an inconsistent set.
Asymmetry to remember
A single row is enough to show that something exists: a row making the sentence false, a row where two sentences differ, a row where all members are true. Showing that no such row exists takes the whole table.
Truth-functional entailment and validity
⊭ means "does not entail", and ⊨ P on its own means ∅ ⊨ P. The empty set entails exactly the truth-functionally true sentences.
Corresponding material conditional
Conjoin the premises in order, (…(P1 & P2) & … & Pn), and make that the antecedent of a conditional whose consequent is the conclusion. An argument with finitely many premises is valid exactly when its corresponding material conditional is truth-functionally true. The section proves this both ways.
Reading the table for validity
Look only at rows where every premise is T. If the conclusion is T in all of them, the argument is valid. A single row with all premises T and the conclusion F is a counterexample, and a shortened table showing it proves invalidity.
If no row makes every premise true, the argument is valid automatically: Chapter 1's special case 3, now in SL.
Truth-functional properties and consistency
Every property in this chapter can be restated as a question about consistency. That matters because Chapter 4's truth-trees test only for consistency, so these results let trees settle everything else too. Each result is proved straight from the definitions.
Γ ∪ {P} is the set containing P and every member of Γ. If P is already in Γ, it's just Γ.
Everything as (in)consistency
| P is TF-false | {P} is inconsistent |
| P is TF-true | {~ P} is inconsistent |
| P is TF-indeterminate | {P} and {~ P} both consistent |
| P, Q equivalent | {~ (P ≡ Q)} is inconsistent |
| Γ ⊨ P | Γ ∪ {~ P} is inconsistent |
| Argument valid | premises plus negated conclusion inconsistent |
Truth-functional property tester
Choose what to test, then type one sentence per line. The tester builds the joint truth-table, marks the rows that decide the answer, and restates the result in terms of consistency as in 3.6.
Practice sets for Chapter 3
Answers are worked out by the same truth-table code as the tester. "Find a row" practises shortened truth-tables: set the letters yourself, then check.
Discussion questions
- A truth-value assignment assigns values to infinitely many sentence letters, yet a table has finitely many rows. Why is that enough?
- Why does a single row suffice to show consistency but not inconsistency? Which other properties have the same asymmetry?
- If Γ is inconsistent, what does Γ entail? Connect this to Chapter 1's special cases.
- Is "truth-functionally true" the same as "logically true"? Find a logically true English sentence whose symbolization in SL is indeterminate.
- Why might the book bother proving that every property reduces to consistency, rather than just testing each one with a table?
Questions for our session
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