Introduction
Deductive logic is about truth-preservation: good deductive reasoning never leads from truths to a falsehood. Reasoning with that property is called valid. Inductive logic sets a weaker standard, that true starting points make the conclusion likely.
The book places its systems in history. Euclid axiomatized geometry; Aristotle's syllogistic logic handles arguments about categories but only with exactly two premises of restricted forms. The whodunit about the maid and the butler has three premises and compound sentences, so syllogistic logic can't handle it. The systems in this book descend from Frege, Hilbert and Russell, and are built from a few intuitive rules for deriving sentences rather than from axioms.
Why study it: it trains the skills of presenting and evaluating arguments, makes you notice how much the exact words matter ("Roberta will pass if she does the homework" versus "only if"), and builds comfort with abstract concepts.
Map of the book
| Ch. 2–5 | Sentential logic: the language SL, truth-tables, truth-trees, derivations |
| Ch. 6 | Metatheory of sentential logic |
| Ch. 7–10 | Predicate logic: the language PL, interpretations, trees, derivations |
| Ch. 11 | Metatheory of predicate logic |
Core concepts of deductive logic
Sentences with truth-values
The book deals only with sentences that are true or false, which means sentences that assert something. Questions, commands and exclamations are left out. A sentence can have a truth-value even if nobody will ever know it, like a claim about the weather on a day a thousand years ago.
Arguments
In standard form the premises are listed, a line is drawn, and the conclusion goes underneath. Spotting arguments in ordinary prose is a skill of its own: the conclusion can come first, last or in the middle, and a passage can contain plenty that isn't part of the argument.
Validity and soundness
Validity is about the relation between premises and conclusion, not about whether they're actually true. A sound argument always has a true conclusion. The definition even counts an unrelated pair of sentences as an argument, so that the tools of logic can show it's a bad one.
Sentences, pairs and sets
A sentence is logically true if it can't be false, logically false if it can't be true, and logically indeterminate otherwise. Logical truths and falsehoods hold by structure alone and say nothing about the world; most everyday sentences are indeterminate.
Two sentences are logically equivalent if one can't be true while the other is false. A set is logically consistent if all its members can be true together. A set logically entails a sentence if the members can't all be true while the sentence is false.
Entailment is more general than validity: an argument needs at least one premise, but the empty set can entail a sentence. It entails exactly the logical truths.
What validity rules out
For a valid argument, only one combination of truth-values is impossible.
| Premises | Conclusion | Valid argument? |
|---|---|---|
| all true | true | possible (and then it's sound) |
| all true | false | impossible |
| some false | true | possible |
| some false | false | possible |
Indicator expressions
They're signals, not proof: "so" and "for" have other jobs too.
Indicator highlighter
Paste or type a passage. Conclusion indicators are marked in blue and premise indicators in yellow. Then decide for yourself what the argument is and write it in standard form.
Special cases of logical concepts
These follow straight from the definitions, even when they feel wrong. Each one comes down to asking what is impossible.
- Logically true conclusion. The argument is valid whatever its premises, because the conclusion can never be false.
- Entailment of logical truths. Every set, including the empty set, entails every logically true sentence.
- Inconsistent premises. If the premises can't all be true, the argument is valid whatever its conclusion, but it can never be sound. Any argument with a logically false premise falls in this case.
- All logical truths are equivalent to one another.
- All logical falsehoods are equivalent to one another. Indeterminate sentences, of course, aren't all equivalent.
The one move behind every case
Validity fails only if there's a possible situation with true premises and a false conclusion. If either half of that situation is impossible on its own, the whole situation is impossible.
These same special cases come back in Chapter 3 for SL, with truth-tables to check them.
Practice sets for Chapter 1
New questions in the style of the 1.2E and 1.3E exercises. Do the book's starred exercises as well.
Discussion questions
- Give an argument that is valid but has a false conclusion. What must be true of its premises?
- Can an invalid argument have true premises and a true conclusion? Make one up.
- Why does the book count the bluebird "argument" as an argument at all? What would we lose with a stricter definition?
- "Every argument with inconsistent premises is valid." Does this show the definition of validity is too generous, or is it a feature?
- The definitions all rest on "possible". What sort of possibility is meant? Is it possible for water not to be H₂O?
- Why is it useful for the empty set to entail sentences? Which sentences does it entail?
Questions for our session
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