PH101 Logic handout Week 13
List of important terms and ideas
I’m using A, B, C . . . for sentences. P, Q, R . . . for predicates. A,B,C. . . for
sets.
Alternative denial Also known as the Sheffer stroke. “One or other of the
sentences is False.” A|B or A ↑ B. Equivalent to ¬(A ∧ B). All other
connectives can be written using only alternative denial.
Antecedent First half of a conditional. The “A” in A → B
Atomic sentence “A, B, C . . .” The smallest units of meaning in propositional logic. The building blocks of logic. Also called “propositional
variables” or “elementary sentences”
Biconditional “If and only if”, sometimes shortened to “Iff”. A ↔ B
Complement A or Ac . The set of things not in A
Compound sentence Sentence composed of several atomic sentences connected by logical connectives (A → B) ∧ (¬C ∨ B) This sentence is
in fact a conjunction. One of its conjuncts is a disjunction and
the other is a conditional.
Conclusion The final sentence of an inference. The end point of an argument
Conditional compound sentence of the form “If . . . then . . . ” e.g. A → B
Conditional Proof A useful proof rule. We assume “for the sake of argument” that A. If, from this, we can prove B, then we can conclude
A → B without the assumption, A.
Conjunct Either half of a conjunction
Conjunction An “and” sentence. A ∧ B
Consequent Second half of a conditional The “B” in A → B
Consistency 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
Constant A placeholder standing for some unspecified but determinate individual
Contingent A sentence whose truth-value 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.
Contradiction A sentence that is always false; regardless of the truth-values
assigned to the atomic sentences. Its negation is a tautology.
PH101 Logic handout Week 13
Counterexample 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 ∆ to conclusion σ is a model of ∆ ∪ {¬σ}
Disjunct Either half of a disjunction
Disjunction An “or” sentence. A ∨ B
Disjunctive syllogism A special case of tautological implication: gets
us from ¬A, A ∨ B to B
Domain 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”
Element a ∈ B: “a is an element of B”, “a is a member of B”
Existential generalisation A proof rule that allows us to deduce ∃xP x
from P x or from P a. The rule allows us to introduce an existential
quantifier
Existential quantifier “There exists” ∃
Existential specification A proof rule that allows us to conclude P α from
premise ∃xP x. This rule allows us to eliminate an existential quantifier
Independence A sentence σ is independent of a set of sentences ∆ when the
inference from ∆ to σ is invalid and so is the inference from ∆ to ¬σ
Inference The logical structure of an argument. An inference starts with
some premises and arrives at some conclusion
Interpretation 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
Intersection A ∩ B. The set of things in both A and in B
Invalid An inference is invalid if there exists a counterexample to it
Joint denial “Both sentences are false” A ↓ B. Equivalent to ¬(A ∨ B). Like
alternative denial, all connectives can be written using only joint
denial
Logical connective Those symbols used to connect atomic sentences into
more elaborate constructions. ∧, ∨, →, ↔ . . .
Logically False/True A sentence is logically false/true if it is false/true in
every interpretation. A bit like a contradiction/tautology but for
predicate logic.
Model An interpretation of a set of sentences that makes all those sentences True
PH101 Logic handout Week 13
Modus ponens A special case of tautological implication: gets us from
A, A → B to B
Modus tollens A special case of tautological implication: gets us from
A → B, ¬B to ¬A
Negation “Not”, “It is not the case that. . . ” ¬A
Predicate P, Q, R . . . a letter that stands for a some property. Read “P a” as
meaning “individual a has property P ”
Predicate logic the formal system that replaces the atomic sentences of
propositional logic with predicates, constants and variables
and augments the language with quantifiers
Premise The starting point of an argument or inference
Proof A list of sentences that starts with 0 or more premises and ends with
a conclusion
Proof rule Something that justifies a particular step in a proof
Propositional logic The formal system that consists only of atomic sentences and logical connectives
Quantifier Something like “for all x” ∀x or “for some x” ∃x
Reductio ad absurdum A special case of tautological implication: allows us to get from A → (B ∧ ¬B) to ¬A
Subset A ⊆ B: A is a subset of B if every element of A is also an element
of B
Tautology A sentence that is always true; regardless of the truth-values assigned to the atomic sentences. Its negation is a contradiction.
Tautological implication A proof rule that allows you to conclude B
from A1 , A2 . . . An if (A1 ∧ A2 . . . ∧ An ) → B is a tautology
Truth value assignment A Truth value assignment assigns to each atomic
sentence a Truth value
Union A ∪ B The set of things that are either in A or in B (or possibly both)
Universal generalisation A proof rule that gets you from P x to ∀xP x.
This rule allows us to introduce a universal quantifier. Handle with
care! Always read the small print
Universal quantifier “For all” ∀x
Universal specification A proof rule that gets you from ∀xP x to P x
or to P a or to P α. The rule that allows us to eliminate a universal
quantifier
Variable A placeholder standing for an individual
Valid 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
PH101 Logic handout Week 13
Logic books
Introductory books There is no recommended textbook for this course.
The notes, available from moodle, should give you all the information you
need. If you want a textbook, here are some suggestions. There are dozens
of introductory logic texts, I only mention a couple. I don’t have first hand
knowledge of all these texts, so I’m working on recommendations from others.
Be warned: You should still read the notes. None of these books
take exactly the same approach to things as we do in this course,
and the exam will test you on the way we’ve done things in the
lectures and lecture notes!
Antony Eagle. Elements of Deductive Logic. Available as PDF here: http:
//users.ox.ac.uk/ ~sfop0118/research.htm. 2008: A book that has the
advantage of being available for free. Might require a little too much mathematical sophistication.
Samuel Guttenplan. The Languages of Logic. Wiley-Blackwell, 1997: Apparently, this is the book used by some other London colleges for their introductory
logic courses.
Wilfrid Hodges. Logic. Penguin, 2001: A short introduction. Available cheap
on Amazon at the moment.
Greg Restall. Logic: An Introduction. Routledge, 2006: Bristol University’s
first year logic course uses this text. It covers semantic trees.
E.J. Lemmon. Beginning Logic. Hackett, 1965: The book I learned from.
Quaintly misogynistic examples like: “If she wears that dress, then she will
never find a husband”.
P. D. Magnus. For All X. Available as PDF here: http://www.fecundity.
com/logic/download.html: Another free logic textbook.
Peter Smith. An Introduction to Formal Logic. Cambridge University Press,
2003: Another logic book.
Paul Tomassi. Logic. Routledge, 1999: This book has been recommended
by several people.
Books for masochists If you really want something more than an introductory book can give you, you might want to try some of these.
Peter Cameron. Sets, Logic and Categories. Springer, 1998: The advanced
logic course (PH217/PH419) uses this book. Also covers set theory.
George Boolos and Richard Jeffrey. Computability and Logic. Cambridge
University Press, 1989: A classic but tremendously challenging book. Relates
logic and theoretical computer science.
Lou Goble, ed. Blackwell guide to Philosophical Logic. Blackwell, 2001: A
more philosophical approach. Introduces lots of extensions to the propositional/predicate logic stuff we do in PH101.
Colin Howson. Logic with Trees. Routledge, 1997: A nice, short book which
treats the basics of logic in an interesting way. Too dense to be an introductory
text.
Theodore Sider. Logic for Philosophy. Oxford University Press, 2010: Covers
much the same ground as the Goble book above, but is much shorter.