# Composite truth tables for sentence relations (entailment, synonymy,etc.)

I'm using John Saeed's 'Semantics'. Now in chap 4 I see he is trying to formalize sentence relations such as entailment, synonymy, contradiction, etc., by some kind of different truth tables he calls composite truth tables(in which he uses arrows to show the direction of inferences for any truth value assigning to propositions. My question is: Can we reconstruct these tables in usual truth tables as in propositional logic? Why? if yes, is there a general algorithm or rule for it?

PS. An example as a (composite) Truth tabular definition for synonymy he suggests the following:

p , q

T -> T

F -> F

T <- T

F <- F

Now my question would be that can we find a composition of formulas containing only p and q and logical propositional connectives?(Or even predicate if necessary) [which is equivalently a usual truth table for it]

For instance defining synonymy as:
(phi is synonym to psi) iff [ (phi <-> psi) is a tautology ]

Without quite answering your question let us remember that propositional logic (or sentential logic) deals with the truth or falsity of propositions. Here we can think of a proposition as anything about which one can say `it is the case that` or `it is not the case that`.

The definition for synonymy that you give can be read:

`if p characterises a situation and p is synonymous with q then q also characterises that same situation in the same way`

and

`if p does not characterise a situation and p is synonymous with q then q also does not characterise that same situation in the same way`

and

`if q characterises a situation and p is synonymous with q then p characterises that same situation`

and finally

`if q does not characterise a situation and p is synonymous with q then p does not characterise that same situation in the same way`

It is being extra verbose in order to show that the relation is symmetric.

Saeed would appear to be trying to capture semantic relations other than truth-functional relations -- as truth-functional relations are well studied. He is using [T]ruth and [F]alsity here to capture the notion of preservation (or not) of meaning across/between lexical/phrasal/sentential units.

So let's take the phrases `the printer is on fire` and `the printer is in flames`. Both of these are propositions and if it is the case that an electrical fault has caused your printer to catch fire then both these propositions will be true. Propositions seem to be always of the form `subject (S) copula (c) predicate (P)` -- See here for a discussion of the copula -- Are they always cast in this form? In truth I am unsure :) Anyway, we cannot say `It is the case that on fire` or `It is the case that in flames` but we can say `It is the case that on fire is synonymous with in flames` which is how the notion of [T]ruth / [F]alsity is being tied with the notion of synonymy.