This is the webpage for the PH101 classes taught at LSE in 20142015. The convener of this course is John Worrall. John devised all the course materials except the quizzes, which were devised by Georgios Zouros, based on John's materials. Everything is hosted on LSE's teaching platform, Moodle. Therefore they are only accessible to registered students. During term time these pages will be updated weekly so please check back regularly. If you find any typos in the lecture notes, exercise sheets, or on these pages please let Alex know.
Some links will only become active later in the term so only contact us about broken links if you think you should already be able to see the hyperlinked files.
If you are not sure when your class is you should consult the official LSE Timetable.
The lecture notes for the course can be found on Moodle and are divided into 10 parts. You can access all of them below:
Although Slides are important, especially when you miss a lecture, they are not a substitute for the Lecture Notes! Nevertheless, you can access the slides for all lectures that have already been delivered below.
The course has not changed much in the last years so the exercises contained in most of the past exam papers are indicative to what you should expect from this year's exam. However, you should be careful that the structure of the final exam has changed in 2012 (the Mock Exam below follows the new format). Some past exams have solutions. The majority do not and will not be provided; however, we encourage you to come to us with any questions you may have.
In addition to the exercises in the Lecture Notes, the weekly assignments and the past exam papers, there are four extra sets of exercises available to you.
All class teachers hold weekly office hours during term time in LAK 1.02. Appointments can be booked via LSE4U by clicking on the button above. (NB: Georgios's office hours are walkin only)
Class Teacher  Office Hour  Location  Email Address 

Camilla Colombo  Tuesday 15:0016:00  LAK.1.02  C.F.Colomboatlse.ac.uk 
Goreti Faria  Tue 11:0012:00  LAK.1.02  M.G.FariaDaCostaatlse.ac.uk 
Alex Marcoci  Friday 14:0015:00 Friday 17:0018:00  LAK.1.02  A.Marcociatlse.ac.uk 
James Nguyen  Friday 11:0012:00 Friday 16:0017:00  LAK.1.02  J.Nguyen1atlse.ac.uk 
Georgios Zouros  Thursday 11:3013:30  LAK.1.02  G.Zourosatlse.ac.uk 
There are no prerequisites for this course. Students are not assumed to have any background in Logic, Mathematics or in Critical Thinking/Thinking Skills.
If you do find that you are struggling to understand something, it is important to sort it out quickly. Here are the steps you should take:
This Glossary has been put together by Seamus Bradley. You can download the .pdf version directly from Seamus's website, together with his recommendations for further reading (note, however, that Seamus is using a slightly different notation).
We are using p, q, r… for sentences, P, Q, R… for predicates and Γ, Δ, Θ for sets. I also use wedge (∧) to denote conjunction instead of ampersand (&).
Also known as the Sheffer stroke. “One or other of the sentences is False.” pq or p ↑ q. Equivalent to ¬(p ∧ q). All other connectives can be written using only alternative denial. 
First half of a conditional. The “p” in p → q 
"p, q, r…” The smallest units of meaning in propositional logic. The building blocks of logic. Also called “propositional variables” or “elementary sentences”

“If and only if”, sometimes shortened to “Iff”. p ↔ q 
Γ or Γ^{c}. The set of things not in Γ 
Sentence composed of several atomic sentences connected by logical connectives (p → q) ∧ (¬r ∨ q) This sentence is in fact a conjunction. One of its conjuncts is a disjunction and the other is a conditional. 
The final sentence of an inference. The end point of an argument 
compound sentence of the form “If …then …” e.g. p → q 
A useful proof rule. We assume “for the sake of argument” that p. If, from this, we can prove q, then we can conclude p → q without the assumption, p. 
Either half of a conjunction 
An “and” sentence. p ∧ q 
Second half of a conditional The “q” in p → q 
A sentence or set of sentences is consistent if they can all be True together. In propositional logic this means that there is a Truth value assignment that makes all the sentences True. In predicate logic this means there is a model of the sentences 
A placeholder standing for some unspecified but determinate individual 
A sentence whose truthvalue can be both True or False depending on the truth values of the atoms is contingent. It is a sentence that is neither a tautology nor a contradiction. 
A sentence that is always false; regardless of the truthvalues assigned to the atomic sentences. Its negation is a tautology. 
A counterexample to an inference is something that shows that the inference is invalid. In propositional logic a counterexample is a Truth value assignment that makes the premises True but the conclusion False. In predicate logic, a counterexample to the inference from premises Δ to conclusion σ is a model of Δ ∪ {¬σ} 
Either half of a disjunction 
An “or” sentence. p ∨ q 
A special case of tautological implication: gets us from ¬p, p ∨ q to q 
The Domain of an interpretation tells you what things the interpretation is about. If the domain were “Animals” then the quantifier “∀x” would be interpreted as meaning “for all animals” 
a ∈ q: “a is an element of q”, “a is a member of q” 
A proof rule that allows us to deduce ∃xPx from Px or from Pa. The rule allows us to introduce an existential quantifier 
“There exists” ∃ 
A proof rule that allows us to conclude Pα from premise ∃xPx. This rule allows us to eliminate an existential quantifier 
A sentence σ is independent of a set of sentences Δ when the inference from Δ to σ is invalid and so is the inference from Δ to ¬σ 
The logical structure of an argument. An inference starts with some premises and arrives at some conclusion 
An interpretation gives meaning to a sentence of predicate logic by assigning to each predicate a property and to each constant the name of an individual 
Γ ∩ Δ. The set of things in both Γ and in Δ 
An inference is invalid if there exists a counterexample to it 
“Both sentences are false” p ↓ q. Equivalent to ¬(p ∨ q). Like alternative denial, all connectives can be written using only joint denial 
Those symbols used to connect atomic sentences into more elaborate constructions. ∧ , ∨ , → , ↔ … 
A sentence is logically false/true if it is false/true in every interpretation. A bit like a contradiction/tautology but for predicate logic. 
An interpretation of a set of sentences that makes all those sentences True 
A special case of tautological implication: gets us from p, p → q to q 
A special case of tautological implication: gets us from p → q, ¬q to ¬p 
P, Q, R… a letter that stands for a some property. Read “Pa” as meaning “individual a has property P” 
the formal system that replaces the atomic sentences of propositional logic with predicates, constants and variables and augments the language with quantifiers 
The starting point of an argument or inference 
A list of sentences that starts with 0 or more premises and ends with a conclusion 
Something that justifies a particular step in a proof 
The formal system that consists only of atomic sentences and logical connectives 
Something like “for all x” ∀x or “for some x” ∃x 
A special case of tautological implication: allows us to get from p → (q ∧ ¬q) to ¬p 
Γ ⊆ Δ: Γ is a subset of Δ if every element of Γ is also an element of Δ 
A sentence that is always true; regardless of the truthvalues assigned to the atomic sentences. Its negation is a contradiction. 
A proof rule that allows you to conclude q from p_{1}, p_{2}…p_{n} if (p_{1} ∧ p_{2}… ∧ p_{n}) → q is a tautology 
A Truth value assignment assigns to each atomic sentence a Truth value 
Γ ∪ Δ The set of things that are either in Γ or in Δ (or possibly both) 
A proof rule that gets you from Px to ∀xPx. This rule allows us to introduce a universal quantifier. Handle with care! Always read the small print 
“For all” ∀x 
A proof rule that gets you from ∀xPx to Px or to Pa or to Pα. The rule that allows us to eliminate a universal quantifier 
A placeholder standing for an individual 
An inference is valid if there is no counterexample to it. That is, if the Truth of the premises guarantees the Truth of the conclusion 
If you think we are making use of other concepts which should be added to this glossary, let Alex know.
The table below lists some of the most important equivalences which you will have to make use of in this course. You can also use any of these equivalences as instantiations of the rule of Tautological Implication. Note that the notation is slightly different from your lecture notes: conjunction is here denoted by the wedge (∧) instead of the ampersand (&). If you have doubts about any of these equivalences we encourage you to construct the appropriate truth table.
p ∨ q ≡ q ∨ p  p ∧ q ≡ q ∧ p  ¬¬p ≡ p  (p ∨ q) ∨ r ≡ p ∨ (q ∨ r) 
(p ∧ q) ∧ r ≡ p ∧ (q ∧ r)  p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)  p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)  p → (q ∧ ¬ q) ≡ ¬p 
p ∧ (p ∨ q) ≡ p  p ∨ (p ∧ q) ≡ p  ¬(p ∧ q) ≡ ¬p ∨ ¬q  ¬(p ∧ q) ≡ ¬p ∨ ¬q 
p ∨ q ≡ ¬(¬p ∧ ¬q)  p ∧ q ≡ ¬(¬p ∨ ¬q)  ¬(p → q) ≡ p ∧ ¬q  p → q ≡ ¬p ∨ q 
p → (q → r) ≡ (p ∧ q) → r  p → q ≡ ¬q → ¬p  p → (q ∨ r) ≡ (p ∧ ¬q) → r  p ↔ q ≡ (p → q) ∧ (q → p) 
p ↔ q ≡ (¬p ∨ q) ∧ (¬q ∨ p)  p ↔ q ≡ (p ∧ q) ∨ (¬p ∧ ¬q)  ¬(p ↔ q) ≡ (p ∧ ¬q) ∨ (¬p ∧ q)  p ∧ q ≡ ¬(p → ¬q) 
(p → q) ∧ (p → r) ≡ p → (q ∧ r)  (p → r) ∧ (q → r) ≡ (p ∧ q) → r  (p → q) ∨ (p → r) ≡ p → (q ∨ r)  (p → r) ∨ (q → r) ≡ (p ∨ q) → r 
In PH101 you only get a glimpse of what logic is. If you wish to explore further there are many excellent logic books on the market, and even some free ones available online. However, in consulting these you should keep in mind that every book is taking a different approach to logic and, more importantly, uses a different notation. For the PH101 exam you are not required to consult any of the below: the exam will test you on the way we have done things in the lectures, classes and, most importantly, in the Lecture Notes!
Graeme Forbes. Modern Logic. Oxford University Press, 1994. The book I used in my very first logic course. Accessible and easy to use. Additional resources available here (includes, solutions, errata and software).
P.D. Magnus. forall x. Online. Introductory and very accessible textbook, similar in spirit to your Lecture Notes. Adapted to the Cambridge syllabus by Tim Button.
Keith Devlin. The Joy of Sets. Springer, 1993. Standard and approachable introduction to set theory. Do not read without the errata.
Theodore Sider. Logic for Philosophy. Oxford, 2010. Survey of several systems of logic with applications to philosophical problems. Currently used in PH217. Errata
Anita Feferman. From Trotsky to Gödel. A. K. Peters, 2000. Biography of Jean van Heijenoort, personal secretary of Trotsky, mathematical logician, editor and translator of several reference works in logic.
Apostolos Doxiadis and Christos Papadimitriou. Logicomix. Bloomsbury, 2009. Illustrated life of Bertrand Russell and the beginnings of modern logic. Short and entertaining.
Finally, when in doubt the first two ports of call should be the Stanford Encyclopedia of Philosophy (for philosophical problems) and Wikipedia (for technical problems). Despite the constant criticism of Wikipedia in academic circles, so far I have had only very good experiences with Wikipedia pages on mathematical and logical notions.
Many to thanks David Makinson, Seamus Bradley and James Nguyen for all of their excellent recommendations.
The Department offers an advanced logic course, PH217 Set Theory and Further Logic taught by Miklós Rédéi (Set Theory, Michaelmas Term) and David Makinson (Further Logic, Lent Term). In 20142015 David Makinson will teach both parts of the course. PH101 is not a prerequisite for taking PH217 but it does provide you with the necessary basics. However, be mindful that:
If you are still unsure whether you should enrol in PH217, I am happy to discuss it further with you (even if you are no longer my student). Of course, you are encouraged to (also) approach the course conveners.
In virtue of what are some symbols logical and others not? Logical symbols seem to be insensitive to the identity of objects. Alfred Tarski suggests that the logical symbols in an interpreted language are those whose meaning is invariant under permutations on the domain of objects.
What determines the meaning of a logical symbol? How about the rules governing its use in inferencemaking? Its introduction and elimination rules. Arthur Prior formulates introduction and elimination rules for a new connective, TONK, with devastating consequences: everything will be derivable from everything.
Lewis Carroll thinks the Tortoise challenged Achilles to produce the reason in virtue of which if she accepts the premises of a valid argument she is compelled to accept the conclusion.
Many classically valid inferences are questionable. But modus ponens seems to be a bedrock of reliable reasoning. Van McGee devises some plausible counterexamples.
FirstOrder Logic cannot express basic mathematical notions such as "finitude" and "countability." Higherorder logics are more expressive. Stewart Shapiro goes SecondOrder.
Nonmonotonic, intuitionistic and classical systems answer differently to the question: "Is argument X valid?" Does that mean that there is no unique answer? J.C. Beall and Greg Restall make the case.
There is a rich literature on all of these and I only direct you to the most influential paper or book on each topic and the most relevant section of the Stanford Encyclopedia of Philosophy . If you are interested in reading more, do contact me.
Three gods A, B, and C are called, in some order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yesno questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for "yes" and "no" are "da" and "ja," in some order. You do not know which word means which.
➥ Boolos calls this the Hardest Logical Puzzle Ever. Give it your best shot! Afterwards you can read Boolos's solution here. A more recent take on the puzzle is due to Rabern and Rabern with a proposed solution by Uzquiano. I also encourage you to read more of Smullyan's puzzles (see Further Readings).
I want to classify all subsets of {1, 2, 3, . . .} as small or large, such that any set with zero or one member is small; any union of two small sets is small; and a set is small if and only if its complement is large. Does there exist a classification scheme satisfying all three rules?
➥ Consider the following possible meanings small might be assigned: "finite," "not containing 1," and "containing at most one element from the set {1,2,3}." Do any of these satisfy all three rules? Read Eric Schechter's explanation of why such a scheme exists and consult Chapter 5 of his book for the technical details.
G. A. Cohen explains in the words of Alfred Tarski:
The German mathematician, David Hilbert came up with this thought experiment to help you grasp the depths of infinity:
Before asking Achilles about the foundations of deduction, the Tortoise challenged the hero to a race. This Zeno's commentary:
In our interpretations we keep referring to numbers 'being prime'. Here is a fascinating recent result on the distance between two primes due to Yitang Zhang.