Phil 155 Final Exam Terms

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Ambiguity

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-a single sentence with two or more meanings
-FOL does not allow for ambiguity

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Analytical consequence or "logical consequence" or "first order consequence"

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A sentence is a logical consequence of a
set of sentences if it is impossible for that sentence to be false when all the sentences in the set
are true

a = c is a logical consequence of a=b ^ b=c

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Antecedent

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The antecedent of a conditional is its first component clause.
In P ! Q, P is the antecedent and Q is the consequent.

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Argument

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The word argument is ambiguous in logic.
1. a sequence of statements in which the conclusion follows from premise
2. variables (only in atomic wffs); LeftOf(x,a) then x and a are the arguments of that

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Atomic sentences

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Atomic sentences in fol correspond to the simplest sentences of English.

i.e. Large(x)

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Boolean connective (Boolean operator)

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-The logical connectives conjunction, disjunction, and negation
-allow us to form complex claims

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Bound variable

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variables with a quantifier

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Conclusion

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statement that is meant to follow from the other statements, or premises.

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Conditional

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The term conditional refers to a wide class of constructions in English including if > then, because>unless, and the like, that express some kind of conditional relationship between the two
parts
only some of the conditionals are truth functional

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Conditional proof

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Conditional proof is the method of proof that allows
one to prove a conditional statement P->Q by temporarily assuming P and proving Q under this additional assumption

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Conjunct

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One of the component sentences in a conjunction. For example,
A and B are the conjuncts of A ^ B.

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Conjunction

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The Boolean connective corresponding to the English word
and. A conjunction of sentences is true if and only if each conjunct is true.

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Connective

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An operator for making new statements out of simpler statements.
Typical examples are conjunction, negation, and the conditional.

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Consequent

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The consequent of a conditional is its second component formula.
In P->Q, Q is the consequent and P is the antecedent.

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Counterexample

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A counterexample to an argument is a possible situation
in which all the premises of the argument are true but the conclusion is false. Finding even a single counterexample is sufficient to show that an argument is not logically valid.

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Contradiction

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Something that cannot possibly be true in any set of
circumstances, for example, a statement and its negation.

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Disjunct

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One of the component sentences in a disjunction

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Disjunction

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The basic Boolean connective corresponding to the English
word or. A disjunction is true if at least one of the disjuncts is true.

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Free variable

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A free variable is an instance of a variable that is not bound, free variables have no quantifiers

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Inclusive disjunction

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only requires one disjunct in a compound sentence to be true in order for the sentence to be true .
Compare Exclusive disjunction.

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Individual constant

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Individual constants, or names, are those symbols of
fol that stand for objects or individuals. In fol is it assumed that each individual constant of the language names one and only one object.

i.e. "a, b, c, d" etc
(x, y, z are variables)

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Informal proof

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written out formal proof

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Lemma

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A lemma is a claim that is proven, like a theorem, but whose primary importance is for proving other claims. Lemmas are of less intrinsic interest than theorems. (See Theorem.)

i.e. DeMorgan's law

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Logical equivalence

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Two sentences are logically equivalent if they have the
same truth values in all possible circumstances.

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Logical truth or logical necessity

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a sentence that is a logical consequence of any set of premises. That is, no matter what the premises may be, it is impossible for the conclusion to be false

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Logical validity

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An argument is logically valid if the conclusion is a logical
consequence of the premises.

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Modus ponens

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The Latin name for the rule that allows us to infer Q from
P and P ->Q. Also known as conditional Elimination.

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Prefix notation

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In prefix notation, the predicate or relation symbol precedes the terms denoting objects in the relation. Larger(a, b) is in prefix notation

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Proof

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A proof is a step-by-step demonstration that one statement (the conclusion) follows logically from some others (the premises).
A formal proof is a proof given in a formal system of deduction; an informal proof is generally given in English, without the benefit of a formal system.

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Proof by contradiction

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To prove :S by contradiction, we assume S and
prove a contradiction. In other words, we assume the negation of what we wish to prove and show that this assumption leads to a contradiction.

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Proposition

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Something that is either true or false. Also called a claim.

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Quantifier

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noun phrase using a determiner such as every, some, three, etc
two quantifiers in FOL: universal and existential

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Sentence

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In propositional logic, atomic sentences are formed by combining names and predicates. Compound sentences are formed by combining
atomic sentences by means of the truth functional connectives. In fol, the definition is a bit more complicated. A sentence of fol is a wff with
no free variables.

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Tautological consequence

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A sentence S is a tautological consequence of
some premises if S follows from the premises simply in virtue of the meanings of the truth-functional connectives. We can check for tautological
consequence by means of truth tables, since S is a tautological consequence of the premises if and only if every row of their joint truth table that assigns true to each of premise also assigns true to S. All
tautological consequences are logical consequences, but not all logical consequences are tautological consequences.

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Tautological equivalence

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A tautology is a logical truth that owes its truth entirely to the meanings of the truth-functional connectives it contains, and not at all to the meanings of the atomic sentences it contains.

Cube(a) ∨ ¬Cube(a)

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Truth-functional connective

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A sentence connective with the property that the truth value of the newly formed sentence is determined solely by the truth value(s) of the constituent sentence(s), nothing more.

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Truth table

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Truth tables show the way in which the truth value of a sentence built up using truth-functional connectives depends on the truth values of the sentence's components.

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Existential quantifier

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used to make claims asserting the existence of some object. In English, we express existentially quantified claims with the use of words like something, at least one thing, a, etc.

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Tautology

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A tautology is a sentence that is logically true in virtue of its truth-functional structure.
This can be checked using truth tables since S is a tautology if and only if every row of the truth table for S assigns true to the main connective.

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Validity

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Validity is used in two ways in logic:
1. Validity as a property of arguments: An argument is valid if the conclusion must be true in any circumstance in which the premises are true.
2. Validity as a property of sentences: A first-order sentence is said to be valid if it is logically true simply in virtue of the meanings of its connectives, quantifiers, and identity. (See First-order validity.)

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Variable

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expressions of fol that function somewhat like pronouns in English. They are like individual constants in that they may be the arguments of predicates, but unlike constants, they can be bound by quantifiers.
Generally letters from the end of the alphabet, x, y, z, etc., are used for variables.

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Well-formed formula (wff):

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the grammatical expressions of fol. They are defined inductively. First, an atomic wff is any binary predicate followed by n terms. Complex wffs are constructed using connectives and quantifiers. The rules for constructing complex wffs are found on page 233. Wffs may have free variables. Sentences of fol are
wffs with no free variables.

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Proof by cases:

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A proof by cases consists in proving some statement S from a disjunction by proving S from each disjunct