PHI3130: Logic

Course Description

PHI3130 Logic is the study of what makes an argument valid — the structure that determines whether a conclusion actually follows from its premises, independently of whether the premises are true.

The course is offered at approximately five Florida institutions, including Florida Gulf Coast University, Florida State University, the University of Florida, the University of South Florida and the University of West Florida.

⚠ Two meaningfully different courses run under this number, and the difference is the single most important thing on this page.

The University of West Florida titles it Modern Logic, places it in the Department of History and Philosophy at 3 semester hours, and describes training and skills of modern symbolic logic and their application to the evaluation of arguments, covering propositional logic and predicate logic. Florida Gulf Coast University titles it simply Logic at 3 credits, describing an introduction to categorical and propositional logic, argumentative fallacies, and the structure of arguments.

UWF's is a formal logic course. It is symbolic, it builds a proof system, and it is closer in method to mathematics than to the rest of philosophy. FGCU's is a broader critical-reasoning course that includes the traditional syllogistic, the informal fallacies, and argument analysis in ordinary language alongside the formal material. Both are legitimate and useful; they are not the same course, and which one you want depends on why you are taking it.

What the two share is the discipline's founding insight, and it is a genuinely powerful one: validity is a property of form, not of content. "All A are B; all B are C; therefore all A are C" is valid whatever A, B and C stand for — and recognising that the reliability of an inference can be assessed without knowing anything about the subject matter is what makes logic a general-purpose tool rather than a branch of any particular field.

The second shared insight is the separation of validity from truth. An argument can be valid with false premises, and a set of true statements can be assembled into an invalid argument. Soundness requires both. Students find this counter-intuitive for about two weeks and then find it indispensable, because it isolates exactly where a disagreement lies: are we disputing the facts, or disputing whether the conclusion follows from them?

The third is that the material is unusually transferable. Logic underlies mathematical proof, computer science, law and any careful argument — and it is one of the few humanities courses whose content is directly and demonstrably applied elsewhere.

Learning Outcomes

Required Outcomes — common to both versions

• Identify arguments in ordinary prose and distinguish them from non-argumentative writing.
• Identify premises and conclusions and reconstruct an argument in standard form.
• Distinguish deductive from inductive reasoning and apply the appropriate standard to each.
• Distinguish validity, soundness, strength and cogency.
• Identify unstated premises and supply them charitably.
• Translate ordinary-language statements into a formal notation.
• Analyse propositional logic — the connectives, well-formed formulas and their semantics.
• Construct and interpret truth tables, and use them to test validity, consistency and equivalence.
• Recognise the standard valid and invalid argument forms, including modus ponens, modus tollens, and the fallacies of affirming the consequent and denying the antecedent.
• Evaluate arguments encountered in real contexts and explain precisely why one fails.

Additional outcomes — the formal (symbolic) version

• Construct natural deduction proofs in propositional logic using inference and replacement rules.
• Apply conditional and indirect proof.
• Translate into and out of predicate logic, using quantifiers, predicates and relations.
• Construct proofs in predicate logic, applying the quantifier rules correctly.
• Demonstrate invalidity by counterexample or interpretation.
• Explain soundness and completeness as properties of a proof system, at least informally.

Additional outcomes — the broader (critical reasoning) version

• Analyse categorical propositions and the traditional syllogism.
• Test syllogisms for validity using Venn diagrams and the rules.
• Identify and explain the informal fallacies in real argument.
• Analyse inductive arguments — generalisation, analogy, causal reasoning — and assess their strength.
• Apply basic probability reasoning and identify common probabilistic errors.
• Analyse definitions and the role of ambiguity and vagueness in argument.

Optional Outcomes

• Apply modal logic — necessity and possibility.
• Explain set theory and its relationship to logic.
• Explain metatheory, decidability and the incompleteness results at an introductory level.
• Apply logic to computing — Boolean algebra, circuits, and logic programming.
• Explain non-classical logics in outline.
• Analyse legal or scientific reasoning as applications.
• Use proof-checking software.

Major Topics

Required Topics

• Arguments and their anatomy. What an argument is, and the practical skill of recognising one in prose that does not announce itself; premise and conclusion indicators, and the cases where there are none; reconstructing an argument in standard form, which is the foundational exercise; enthymemes — arguments with a suppressed premise — and the principle of charity, which requires supplying the premise that makes the argument strongest rather than weakest; distinguishing arguments from explanations, descriptions and assertions.
• The central distinctions. Deductive and inductive, and the different standards each is held to; validity as the impossibility of true premises with a false conclusion, stated precisely; soundness as validity plus true premises; strength and cogency for inductive arguments; the recurring student difficulty that a valid argument can have false premises and a false conclusion, which is exactly right and takes a fortnight to internalise; the practical payoff — separating a factual disagreement from a logical one.
• Propositional logic. Simple and compound statements; the connectives — negation, conjunction, disjunction, the conditional and the biconditional — with their truth conditions; the material conditional and the well-known ways it diverges from the ordinary-language "if… then", which is worth understanding rather than glossing; translation from English, which is where most of the real difficulty lies, since ordinary language is ambiguous and the formalisation is a judgement; well-formed formulas and scope; truth tables — constructing them, and using them to test validity, tautology, contradiction, consistency and equivalence; the standard forms and the standard traps — modus ponens and modus tollens against affirming the consequent and denying the antecedent, which are the two most common invalid patterns in real argument.
• Evaluating real arguments. Applying the machinery to editorials, advertisements, policy claims and everyday reasoning; the discipline of identifying precisely where an argument fails rather than merely rejecting its conclusion; the distinction between a bad argument and a false conclusion, since a true claim can be badly argued and it is worth being able to say so.

Required Topics — the formal (symbolic) version

• Natural deduction in propositional logic. Rules of inference — modus ponens, modus tollens, hypothetical and disjunctive syllogism, simplification, conjunction, addition, constructive dilemma; rules of replacement — De Morgan's, commutation, association, distribution, transposition, material implication, exportation, tautology, double negation; constructing a proof, which is the course's central activity and is a genuine skill: there is no algorithm, and finding a derivation requires working forwards from the premises and backwards from the goal at the same time; conditional proof and indirect proof (reductio), which make many otherwise intractable derivations straightforward; proving theorems from no premises.
• Predicate logic. Why propositional logic is insufficient — it cannot represent the internal structure that makes "All humans are mortal; Socrates is human; therefore Socrates is mortal" valid; predicates, individual constants and variables; the universal and existential quantifiers; translation, which is substantially harder here — the interaction of quantifiers with conditionals and conjunctions is the standard sticking point, and "all A are B" takes a conditional while "some A are B" takes a conjunction for reasons worth understanding rather than memorising; scope and binding; multiple and nested quantifiers, and the fact that order matters — "everyone loves someone" and "someone is loved by everyone" are different claims; relations and identity; the quantifier rules for proof and their restrictions, which exist to block invalid inferences and are the most error-prone part of the course; demonstrating invalidity by constructing an interpretation.
• Metatheoretical ideas, usually informally. What it means for a proof system to be sound (it proves only valid things) and complete (it proves all of them); decidability, and the fact that propositional logic is decidable and predicate logic is not; a brief and honest gesture at the incompleteness results, with the caution that they are among the most misrepresented results in mathematics and that a serious treatment requires a further course.

Required Topics — the broader (critical reasoning) version

• Categorical logic. The four categorical propositions (A, E, I, O) and their standard forms; translating ordinary sentences into them; the square of opposition and the immediate inferences — conversion, obversion, contraposition; Venn diagrams for propositions and for syllogisms; the categorical syllogism, mood and figure, and the rules for validity; the existential import question, which is where the traditional and modern treatments diverge; the historical point that this system was essentially the whole of logic for two thousand years, which is worth knowing and also worth knowing was superseded.
• Informal fallacies. Fallacies of relevance — ad hominem in its several forms, appeal to force, appeal to pity, appeal to the people, straw man, red herring, appeal to inappropriate authority; fallacies of presumption — begging the question, complex question, false dichotomy, suppressed evidence; fallacies of ambiguity — equivocation and amphiboly; fallacies of weak induction — hasty generalisation, false cause, slippery slope, weak analogy; ⚠ and the important qualification a good course supplies: fallacy labels are diagnostic tools, not conversation-enders. Not every appeal to authority is fallacious — deferring to genuine expertise is normally rational — and the skill is explaining why a particular instance fails, not naming it in Latin. Fallacy-spotting used as a rhetorical weapon is itself a poor habit the course should discourage.
• Inductive reasoning. Generalisation from samples and what makes a sample adequate; argument from analogy and the criteria for assessing it; causal reasoning — Mill's methods, and the distinction between correlation and causation treated properly rather than as a slogan, including confounding and reverse causation; probability — the basic rules, conditional probability, and the common errors, particularly the conjunction fallacy and base rate neglect, both of which are well documented and both of which most people commit; statistical reasoning and how statistics mislead.
• Language, meaning and definition. The functions of language beyond the informative; ambiguity and vagueness, and the difference; emotive language and its effect on argument; definitions — their types and purposes, the criteria for a good one, and persuasive definition as a rhetorical move that smuggles a conclusion into a premise; the practical skill of noticing when a dispute is verbal rather than substantive, which resolves a surprising number of disagreements.

Optional Topics

• Modal logic — necessity, possibility and possible worlds.
• Set theory and its relationship to logic (see MHF3202).
• Metatheory — soundness and completeness proofs, decidability, Gödel's results.
• Boolean algebra, digital circuits and logic in computing.
• Non-classical logics — intuitionistic, many-valued, fuzzy, relevance.
• The history of logic from Aristotle through Frege to the twentieth century.
• Legal and scientific reasoning as applications.
• Automated proof checking and theorem proving.

Resources & Tools

• Introduction to Logic by Irving Copi, Carl Cohen and Kenneth McMahon — the long-standing standard, covering both the categorical and symbolic material; the likeliest assignment in a broad-scope section.
• A Concise Introduction to Logic by Patrick Hurley — the most widely adopted undergraduate text, exceptionally clear, with an enormous exercise bank and good software support.
• The Logic Book by Bergmann, Moor and Nelson — more rigorous and formal; suited to a symbolic-logic section.
• Symbolic Logic by Copi; Language, Proof and Logic by Barwise and Etchemendy — the latter comes with the Tarski's World and Fitch software and is excellent for learning to build proofs with immediate feedback.
• Free and genuinely good:
  • forall x by P.D. Magnus and its several institutional adaptations — a free, complete, well-regarded open-access logic textbook, used at many universities. If your course does not assign a text you can afford, this is the answer.
  • Carnap (carnap.io) — free browser-based proof software with automatic checking, which is transformative for learning natural deduction because the feedback is immediate.
  • The Stanford Encyclopedia of Philosophy for the conceptual and historical entries; the Internet Encyclopedia of Philosophy's fallacy entries.
  • Logic proof checkers — several free ones exist; a checker that tells you a step is invalid is worth more than a solutions manual, because it makes you find the error.
• ⚠ The study method that works, and it is not negotiable: do the exercises. Logic is learned the way mathematics is learned — by working problems until the patterns are automatic. Reading a completed proof produces the sensation of understanding and almost none of the ability, and the gap between the two becomes obvious in an examination where you are given a blank page. Twenty proofs a week is not excessive.
• Professional context: the Association for Symbolic Logic for students continuing in the field; logic is also central to the LSAT, and students preparing for law school should know that this course is the most direct academic preparation available for the reasoning sections.

Career Pathways

Logic is not a job and it is one of the most portable skills in a humanities education, with several unusually direct applications.

• Law (SOC 23-1011) — the most direct application. Legal reasoning is argument reconstruction and validity assessment, and the LSAT's logical reasoning and analytical reasoning sections test precisely this material. Philosophy majors post among the highest average LSAT scores of any field, and logic is a substantial part of why. Florida has eleven law schools.
• Software Developers and Computer Scientists (SOC 15-1252, 15-1221) — logic is foundational to computing: Boolean algebra in circuits, propositional and predicate logic in program specification and verification, and the direct lineage from predicate logic to database query languages and to logic programming.
• Mathematicians and Statisticians (SOC 15-2021, 15-2041) — proof is applied logic; see MHF3202, the mathematics department's version of much of this material.
• Data and business analysts (SOC 13-1161, 15-2051) — query construction is applied predicate logic, and the inductive and probabilistic reasoning content is directly relevant.
• Information Security Analysts (SOC 15-1212) — formal reasoning about system properties and about what an attacker can infer.
• Technical writers, editors and journalists (SOC 27-3042, 27-3041, 27-3023) — the ability to identify what an argument actually claims and where it fails is the craft.
• Policy analysts and consultants (SOC 19-3094, 13-1111).
• Teaching (SOC 25-2031, 25-1126); graduate study in philosophy, mathematics, computer science, linguistics or law.
• Any role requiring the evaluation of arguments, which is most professional work and none of the job titles.

⚠ The practical advice, and it is specific: if you are considering law school, take the symbolic version and take it early. The formal reasoning transfers directly to the LSAT's analytical sections, the transfer is well documented, and doing it two years before you sit the test is considerably more useful than a commercial preparation course three months before.

And for computing students: take it even if it is not required. The predicate logic content is the same material that underlies database queries, program specification, type systems and automated reasoning, and a computing graduate who understands quantifier scope has an advantage that shows up in unexpected places.

Special Information

⚠⚠ Two different courses — check which one your institution runs

	Formal / symbolic (UWF, "Modern Logic")	Broad / critical reasoning (FGCU, "Logic")
Covers	Propositional and predicate logic, symbolic translation, natural deduction proofs	Categorical and propositional logic, informal fallacies, argument structure
Feels like	Mathematics — notation, rules, proof construction	Philosophy and critical thinking — analysis of real arguments
Best for	Philosophy, mathematics, computing, law school preparation	General critical reasoning, a broader audience
Difficulty	Steeper, and cumulative — falling behind is costly	More evenly distributed

Both are 3 credits and both transfer under the same number. ⚠ But a student who took the broad version and transfers into a programme where PHI3130 is the symbolic course will not have done natural deduction or predicate logic — and will meet the gap in any later course that assumes them, which in philosophy and computing is several. The reverse student has done no fallacy analysis.

Read the description rather than the title, and keep the syllabus. Note also that many institutions carry BOTH — a lower-division critical thinking or informal logic course and an upper-division symbolic logic course at different numbers — so check what else is available before assuming this is your only option.

⚠ Logic in two departments — this course and MHF3202

Logic is taught in philosophy and in mathematics, and the two versions overlap substantially. Florida carries both: PHI3130 in philosophy, and MHF3202 Sets and Logic in mathematics.

	PHI3130	MHF3202
Department	Philosophy	Mathematics
Prerequisite	Typically none	Calculus II at UWF
Emphasis	Argument evaluation; logic as a tool for reasoning	Set theory and proof technique; preparation for upper-division mathematics

The material genuinely overlaps — propositional and predicate logic appear in both — but the purposes differ. Philosophy asks you to evaluate arguments; mathematics asks you to construct proofs about mathematical objects. ⚠ Check whether your programme accepts one for the other — some do, many do not, and a mathematics major generally needs the mathematics version because it also teaches the set-theoretic vocabulary the rest of the curriculum uses.

Prerequisites and position in the curriculum

Neither UWF nor FGCU lists a prerequisite for PHI3130, which is typical and is one of the course's virtues: it is genuinely open.

No mathematics is required, and this is worth saying because students see the symbols and assume otherwise. Symbolic logic uses no arithmetic — it is a system of rules applied to symbols, and students who dislike mathematics frequently do well at it while some who are strong at mathematics find the translation from English harder than the formal work. What it requires is care, patience and attention to detail.

PHI3130 is an upper-division course despite assuming nothing, normally taken at any point after the first year. It is typically required for the philosophy major and is a common elective for mathematics, computer science, pre-law, linguistics and English students. It is the most useful preparation for ethical theory, philosophy of mind and epistemology, all of which turn on arguments whose structure is much easier to assess with logic in hand.

Course format and workload

Taught as a lecture with extensive problem work, frequently with in-class exercises. Assessment is overwhelmingly problem-based — problem sets and examinations consisting of translations, truth tables and derivations — with little or no essay writing in the symbolic version.

Expect five to nine hours a week outside class, and the distribution matters more than the total: steady daily work beats concentrated effort, because the skill is procedural and builds by repetition.

⚠ This is the most cumulative course in a philosophy curriculum, and it is where students most often come to grief. Predicate logic proofs assume propositional proofs, which assume the rules, which assume translation. A student who does not understand week four cannot do week eight, and unlike a reading-based course there is no way to catch up by working harder at the end. If you are lost, get help in the week it happens. Instructors expect this and office hours in a logic course are unusually productive, because a specific confusion can usually be fixed in ten minutes.

⚠ What students find hardest, in order.

• Translation from English, which is harder than the formal work because ordinary language is ambiguous and the formalisation is a judgement. "Unless", "only if", "if and only if" and the several senses of "or" cause the most trouble, and it is worth learning them explicitly rather than by feel.
• Finding a proof. There is no algorithm, and beginners stare at premises with no idea where to start. The technique is to work backwards from the conclusion and forwards from the premises simultaneously, and to recognise which rule produces the shape you need. It becomes pattern recognition with practice and feels impossible before it does.
• Quantifier rules and their restrictions, which are fiddly and which exist to block invalid inferences — the restrictions are the point rather than an annoyance.
• Accepting that validity ignores content. Students resist calling an argument valid when they think the conclusion is false, and separating those judgements is the course's central conceptual demand.

Articulation and transfer

PHI3130 carries the same SCNS number across Florida public institutions and SCNS equivalency governs transfer of the credit. As an upper-division course it does not appear in A.A. programmes, though most Florida state colleges carry a lower-division critical thinking or introductory logic course that transfers separately and that may or may not satisfy the same requirement.

⚠ The scope divergence above is the substantive transfer issue, and it is invisible on a transcript. Keep the syllabus, particularly if the course is meant to satisfy a philosophy major requirement or a computing programme's logic requirement, since those generally assume the symbolic content.

AI Integration

Logic has an unusually direct and interesting relationship with these tools, in three separate ways.

Where the tools help a student. Explaining a rule or a concept — the difference between conditional and indirect proof, why the material conditional behaves as it does — which is genuinely useful. Checking a translation you have already attempted. Generating practice problems, which is valuable in a course where problem volume is the study method. And explaining a step in a proof you have found but do not understand.

⚠ Where they fail, and the failure here is unusually instructive.

Generated proofs are frequently invalid, and confidently so. A model asked for a natural deduction derivation will produce something that looks exactly like a proof — numbered lines, cited rules, a conclusion — and contains steps that do not follow from the rules cited. The formatting is right and the logic is wrong.

And this is the pedagogically valuable part: you can check. Logic is one of the very few subjects where correctness is mechanically verifiable — a proof either follows the rules or it does not, and a free proof checker such as Carnap will tell you in seconds. Running a generated proof through a checker is one of the most instructive exercises available in this course, because it demonstrates concretely, in a domain where the answer is not a matter of opinion, that fluent output and correct output are different things.

Translation errors are subtle. Formalising "unless", scope ambiguities, and quantifier order are exactly the judgements that go wrong, and a wrong translation produces a formally impeccable derivation of the wrong claim.

The subject-matter connection, which belongs in this course. Logic is foundational to the symbolic tradition in artificial intelligence — automated theorem provers, logic programming, formal verification and knowledge representation are applied logic, and they are sound in a way current language models are not: a theorem prover's output is guaranteed by construction.

Language models work differently, and this course gives a student the vocabulary to say how. They generate probable continuations rather than derive conclusions. There is no validity-preserving step anywhere in the process, which is why they can produce an invalid proof with complete confidence — confidence and correctness are unconnected in the architecture. A student who has spent a term on the difference between a valid inference and a plausible one is unusually well placed to understand that, and to explain it to someone who has not.

The current research direction is worth knowing: combining the two — using a language model to propose and a formal system to verify — which is an explicit acknowledgement that generation and validation are different operations and that the second is what logic provides.

Academic integrity. Read your instructor's policy. The point specific to this course: the problem sets build a procedural skill that examinations test on a blank page, and in a cumulative subject the deficit compounds weekly rather than staying local. Generated solutions to logic problems are also, conveniently, the easiest kind of academic dishonesty to detect — because a student who cannot reproduce a derivation they submitted is apparent in about thirty seconds of conversation.