- accessible (lines of a derivation)
- The accessible lines at line n of a derivation are those lines < n which are not within a terminated subderivation.
- antecedent
- The "if" clause of a conditional sentence. Example: In 'A>B', 'A' is the antecedent.
- argument
- An argument is a collection of statements some of which (the premises) are given as reasons for another member of the collection (the conclusion).
- atomic component
- The atomic components of a sentence of SL are all atomic sentences used to construct that sentence.
- atomic formula
- In PL, the atomic formulas include the atomic sentences of SL -- upper case letters excluding 'V', possibly subscripted -- but these may be followed by any number of individual terms. Example: 'R2ax'.
- atomic sentence
- In SL, the atomic sentences are all capital letters except 'V' possibly followed by a subscript. Example: 'A 1 '. In PL, the atomic sentences are atomic formulas which are sentences. Example: 'R2ab'.
- binary connective
- A connective which takes a pair of sentences or formulas and produces a new one. The binary connectives of SL and PL are '&', 'v', '>', and '='.
- bound variable
- An instance of a variable x in a formula of PL is bound if and only if it is within the scope of an x-quantifier. Otherwise we say it is free.
- categorical logic
-
Categorical logic treats relationships between the types of things (categories) which satisfy one-place predicates.
Categorical logic recognizes four main forms of statement:
Type English Form PL Form A-form: All S are P (^x)(Sx>Px) E-form: No S are P (^x)(Sx>~Px) or ~(%x)(Sx&Px) I-form: Some S are P
(%x)(Sx&Px) O-form: Some S are not-P (%x)(Sx&~Px)
See also the square of opposition.
- conclusion indicator
- A word or phrase which is often followed by a conclusion. For example, the words "because" and "since" are premise indicators. Note that a conclusion need not have an indicator.
- conditional (or conditional sentence)
- Any sentence that makes a claim based on a condition holding true. They are often explicitly of the form "if...then...". In SL or PL they have horseshoe as main connective.
- connective
- An object of a language's lexicon which operates on formulas or sentences of that language to produce a new formula or sentence. In SL, the connectives are '&', 'v', '>', '=' and '~'. In PL, there are two new connectives: the quantifiers '^' and '%'. (Note that some instructors call quantifiers "operators" rather than "connectives".)
- consequent
- The "then" clause of a conditional sentence. Example: In 'A>B', 'B' is the antecedent.
- consistency or logical consistency
- A set of sentences is logically consistent if and only if it is possible for all members of the set to be true together.
- deductive
- An argument is deductive if and only if its premises are intended to lead to the conclusion in a valid way.
- derivable
- To show that a sentence P is derivable from set \ of sentences, simply take the members of \ as the premises of a derivation and derive P using only the rules.
- derivation
- A vertical list giving step-by-step deductions from premises or assumptions.
- disjunct
- Either of the two components of a disjunction. Example, in 'AvB' both 'A' and 'B' are the disjuncts.
- disjunction
- An "either...or..." sentence. In SL or PL, a sentence with wedge as main connective.
- entails or logically entails
- If one sentence entails another, then the argument from the first as premise to the second as conclusion, is valid. In other words, it's impossible for the first to be true while the second is false.
- equivalence or logical equivalence
- The two members of a pair of sentences are logically equivalent if and only if it is not possible for one of the pair to be true while the other is false.
- existential form
- In English, the form "Some S are P". PL's way of representing such a form is: '(%x)(Sx&Px)' where the metavariables are to be replaced by a PL constructions.
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- formula
- In PL, a formula is constructed in accordance with set rules from atomic formulas (e.g. 'Pxyc'), the connectives from SL, and quantifier phrases (e.g., '(^x)'). If a formula has no free variables, then it is a sentence.
- free variable
- An instance of a variable x in a formula of PL is free if and only if it is not within the scope of an x-quantifier. Otherwise we say it is bound.
- functor
- In PLIF, the functors are the symbols, '*','-','+',''|', which represent functions. They are used to build "complex" terms.
- immediate component
- Pertaining to a sentence of SL. An immediate component P of an SL sentence Q is a sentential component used in the final stage of Q's construction.
- immediate subformula
- Pertaining to PL. P is an immediate subformula of a formula Q if and only if P is a subformula of Q and is used in the final step of the building process of Q.
- inconsistency or logical inconsistency
- A set of sentences is logically inconsistent if and only if it is impossible for all members of the set to be true together.
- individual term
- A name or variable of PL.
- inductive
- An argument is inductive if and only if its premises are intended to lead to its conclusion with high probability.
- interpretation
- An interpretation for PL is an assignment of meaning which specifies a) a universe of discourse, b) the members of the universe of discourse to which any one-place predicates apply, c) the relationships between members of the universe of discourse to which any 2 or more-place predicates apply, d) the truth value of any 0-place predicate letters and e) the objects named by any individual constants.
- invalid
- An argument is invalid if and only if it is not valid.
- lexicon
- The lexicon of a language is the collection of basic symbols it uses as components for the construction of expressions.
- logical consistency
- A set of sentences is logically consistent if and only if it
is possible for all members of the set to be true together.
(In SL, a set is logically consistent if and only if there is a truth value assignment making all its members true. In PL, a set is logically consistent if and only if there is an interpretation making all its members true.) - logically entails
- If one sentence entails another, then the argument from the first
as premise to the second as conclusion, is valid. In other words, it's
impossible for the first to be true while the second is false.
(In SL, a sentence logically entails another if and only if there is no truth value assignment making the first sentence true and the second false. In PL, a sentence logically entails another if and only if there is no interpretation making the first sentence true and the second false.) - logical equivalence
- The two members of a pair of sentences are logically equivalent
if and only if it is not possible for one of the pair to be true while
the other is false.
(In SL, a pair of sentences are logically equivalent if and only if there is no truth value assignment making one of the pair true and the other false. In PL, a pair of sentences are logically equivalent if and only if there is no interpretation making one of the pair true and the other false.) - logically false
- A sentence which could not be true is called "logically false".
Example: "No females are female".
(In SL, a sentence is logically false if and only if there is no truth value assignment making it true. In PL, a sentence is logically false if and only if there is no interpretation making it true.) - logical inconsistency
- A set of sentences is logically inconsistent if and
only if it is impossible for all members of the set to be true together.
(In SL, a set is logically inconsistent if and only if there is no truth value assignment making all its members true. In PL, a set is logically inconsistent if and only if there is no interpretation making all its members true.) - logically indeterminate
- A sentence which is neither logically true nor logically false. In
other words, a sentence is logically indeterminate iff it is possibly
true and also possibly false. For example: "The first U.S.
President was male."
(In SL, a sentence is logically indeterminate if and only if there is a truth value assignment making it true but also one making it false. In PL, a sentence is logically indeterminate if and only if there is an interpretation making it true but also one making it false.) - logically true
- A sentence which couldn't be false is called "logically true".
Example: "All male monkeys are male".
(In SL, a sentence is logically true if and only if there is no truth value assignment making it false. In PL, a sentence is logically true if and only if there is no interpretation making it false.)
- main connective
- In SL or PL, the main connective of a sentence (or, in PL, a formula) is the last connective used to construct that sentence (or formula).
- material conditional
- A conditional defined to be false only when its antecedent is true and consequent false and true otherwise.
- metavariable
- A variable used in English or other natural language (the "metalanguage") to refer to elements and expressions of symbolic languages (SL and PL) which are sometime called "object languages" because they are the object of our interest in logic.
- name
- Any expression of a language meant to refer to a single object. In PL, the names are 'a' - 'v', possibly subscripted. Any interpretation assigns a member of the universe of discourse to each name.
- negate
- That which is negated in a negation. For example, in the negation '~(B&C)', the negate is the component negated: '(B&C)'
- negation
- A sentence of any language which is a denial. For instance, any sentence of SL or PL with main connective tilde, or any English sentence of the form "It's not the case that ____".
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- PL
- Predicate Logic: The logic utilizing names and variables, the connectives of SL together with quantifiers.
- PLI
- Predicate Logic with Identity: PL together with constraints on the two place 'I' relation so that it is interpreted as identity. The associated derivation system, PDI, involves additional rules for identity.
- PLIF
- Predicate Logic with Identity and Functions. This is mathematical logic. See functors
- possible
- In this text, we restrict attention to logical possibility. Roughly what is logically possible is what the language allows to be true: sentences which are not logically false. A possible situation is described by an interpretation. (In SL, the interpretation just amounts to a truth value assignment.)
- premise indicator
- A word or phrase which precedes what is often a premise. For example, the words "because" and "since" are premise indicators. Note that a premise need not have an indicator.
- rules of inference
- Rules of derivation which allow one to draw a conclusion (the "output") from specified "input". All rules of inference work only with the main connectives of sentences: for example the rule of inference >E allows one to derive 'B' from 'A>B' and 'A' but nothing from '(A>B)&(C>D)' and 'A' (because '>' is not the main connective of '(A>B)&(C>D)'.
- rules of replacement
- Rules of derivation which allow one to replace any one sentence or sentential component on a line (the "input" line) with a particular equivalent expression on another line (the "output" line). For example, if line 1 is a premise '(AvB)>C', then any line below it can be '(BvA)>C' justified by the rule of replacement called "commutation" or "CM".
- scope
- The scope of a quantifier in a PL formula P is the subformula of P for which that quantifier is main connective.
- sentence
- A sentence of SL is an expression involving upper case letters, connectives and parentheses according to the recursive definition of chapter 2. A sentence of PL is likewise constructed but from a fuller lexicon including names, predicates and quantifiers. In addition, a PL sentence has no free variables.
- sentence-form
- A particular pattern which many sentences can instantiate. Example: P>~Q is the form of any SL or PL sentence having horseshoe as main connective and consequent a negation.
- sentential component
- The sentential components of a sentence of SL are all sentences used in the building process in order to construct that sentence.
- sound
- An argument is sound if and only if it is both valid and has only true premises.
- square of opposition
- An aspect of categorical logic. The opposites on
the square are contradictory: if one is true then the other is false.
- subderivation
- A derivation within a derivation started by an assumption. Everything derived within a subderivation counts as based on the assumption as well as on the premises. Thus, one canNOT site from within a subderivation once it is terminated; only the whole subderivation can be cited.
- subfomula
- A subfomula of a formula P is any formula used or produced in the building of P.
- substitution instance
- If a sentence is of the form '(^x)P' or '(%x)P', then the substitution instance, P(a/x), of the quantified sentence is the result of taking P and replacing every occurrence of x with a.
- term
- In PL, the terms include all names ('a' - 'u') and the variables ('w' - 'z') possibly subscripted. In PLIF, terms may become complex when functors are allowed.
- termination (of a subderivation)
- A subderivation is terminated when one is finished utilizing the assumption which begins the subderivation. It is marked off as terminated and thereafter no lines from within the subderivation may be cited: they are not "accessible". Only the whole of the subderivation may be cited.
- truth functional connective
- A connective is used truth functionally to form a sentence from components if and only if that sentence's truth value depends only on the truth value of the components. Otherwise, it is used non-truth functionally.
- truth table definitions of the connectives
-
P Q P&Q PvQ P>Q P=Q ~P T T T T T T F T F F T F F F F T F T T F T F F F F T T T - truth value assignment
- An association of a truth value (true or false) to each sentence of SL. (In SL, this is all there is to an interpretation .)
- unary connective
- A connective which takes a single sentence or formula and produces a new one. In SL, the tilde, '~' is the only unary connective. Both PL's quantifiers are unary.
- universal form
- In English, the form "All S are P". PL's way of representing such a form is: '(^x)(Sx>Px)' where the metavariables are to be replaced by a PL constructions.
- universe of discourse
- The part of a PL interpretation which is the collection of all objects considered.
- unsound
- An argument is unsound if and only if it is not sound.
- valid
- An argument is valid just in case it is not possible that its conclusion be false while its premises are all true. For our formalized languages, we say that an argument in SL or PL is valid just in case there is no interpretation (or truth value assignment) assigning the argument's premises true and conclusion false.
- variable
- A variable is grammatically similar to a name but does not refer to any particular thing. Instead, roughly, it can be thought to vary in it's reference, to be a stand in for names of different things. In PL, 'w','x','y','z' (possibly subscripted) serve as the variables. In this language variables range over all elements of the universe of discourse. See also metavariable for the variables we use as part of our English description of SL or PL.
- wedge
- The two place connective of SL and PL which represents disjunction. A sentence with wedge as main connective is true except when both its immediate components are false.