Logic & Philosophy of Language — Quick Reference Notes

LOGIC

Logic: overlap between mathematics and philosophy; study of correct reason; formalizes natural languages via symbolic notation; clarifies vague/ambiguous language using logical symbolism.

Learn the symbolic alphabet; translate English into symbols. Example: "all humans are mortals" → \forall x\,(Human(x)\rightarrow Mortal(x)). This disambiguates meaning by symbolizing terms (e.g., Human → H, Mortal → M).

Active parts of logic function like verbs; use deductive component and operators: \land,\; \lor,\; \neg,\; \rightarrow (and others). These tools extract meaning and support deductive arguments.
Example (valid by meaning and form): "All bachelors are unmarried men; Tom is a bachelor; therefore Tom is unmarried." Expressed as: \forall x\,(Bachelor(x)\rightarrow Unmarried(x))\land Bachelor(Tom)\Rightarrow Unmarried(Tom).
Premises: reasons for the conclusion. Conclusion: what follows.
Validity: an argument is valid if its conclusion follows from the premises by meaning and form.
Soundness: if an argument is valid and the premises are true, then the conclusion is true.

Logic now handles qualifiers like \forall (all) and \exists (some); used to reason about existence and metaphysical questions; serves as a tool of scientific reasoning.

Inductive logic: uses language in its true form and derives meaning from propositions.
Deductive logic: translates language into universally usable formulas to test validity.
Validity condition: an argument is valid iff premises cannot all be true while the conclusion is false. Premises are the building blocks; if valid and premises true, then the conclusion is true (soundness).

Aristotle identified forms of arguments that are always valid; logic courses emphasize turning invalid arguments into valid ones and proving validity/invalidity.

Formal logic uses clear symbols; relation often shown as: "If I want to pass my classes, then I need to do well on all major assignments. I want to pass my classes. Therefore, I need to do well on all of the major assignments." Translated: p\rightarrow q,\ p\vdash q. (Modus Ponens)
Correct common form: p\rightarrow q,\ p\vdash q.
Incorrect form (invalid): p\rightarrow q,\ q\nvdash p. i.e., from q you cannot infer p.

Emerged in the mid-19th century (Linguistic Turn); study meaning in logic; how language relates to logic and truth.

Reference theories focus on what words refer to; Sense adds abstract content connecting word to reference via word → sense → reference.

The present King of France is bald: paradox shows reference grounds meaning but references may not exist in reality; reference can outrun sense in some cases.

Fiction challenges: formal language struggles with completely informal or fictional entities; grounding is problematic when no real-world referent.

Modality raises existence questions: would senses/references hold in another world identical to Earth? Deals with permanency and content; meaning can change over time.

Links to epistemology: relation between knowledge and linguistic identifiers; connotative vs denotative meaning; uttered vs implied meanings.

Paraphrase #3: work with paraphrase group; compile notes; share a Google Doc submission on Canvas; slides available on Canvas for review.