Neil Tennant

NEIL W. TENNANT

tennant.9@osu.edu

If you email me, please use the header PHIL 250: YOURNAME.

Professor
Department of Philosophy

Definitions of important notions

Term
A term is a well-formed expression whose semantic role is to denote a single object. Hence terms are also called singular terms.
The simplest example of a term is a proper name, such as John, London or 0.
There are also complex terms. These are of two main kinds: functional terms, and abstractive terms.
Examples of functional terms are John's father (or the father of John), symbolized as f(j); and a+b. f( ) is a one-place function sign; it takes one term to form a term. + is a two-place function sign; it takes two terms to form a term.
Examples of abstractive terms are the F, the number of Fs, and the set of all Fs (where F is a predicate). We symbolize these as follows:

the F : ιxF(x)

the number of Fs : nxF(x)

the set of all Fs : {x|F(x)}

Note that these abstractive terms involve variable-binding. They are formed by means of the variable-binding operators ι (the ... ), n (the number of ...s) and { | } (the set of ...s), applied to a formula with suitable free occurrences of a variable, which are thereby bound once the term is formed. So these operators turn predicates into terms.
Well-formed formula (wff)
In propositional languages, we define well-formed formulae as follows. (This is called an inductive definition.)

Any atomic sentence is a wff
If f is a wff, then so is ¬f
If f and y are wff, then so are (f∧y), (f∨y), and (f→y)
A thing is a wff only if it can be shown to be so by means of the preceding rules
In first-order languages, we define well-formed formulae and terms simultaneously as follows. (This is called a co-inductive definition.) Complications are introduced because of the need to keep precise track of which occurrences of variables are free within any formula or term.

Any variable is a term, with itself as its only free occurrence of a variable
Any name is a term, with no free occurrences of variables
Any primitive n-place function-sign f and any n terms t1,...,tn can be combined to form the term f(t1,...,tn), whose free occurrences of variables are those of the constituent terms t1,...,tn
If f is a wff with a free occurrence of the variable x, then ιxf is a term whose free occurrences of variables are those of f, except for f's free occurrences of x, which are now bound by the dominant occurrence of ι in ιxf
If f is a wff with a free occurrence of the variable x, then nxf is a term whose free occurrences of variables are those of f, except for f's free occurrences of x, which are now bound by the dominant occurrence of n in nxf
If f is a wff with a free occurrence of the variable x, then {x|f} is a term whose free occurrences of variables are those of f, except for f's free occurrences of x, which are now bound by the dominant occurrence of { | } in {x|f}
[Note how in each of these last three clauses we have used a wff here to build a term. This is what makes this a co-inductive definition of term and of wff.]
Any primitive n-place predicate P and any n terms t1,...,tn can be combined to form the atomic wff P(t1,...,tn)
If f is a wff, then so is ¬f, whose free occurrences of variables are those of f
If f and y are wff, then so are (f∧y), (f∨y), and (f→y), whose free occurrences of variables are those that are free in f and those that are free in y
If f is a wff with a free occurrence of the variable x, then ∀xf is a wff, whose free occurrences of variables are those that are free in f, except for the free occurrences of x in f, which are now bound by the dominant occurrence of ∀ in ∀xf
If f is a wff with a free occurrence of the variable x, then ∃xf is a wff, whose free occurrences of variables are those that are free in f, except for the free occurrences of x in f, which are now bound by the dominant occurrence of ∃ in ∃xf
A thing is a term or a wff only if it can be shown to be so by means of the preceding rules
A closed term is a term with no free occurrences of variables.
A sentence is a wff with no free occurrences of variables.

Truth-value assignment (in propositional logic)
A truth-value assignment is a mapping that assigns to each atomic sentence one of the truth-values T (truth) or F (falsity).
A truth-value assignment corresponds to a row in a truth-table.
Interpretation/model (in first-order logic)
A model for a first-order language consists of a domain of individuals, along with a denotation or extension for each item of non-logical vocabulary in the language. Call the model M. The denotation of a name a in M, written M(a), is a single individual in the domain of M. The denotation of an n-place function-sign, written M(f), is a function defined on the domain of M, with n argument-places. The extension of an n-place predicate P, written M(P), is a set of n-tuples of members of the domain of P.
Truth-value of a complex sentence under an interpretation/model
The truth-value of a complex sentence is worked out from its logical structure and from the interpretation given.
In the propositional case, the interpretation is a truth-value assignment. This 'plugs in' truth-values for atomic sentences. The overall truth-value of the complex sentence is then determined in the obvious stage-by-stage fashion by the truth-table mappings for the connectives occurring in the sentence.
In the first-order case, the interpretation is a model M, as just described. The presence of quantifiers complicates matters. Since sentences are special cases of formulae (namely, formulae with no free occurrences of variables), we make the truth-value determination for sentences drop out as a special case of truth-value determination for a formula, relative to a denotation-assignment, to its free variables, of individuals from the domain of M. Such a denotation-assignment makes each free variable in the formula behave, temporarily, like a name, by standing for an individual in the domain of M. We shall use the symbol d for a denotation assignment dealing with just the free variables in the formula f. So d(x), where x is free in f, is an individual in the domain of M. We now extend the function d to other terms as follows:
if a is a name, then d(a) = M(a);
d(f(t1,...,tn)) = M(f)(d(t1),...,d(tn)).
[Note: We do not deal here with terms formed by means of variable-binding term-forming operators; they present special problems beyond the scope of PHIL 250.]
By thus extending the function d, we have secured a denotation in the domain of M for:
(i) each variable dealt with by d;
(ii) each name; and
(iii) each functional term whose free variables are dealt with by d.
Some technical preliminaries:

If y is a subformula of f, then by d[y] we mean the denotation-assignment obtained from d by restricting it to just the variables that are free in y.
If d deals with exactly the variables free in f, and x is not free in f, then by d[x/a] we mean the extension of d to the denotation-assignment that assigns the individual a to x.
We are now in a position to deal with the notion of d making a formula true in M.

d makes an atomic formula P(t1,...,tn) true in M if and only if the n-tuple is in the extension M(P).
d makes ¬f true in M if and only if d does not make f true in M.
d makes (f∧y) true in M if and only if d[f] makes f true in M and d[y] makes y true in M.
d makes (f ∨y) true in M if and only if either d[f] makes f true in M or d[y] makes y true in M.
d makes (f→y) true in M if and only if either d[f] does not make f true in M or d[y] makes y true in M.
d makes ∀xf true in M if and only if for every individual a in the domain of M, d[x/a] makes f true in M.
d makes ∃xf true in M if and only if for some individual a in the domain of M, d[x/a] makes f true in M.
A sentence f is true in M if and only if f is made true in M by the empty denotation-assignment. We also say that the model M makes f true.
Logical consequence
f is a logical consequence of D (written D|=f) if and only if every model that makes every member of the set D true makes f true also.
When f is a logical consequence of D, we also say that the argument D/f is valid.
Logical deducibility
f is a logically deducible from D (written D|-f) if and only if there is a proof, in our system of logical rules of inference, of the conclusion f from undischarged assumptions drawn from the set D.
Logical truth
A sentence is logically true if and only if it is true in every model.
(Logical truth is a special case of logical consequence, namely logical consequence from the empty set of premises.)
Theoremhood
A sentence f is a theorem if and only if f is logically deducible from the empty set of assumptions.
(Theoremhood is a special case of deducibility, namely deducibility from the empty set of premises.)
Consistency
A set D of sentences is consistent if and only if ⊥ (absurdity) is not logically deducible from D.
Soundness of a logical system of proof
A system of logical proof is sound if and only if:
for all D, for all f, if D|-f, then D|=f
This says that one can prove only valid arguments.
Completeness of a logical system of proof
A system of logical proof is complete if and only if:
for all D, for all f, if D|=f, then D|-f
This says that one can prove every valid argument.

the F	:	ιxF(x)
the number of Fs	:	nxF(x)
the set of all Fs	:	{x\|F(x)}

NEIL W. TENNANT

Professor Department of Philosophy

Definitions of important notions

Professor
Department of Philosophy