# Synonomy syllogisms

How do you get from understanding the uninformative proposition ‘No unmarried man is married’ to understanding the informative proposition ‘No bachelor is married’? The move can be expressed in a syllogism.

1. No unmarried man is married.
2. A bachelor is an unmarried man.
3. Therefore, no bachelor is married.

This syllogism makes explicit what is implicit in the informative ‘No bachelor is married’ – foremost, premiss (2): that a bachelor is an unmarried man. Premiss (2) contains the information. Premiss (2) nominates the synonymy of ‘bachelor’ and ‘unmarried man’. Premiss (2) therefore legitimates the synonym substitution of ‘bachelor’ for ‘unmarried man’. But this does not explain the identity of ‘bachelor’ and ‘unmarried man’, so let’s assume a pure nominalism whereby we establish that to be a bachelor is to be an unmarried man. Establishing synonymy, this form of syllogism can thus be called a syllogism establishing synonymy.

Synonymy syllogisms establishing that a positive term is synonymous with a negative take the form:

1. ~(~p · p)
2. q = ~p
3. ∴ ~(q · p)

If we make ‘unmarried’ ~p, ‘married’ p and q ‘bachelor’, we find that premiss (1) establishes that one cannot be unmarried and married. Premiss (2) establishes that to be a bachelor is to be unmarried. The conclusion, (3), therefore is that one cannot be a bachelor and married.

The synonymy syllogism combines the law of non-contradiction and a term in premiss (1) with the synonym of that term in premiss (2) to get the law of non-contradiction as it applies to the synonymous terms in premiss (3). This is the explicit form of the (enthymematic) analytic statement ‘No bachelor is married’. ‘No bachelor is married’ is informative because it is a syllogism containing an identity statement as the minor premiss. If ‘No bachelor is married’ is really an enthymematic syllogism, should we consider it to be a primitive axiom?

Synonymy syllogisms may take two other forms. One may wish to establish the synonymy of two positive terms, such as ‘brother’ and ‘male sibling’. The form for this synonymy syllogism would thus be:

1. ~(p · ~p)
2. q = p
3. ∴ ~(q · ~p)

or

1. Every brother is a brother / One cannot both be a brother and not be a brother.
2. A male sibling is a brother.
3. Therefore, Every male sibling is a brother / One cannot both be a male sibling and not be a brother.

One may wish to establish the synonymy of two negative terms, such as ‘unmarried’ and ‘unwed’. The form of this synonymy syllogism would thus be:

1. ~(p · ~p)
2. ~q = ~p
3. ∴ ~(q · ~p)

or

1. No married person is unmarried / One cannot be both married and unmarried.
2. To be unwed is to be unmarried.
3. Therefore, no married person is unwed / One cannot be both married and unwed.

But this all brings us to the major problem. I have had to beg the identity of the terms such as ‘bachelor’ and ‘unmarried man’ to make my case. In essence, I have shown how we get from knowing, according to the law of non-contradiction, that one cannot be both married and unmarried – an uninformative proposition – but I have not explained why (or why not) ‘bachelor’ and ‘unmarried man’ are synonymous. As such, I’ve had to assume an answer to the larger problem of why are ‘bachelor’ and ‘unmarried man’ synonymous. But if I always intended to invoke the nominalist idea that ‘bachelor’ equals ‘unmarried man’ because I say it equals ‘unmarried man’, then I never needed to explain how we get from understanding that no unmarried man is married to understanding that no bachelor is married. It would have established as much about the larger problem if I had merely said ‘I known no bachelor is married because “bachelor” equals “unmarried man”‘. I’ll have to leave the larger problem to Quine, Grice and Strawson et al. But I hope this little rumination on synonym was of some use.