Intro to FOL: constants & predicates Flashcards
What does FOL stand for?
First Order Logic
What is a declarative sentence?
Declarative sentences say that things are a certain way, and are true or false (in a given situation) according to how things are (in that situation).
What are (individual) constants?
- (Individual) constants are analogous to names in English.
- Like names, their role is to refer to particular objects.
- In FOL, (individual) constants are always lower-case letters from the beginning of the alphabet, which can be followed by a numerical index.
What are lower case letters used for in FOL?
- In FOL, (individual) constants are always lower-case letters from the beginning of the alphabet, which can be followed by a numerical index.
- Lower case letters from the end of the alphabet are used as variables.
What are predicates?
- Predicates are analogous to English predicates (e.g., ‘is smiling’)
- The predicate is that which is said of a grammatical subject, or rather the part of a sentence or clause in which something is said about the subject(s).
- Their role is, roughly, to express a property (of objects) or a relation (between objects).
- They are strings of letters and numerals, always starting with a capital letter, followed by a number of variables.
What is the role of a predicate?
To roughly express a property (of objects) or a relation (between objects).
What is the name for symbols which are lower case in FOL?
( individual) Constants
What is the name for strings of letters and numerals which always start with a capital letter?
Predicates
What are atomic sentences?
- Atomic sentences are the result of combining an n-place predicate (a predicate of arity n) with no occurrences of (individual constants).
What is an ‘arity’ of a predicate?
The ‘number of places’ / different names it can be combined with to get a sentence. (each predicate has a fixed arity)
How do we dictate the arity/ number of places of different variables?
By the number of different variables within a bracket.
What is the grammar (syntax) rule of atomic sentences?
Any combination of an n-place predicate and n constant is an atomic sentence.
What are semantics?
meanings
What are truth values/ conditions determined by?
- the truth conditions of the sentence &
- the situation
In the case of atomic sentences, what are the truth conditions determined by?
- a domain of objects: the property/relation expressed by the predicate from which it was composed &
- An interpretation: the reference(s) of the name(s) that occur(s) in it.
e.g. ‘John is smiling’ is true in a given situation iff in that situation, John (the object which the name ‘John’ refers to) is smiling (has the property expressed by the predicate ‘is smiling’).
What is the identity symbol?
=
What does the identity symbol do? (=)
- Like other 2-place predicates, ‘=’ combines with two (occurrences of) constants to form atomic sentences.
- Not written in a standard way as it is flanked by the constants (without any brackets)
- It has an unchanged interpretation.
What is the unchanged interpretation of ‘=’?
- numerically identical to
- So given any two constants, the objects assigned are one of the same.
